List of integrals of rational functions
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integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s (
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
functions) of
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s. Any rational function can be integrated by
partial fraction decomposition In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
of the function into a sum of functions of the form: : \frac, and \frac. which can then be integrated term by term. For other types of functions, see
lists of integrals Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, ...
.


Miscellaneous integrands

:\int\frac \, dx= \ln\left, f(x)\ + C :\int\frac \, dx = \frac\arctan\frac\,\! + C :\int\frac \, dx = \frac\ln\left, \frac\ + C = \begin \displaystyle -\frac\,\operatorname\frac + C = \frac\ln\frac + C & \text, x, < , a, \mbox \\
2pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\displaystyle -\frac\,\operatorname\frac + C = \frac\ln\frac + C & \text, x, > , a, \mbox \end :\int\frac \, dx = \frac\ln\left, \frac\ + C = \begin \displaystyle \frac\,\operatorname\frac + C = \frac\ln\frac + C & \text, x, < , a, \mbox \\
2pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\displaystyle \frac\,\operatorname\frac + C = \frac\ln\frac + C & \text, x, > , a, \mbox \end : \int \frac = \frac\sum_^ \sin \left(\frac\pi\right) \arctan\left left(x - \cos \left(\frac\pi \right) \right ) \csc \left(\frac\pi \right) \right- \frac \cos \left(\frac\pi \right) \ln \left , x^2 - 2 x \cos \left(\frac\pi \right) + 1 \right , + C


Integrands of the form ''x''''m''(''a x'' + ''b'')''n''

Many of the following antiderivatives have a term of the form ln , ''ax'' + ''b'', . Because this is undefined when ''x'' = −''b'' / ''a'', the most general form of the antiderivative replaces the
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected ...
with a
locally constant function In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function. ...
.Reader Survey: log, ''x'', + ''C''
, Tom Leinster, ''The ''n''-category Café'', March 19, 2012 However, it is conventional to omit this from the notation. For example, :\int\frac \, dx= \begin \dfrac\ln(-(ax + b)) + C^- & ax+b<0 \\ \dfrac\ln(ax + b) + C^+ & ax+b>0 \end is usually abbreviated as :\int\frac \, dx= \frac\ln\left, ax + b\ + C, where ''C'' is to be understood as notation for a locally constant function of ''x''. This convention will be adhered to in the following. :\int (ax + b)^n \, dx= \frac + C \qquad\text n\neq -1\mbox ( Cavalieri's quadrature formula) :\int\frac \, dx= \frac - \frac\ln\left, ax + b\ + C :\int\frac \, dx= \frac x + \frac\ln\left, ax + b\ + C :\int\frac \, dx= \frac + \frac\ln\left, ax + b\ + C :\int\frac \, dx= \frac + C \qquad\text n\not\in \\mbox :\int x(ax + b)^n \, dx= \frac (ax + b)^ + C \qquad\textn \not\in \\mbox :\int\frac \, dx= \frac+\frac + C :\int\frac \, dx= \frac\left(ax - 2b\ln\left, ax + b\ - \frac\right) + C :\int\frac \, dx= \frac\left(\ln\left, ax + b\ + \frac - \frac\right) + C :\int\frac \, dx= \frac\left(-\frac + \frac - \frac\right) + C \qquad\text n\not\in \\mbox :\int\frac \, dx = -\frac\ln\left, \frac\ + C :\int\frac \, dx = -\frac + \frac\ln\left, \frac\ + C :\int\frac \, dx = -a\left(\frac + \frac - \frac\ln\left, \frac\\right) + C


Integrands of the form ''x''''m'' / (''a x''2 + ''b x'' + ''c'')''n''

