List of Integrals of Rational Functions - Integrands of The Form (A + B X) (a + b X)m (c + d X

Integrands of The Form (A + B X) (a + b X)m (c + d X

  • 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 by setting B to 0.
\int (A+B\,x) (a+b\,x)^m (c+d\,x)^n (e+f\,x)^p dx= -\frac{(A\,b-a\,B)(a+b\,x)^{m+1} (c+d\,x)^n(e+f\,x)^{p+1}}{b (m+1) (a\,f-b\,e)}\,+\, \frac{1}{b (m+1) (a\,f-b\,e)}\,\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)^{m+1} (c+d\,x)^{n-1}(e+f\,x)^p dx
\int (A+B\,x) (a+b\,x)^m (c+d\,x)^n (e+f\,x)^p dx= \frac{B(a+b\,x)^m (c+d\,x)^{n+1}(e+f\,x)^{p+1}}{d\,f(m+n+p+2)}\,+\, \frac{1}{d\,f(m+n+p+2)}\,\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)^{m-1} (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{(A\,b-a\,B)(a+b\,x)^{m+1} (c+d\,x)^{n+1}(e+f\,x)^{p+1}}{(m+1)(a\,d-b\,c)(a\,f-b\,e)}\,+\, \frac{1}{(m+1)(a\,d-b\,c)(a\,f-b\,e)}\,\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)^{m+1} (c+d\,x)^n(e+f\,x)^p dx

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