List of Integrals of Rational Functions - Integrands of The Form xm (a + b Xn)p | Integrands Form xm<> (a<> bn)p<>

Integrands of The Form xm (a + b Xn)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.

$\int x^m \left(a+b\,x^n\right)^p dx = \frac{x^{m+1} \left(a+b\,x^n\right)^p}{m+n\,p+1}\,+\, \frac{a\,n\,p}{m+n\,p+1}\int x^m \left(a+b\,x^n\right)^{p-1}dx$

$\int x^m \left(a+b\,x^n\right)^p dx = -\frac{x^{m+1} \left(a+b\,x^n\right)^{p+1}}{a\,n (p+1)}\,+\, \frac{m+n (p+1)+1}{a\,n (p+1)}\int x^m \left(a+b\,x^n\right)^{p+1}dx$

$\int x^m \left(a+b\,x^n\right)^p dx = \frac{x^{m+1} \left(a+b\,x^n\right)^p}{m+1}\,-\, \frac{b\,n\,p}{m+1}\int x^{m+n} \left(a+b\,x^n\right)^{p-1}dx$

$\int x^m \left(a+b\,x^n\right)^p dx = \frac{x^{m-n+1} \left(a+b\,x^n\right)^{p+1}}{b\,n (p+1)}\,-\, \frac{m-n+1}{b\,n (p+1)}\int x^{m-n} \left(a+b\,x^n\right)^{p+1}dx$

$\int x^m \left(a+b\,x^n\right)^p dx = \frac{x^{m-n+1} \left(a+b\,x^n\right)^{p+1}}{b (m+n\,p+1)}\,-\, \frac{a (m-n+1)}{b (m+n\,p+1)}\int x^{m-n}\left(a+b\,x^n\right)^pdx$

$\int x^m \left(a+b\,x^n\right)^p dx = \frac{x^{m+1} \left(a+b\,x^n\right)^{p+1}}{a (m+1)}\,-\, \frac{b (m+n (p+1)+1)}{a (m+1)}\int x^{m+n}\left(a+b\,x^n\right)^pdx$

Read more about this topic: List Of Integrals Of Rational Functions

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