For a\neq 0:
:\int\frac dx = \begin \displaystyle \frac\arctan\frac + C & \text4ac-b^2>0\mbox \\
2pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\displaystyle \frac\ln\left, \frac\ + C = \begin \displaystyle -\frac\,\operatorname\frac + C &\text, 2ax+b, <\sqrt\mbox \\ pt\displaystyle -\frac\,\operatorname\frac + C &\text \end & \text4ac-b^2<0\mbox \\
2pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\displaystyle -\frac + C & \text4ac-b^2=0\mbox \end :\int\frac \, dx = \frac\ln\left, ax^2+bx+c\-\frac\int\frac + C :\int\frac \, dx = \begin \displaystyle \frac\ln\left, ax^2+bx+c\+\frac\arctan\frac + C &\text4ac-b^2>0\mbox \\
2pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\displaystyle \frac\ln\left, ax^2+bx+c\+\frac\ln\left, \frac\ + C = \begin \displaystyle \frac\ln\left, ax^2+bx+c\-\frac\,\operatorname\frac + C &\text, 2ax+b, <\sqrt\mbox \\ pt\displaystyle \frac\ln\left, ax^2+bx+c\-\frac\,\operatorname\frac + C &\text \end & \text4ac-b^2<0\mbox \\
2pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\displaystyle \frac\ln\left, ax^2+bx+c\-\frac + C = \frac\ln\left, x+\frac\-\frac + C &\text4ac-b^2=0\mbox\end : \int\frac \, dx= \frac+\frac\int\frac \, dx + C : \int\frac \, dx= -\frac-\frac\int\frac \, dx + C : \int\frac \, dx= \frac\ln\left, \frac\-\frac\int\frac \, dx + C


Integrands of the form ''x''''m'' (''a'' + ''b x''''n'')''p''

* The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents ''m'' and ''p'' toward 0. * These reduction formulas can be used for integrands having integer and/or fractional exponents. : \int x^m \left(a+b\,x^n\right)^p dx = \frac\,+\, \frac\int x^m \left(a+b\,x^n\right)^dx : \int x^m \left(a+b\,x^n\right)^p dx = -\frac\,+\, \frac\int x^m \left(a+b\,x^n\right)^dx : \int x^m \left(a+b\,x^n\right)^p dx = \frac\,-\, \frac\int x^ \left(a+b\,x^n\right)^dx : \int x^m \left(a+b\,x^n\right)^p dx = \frac\,-\, \frac\int x^ \left(a+b\,x^n\right)^dx : \int x^m \left(a+b\,x^n\right)^p dx = \frac\,-\, \frac\int x^\left(a+b\,x^n\right)^pdx : \int x^m \left(a+b\,x^n\right)^p dx = \frac\,-\, \frac\int x^\left(a+b\,x^n\right)^pdx


Integrands of the form (''A'' + ''B x'') (''a'' + ''b x'')''m'' (''c'' + ''d x'')''n'' (''e'' + ''f x'')''p''

* The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents ''m'', ''n'' and ''p'' toward 0. * These reduction formulas can be used for integrands having integer and/or fractional exponents. * Special cases of these reductions formulas can be used for integrands of the form (a+b\,x)^m (c+d\,x)^n (e+f\,x)^p by setting ''B'' to 0. : \int (A+B\,x) (a+b\,x)^m (c+d\,x)^n (e+f\,x)^p dx= -\frac\,+\, \frac\,\cdot :: \int (b\,c(m+1) (A\,f-B\,e)+(A\,b-a\,B) (n\,d\,e+c\,f(p+1))+d(b(m+1) (A\,f-B\,e)+f(n+p+1) (A\,b-a\,B))x)(a+b\,x)^ (c+d\,x)^(e+f\,x)^p dx : \int (A+B\,x) (a+b\,x)^m (c+d\,x)^n (e+f\,x)^p dx= \frac\,+\, \frac\,\cdot :: \int (A\,a\,d\,f(m+n+p+2)-B (b\,c\,e\,m+a(d\,e(n+1)+c\,f(p+1)))+(A\,b\,d\,f(m+n+p+2)+B (a\,d\,f\,m-b(d\,e(m+n+1)+c\,f(m+p+1)))) x)(a+b\,x)^ (c+d\,x)^n(e+f\,x)^p dx : \int (A+B\,x) (a+b\,x)^m (c+d\,x)^n (e+f\,x)^p dx= \frac\,+\, \frac\,\cdot :: \int ((m+1) (A (a\,d\,f-b(c\,f+d\,e))+B\,b\,c\,e)-(A\,b-a\,B) (d\,e(n+1)+c\,f(p+1))-d\,f(m+n+p+3) (A\,b-a\,B)x)(a+b\,x)^ (c+d\,x)^n(e+f\,x)^p dx


Integrands of the form ''x''''m'' (''A'' + ''B x''''n'') (''a'' + ''b x''''n'')''p'' (''c'' + ''d x''''n'')''q''

* The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents ''m'', ''p'' and ''q'' toward 0. * These reduction formulas can be used for integrands having integer and/or fractional exponents. * Special cases of these reductions formulas can be used for integrands of the form \left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^q and x^m\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^q by setting ''m'' and/or ''B'' to 0. : \int x^m\left(A+B\,x^n\right)\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^qdx= -\frac\,+\, \frac\,\cdot :: \int x^m\left(c (A\,b\,n (p+1)+(A\,b-a\,B) (m+1))+d (A\,b\,n (p+1)+(A\,b-a\,B) (m+n\,q+1)) x^n\right)\left(a+b\,x^n\right)^\left(c+d\,x^n\right)^dx : \int x^m\left(A+B\,x^n\right)\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^qdx= \frac\,+\, \frac\,\cdot :: \int x^m\left(c ((A\,b-a\,B) (1+m)+A\,b\,n (1+p+q))+(d(A\,b-a\,B) (1+m)+B\,n\,q(b\,c-a\,d)+A\,b\,d\,n (1+p+q))\,x^n\right)\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^dx : \int x^m\left(A+B\,x^n\right)\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^qdx= -\frac\,+\, \frac\,\cdot :: \int x^m\left(c(A\,b-a\,B)(m+1)+A\,n (b\,c-a\,d)(p+1)+d(A\,b-a\,B) (m+n (p+q+2)+1) x^n\right)\left(a+b\,x^n\right)^\left(c+d\,x^n\right)^qdx : \int x^m\left(A+B\,x^n\right)\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^qdx= \frac\,-\, \frac\,\cdot :: \int x^\left(a\,B\,c (m-n+1)+(a\,B\,d (m+n\,q+1)-b (-B\,c (m+n\,p+1)+A\,d (m+n (p+q+1)+1))) x^n\right)\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^qdx : \int x^m\left(A+B\,x^n\right)\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^qdx= \frac\,+\, \frac\,\cdot :: \int x^\left(a\,B\,c (m+1)-A (b\,c+a\,d) (m+n+1)-A\,n (b\,c\,p+a\,d\,q)-A\,b\,d (m+n (p+q+2)+1) x^n\right)\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^qdx : \int x^m\left(A+B\,x^n\right)\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^qdx= \frac\,-\, \frac\,\cdot :: \int x^\left(c(A\,b-a\,B)(m+1)+A\,n (b\,c (p+1)+a\,d\,q)+d ((A\,b-a\,B) (m+1)+A\,b\,n (p+q+1)) x^n\right)\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^dx : \int x^m\left(A+B\,x^n\right)\left(a+b\,x^n\right)^p\left(c+d\,x^n\right)^qdx= \frac\,-\, \frac\,\cdot :: \int x^\left(c(A\,b-a\,B)(m-n+1)+(d(A\,b-a\,B)(m+n\,q+1)-b\,n(B\,c-A\,d)(p+1)) x^n\right)\left(a+b\,x^n\right)^\left(c+d\,x^n\right)^qdx


Integrands of the form (''d'' + ''e x'')''m'' (''a'' + ''b x'' + ''c x''2)''p'' when ''b''2 − 4 ''a c'' = 0

* The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents ''m'' and ''p'' toward 0. * These reduction formulas can be used for integrands having integer and/or fractional exponents. * Special cases of these reductions formulas can be used for integrands of the form \left(a+b\,x+c\,x^2\right)^p when b^2-4\,a\,c=0 by setting ''m'' to 0. : \int (d+e\,x)^m \left(a+b\,x+c\,x^2\right)^pdx= \frac\,-\, \frac\,+\, \frac \int (d+e\,x)^\left(a+b\,x+c\,x^2\right)^dx : \int (d+e\,x)^m \left(a+b\,x+c\,x^2\right)^pdx= \frac\,-\, \frac\,+\, \frac \int (d+e\,x)^ \left(a+b\,x+c\,x^2\right)^dx : \int (d+e\,x)^m\left(a+b\,x+c\,x^2\right)^pdx= -\frac\,+\, \frac\,+\, \frac \int (d+e\,x)^ \left(a+b\,x+c\,x^2\right)^dx : \int (d+e\,x)^m \left(a+b\,x+c\,x^2\right)^pdx= -\frac\,+\, \frac\,+\, \frac \int (d+e\,x)^ \left(a+b\,x+c\,x^2\right)^dx : \int (d+e\,x)^m \left(a+b\,x+c\,x^2\right)^pdx= \frac\,-\, \frac\,+\, \frac \int (d+e\,x)^m \left(a+b\,x+c\,x^2\right)^dx : \int (d+e\,x)^m \left(a+b\,x+c\,x^2\right)^pdx= -\frac\,+\, \frac\,+\, \frac \int (d+e\,x)^m \left(a+b\,x+c\,x^2\right)^dx : \int (d+e\,x)^m \left(a+b\,x+c\,x^2\right)^pdx= \frac\,+\, \frac \int (d+e\,x)^\left(a+b\,x+c\,x^2\right)^pdx : \int (d+e\,x)^m\left(a+b\,x+c\,x^2\right)^pdx= -\frac\,+\, \frac \int (d+e\,x)^ \left(a+b\,x+c\,x^2\right)^pdx


Integrands of the form (''d'' + ''e x'')''m'' (''A'' + ''B x'') (''a'' + ''b x'' + ''c x''2)''p''

* The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents ''m'' and ''p'' toward 0. * These reduction formulas can be used for integrands having integer and/or fractional exponents. * Special cases of these reductions formulas can be used for integrands of the form \left(a+b\,x+c\,x^2\right)^p and (d+e\,x)^m \left(a+b\,x+c\,x^2\right)^p by setting ''m'' and/or ''B'' to 0. : \int (d+e\,x)^m (A+B\,x) \left(a+b\,x+c\,x^2\right)^pdx= \frac\,+\, \fracp\,\cdot :: \int (d+e\,x)^ (B (b\,d+2 a\,e+2 a\,e\,m+2 b\,d\,p)-A\,b\,e (m+2 p+2)+(B (2 c\,d+b\,e+b\,e m+4 c\,d\,p)-2 A\,c\,e (m+2 p+2))x)\left(a+b\,x+c\,x^2\right)^dx : \int (d+e\,x)^m (A+B\,x) \left(a+b\,x+c\,x^2\right)^pdx= \frac\,+\, \frac\,\cdot :: \int (d+e\,x)^(B (2 a\,e\,m+b\,d (2 p+3))-A (b\,e\,m+2 c\,d (2 p+3))+e(b\,B-2 A\,c) (m+2 p+3) x)\left(a+b\,x+c\,x^2\right)^dx : \int (d+e\,x)^m (A+B\,x) \left(a+b\,x+c\,x^2\right)^pdx= \frac\,-\, \frac\,\cdot :: \int (d+e\,x)^m (A\,c\,e (b\,d-2 a\,e) (m+2 p+2)+B (a\,e (b\,e-2 c\,d\,m+b\,e\,m)+b\,d (b\,e\,p-c\,d-2 c\,d\,p))+ ::: \left(A\,c\,e (2 c\,d-b\,e) (m+2 p+2)-B \left(-b^2 e^2 (m+p+1)+2 c^2 d^2 (1+2 p)+c\,e (b\,d (m-2 p)+2 a\,e (m+2 p+1))\right)\right) x)\left(a+b\,x+c\,x^2\right)^dx : \int (d+e\,x)^m (A+B\,x) \left(a+b\,x+c\,x^2\right)^pdx= \frac\,+ :: \frac\,\cdot ::: \int (d+e\,x)^m (A \left(b\,c\,d\,e (2 p-m+2)+b^2 e^2 (m+p+2)-2 c^2 d^2 (3+2 p)-2 a\,c\,e^2 (m+2 p+3)\right)- :::: B (a\,e (b\,e-2 c\,d m+b\,e\,m)+b\,d (-3 c\,d+b\,e-2 c\,d\,p+b\,e\,p))+c\,e(B (b\,d-2 a\,e)-A (2 c\,d-b\,e)) (m+2 p+4) x)\left(a+b\,x+c\,x^2\right)^dx : \int (d+e\,x)^m (A+B\,x) \left(a+b\,x+c\,x^2\right)^pdx= \frac\,+\, \frac\,\cdot :: \int (d+e\,x)^ (m(A\,c\,d-a\,B\,e)-d(b\,B-2 A\,c)(p+1) +((B\,c\,d-b\,B\,e+A\,c\,e) m-e(b\,B-2 A\,c)(p+1))x) \left(a+b\,x+c\,x^2\right)^pdx : \int (d+e\,x)^m (A+B\,x) \left(a+b\,x+c\,x^2\right)^pdx= -\frac\,+\, \frac\,\cdot :: \int (d+e\,x)^ ((A\,c\,d-A\,b\,e+a\,B\,e) (m+1)+b (B\,d-A\,e) (p+1)+c (B\,d-A\,e) (m+2 p+3) x)\left(a+b\,x+c\,x^2\right)^pdx


Integrands of the form ''x''''m'' (''a'' + ''b x''''n'' + ''c x''2''n'')''p'' when ''b''2 − 4 ''a c'' = 0

* The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents ''m'' and ''p'' toward 0. * These reduction formulas can be used for integrands having integer and/or fractional exponents. * Special cases of these reductions formulas can be used for integrands of the form \left(a+b\,x^n+c\,x^\right)^p when b^2-4\,a\,c=0 by setting ''m'' to 0. : \int x^m \left(a+b\,x^n+c\,x^\right)^p dx= \frac\,+\, \frac\,-\, \frac \int x^ \left(a+b\,x^n+c\,x^\right)^dx : \int x^m \left(a+b\,x^n+c\,x^\right)^p dx= \frac\,+\, \frac\,+\, \frac \int x^ \left(a+b\,x^n+c\,x^\right)^dx : \int x^m \left(a+b\,x^n+c\,x^\right)^p dx= \frac\,-\, \frac\,-\, \frac \int x^ \left(a+b\,x^n+c\,x^\right)^dx : \int x^m \left(a+b\,x^n+c\,x^\right)^p dx= -\frac\,-\, \frac\,+\, \frac \int x^ \left(a+b\,x^n+c\,x^\right)^dx : \int x^m \left(a+b\,x^n+c\,x^\right)^p dx= \frac\,+\, \frac\,+\, \frac \int x^m \left(a+b\,x^n+c\,x^\right)^dx : \int x^m \left(a+b\,x^n+c\,x^\right)^p dx= -\frac\,-\, \frac\,+\, \frac \int x^m \left(a+b\,x^n+c\,x^\right)^dx : \int x^m\left(a+b\,x^n+c\,x^\right)^p dx= \frac\,-\, \frac \int x^ \left(a+b\,x^n+c\,x^\right)^p dx : \int x^m\left(a+b\,x^n+c\,x^\right)^p dx= \frac\,-\, \frac \int x^ \left(a+b\,x^n+c\,x^\right)^p dx


Integrands of the form ''x''''m'' (''A'' + ''B x''''n'') (''a'' + ''b x''''n'' + ''c x''2''n'')''p''

* The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents ''m'' and ''p'' toward 0. * These reduction formulas can be used for integrands having integer and/or fractional exponents. * Special cases of these reductions formulas can be used for integrands of the form \left(a+b\,x^n+c\,x^\right)^p and x^m \left(a+b\,x^n+c\,x^\right)^p by setting ''m'' and/or ''B'' to 0. : \int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^\right)^pdx= \frac\,+\, \frac\,\cdot :: \int x^ \left(2 a\,B (m+1)-A\,b (m+n (2 p+1)+1)+(b\,B (m+1)-2\,A\,c (m+n (2 p+1)+1)) x^n\right)\left(a+b\,x^n+c\,x^\right)^dx : \int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^\right)^pdx= \frac\,+\, \frac\,\cdot :: \int x^\left((m-n+1)(2 a\,B-A\,b)+(m+2n (p+1)+1) (b\,B-2 A\,c) x^n\right)\left(a+b\,x^n+c\,x^\right)^dx : \int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^\right)^pdx= \frac\,+\, \frac\,\cdot :: \int x^m \left(2 a\,A\,c (m+n (2 p+1)+1)-a\,b\,B (m+1)+\left(2 a\,B\,c (m+2 n\,p+1)+A\,b\,c (m+n (2 p+1)+1)-b^2 B (m+n\,p+1)\right) x^n\right)\left(a+b\,x^n+c\,x^\right)^dx : \int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^\right)^pdx= -\frac\,+\, \frac\,\cdot :: \int x^m \left((m+n (p+1)+1) A\,b^2-a\,b\,B(m+1)-2(m+2n (p+1)+1)a\,A\,c+(m+n (2p+3)+1)(A\,b-2 a\,B) c\,x^n\right)\left(a+b\,x^n+c\,x^\right)^dx : \int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^\right)^pdx= \frac\,-\, \frac\,\cdot :: \int x^ \left(a\,B (m-n+1)+(b\,B (m+n\,p+1)-A\,c (m+n (2 p+1)+1)) x^n\right) \left(a+b\,x^n+c\,x^\right)^pdx : \int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^\right)^pdx= \frac\,+\, \frac\,\cdot :: \int x^ \left(a\,B (m+1)-A\,b (m+n (p+1)+1)-A\,c (m+2 n(p+1)+1) x^n\right)\left(a+b\,x^n+c\,x^\right)^pdx


References

{{Lists of integrals
Rational functions In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rati ...