Optimal. Leaf size=108 \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{5 e^4 (d+e x)^5}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{6 e^4 (d+e x)^6}+\frac{c (3 B d-A e)}{4 e^4 (d+e x)^4}-\frac{B c}{3 e^4 (d+e x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0679083, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {772} \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{5 e^4 (d+e x)^5}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{6 e^4 (d+e x)^6}+\frac{c (3 B d-A e)}{4 e^4 (d+e x)^4}-\frac{B c}{3 e^4 (d+e x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 772
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )}{(d+e x)^7} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )}{e^3 (d+e x)^7}+\frac{3 B c d^2-2 A c d e+a B e^2}{e^3 (d+e x)^6}+\frac{c (-3 B d+A e)}{e^3 (d+e x)^5}+\frac{B c}{e^3 (d+e x)^4}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2+a e^2\right )}{6 e^4 (d+e x)^6}-\frac{3 B c d^2-2 A c d e+a B e^2}{5 e^4 (d+e x)^5}+\frac{c (3 B d-A e)}{4 e^4 (d+e x)^4}-\frac{B c}{3 e^4 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0465528, size = 87, normalized size = 0.81 \[ -\frac{10 a A e^3+2 a B e^2 (d+6 e x)+A c e \left (d^2+6 d e x+15 e^2 x^2\right )+B c \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )}{60 e^4 (d+e x)^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 110, normalized size = 1. \begin{align*} -{\frac{Bc}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{c \left ( Ae-3\,Bd \right ) }{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{-2\,Acde+aB{e}^{2}+3\,Bc{d}^{2}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{aA{e}^{3}+Ac{d}^{2}e-aBd{e}^{2}-Bc{d}^{3}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0513, size = 207, normalized size = 1.92 \begin{align*} -\frac{20 \, B c e^{3} x^{3} + B c d^{3} + A c d^{2} e + 2 \, B a d e^{2} + 10 \, A a e^{3} + 15 \,{\left (B c d e^{2} + A c e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e + A c d e^{2} + 2 \, B a e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.78554, size = 332, normalized size = 3.07 \begin{align*} -\frac{20 \, B c e^{3} x^{3} + B c d^{3} + A c d^{2} e + 2 \, B a d e^{2} + 10 \, A a e^{3} + 15 \,{\left (B c d e^{2} + A c e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e + A c d e^{2} + 2 \, B a e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 35.657, size = 172, normalized size = 1.59 \begin{align*} - \frac{10 A a e^{3} + A c d^{2} e + 2 B a d e^{2} + B c d^{3} + 20 B c e^{3} x^{3} + x^{2} \left (15 A c e^{3} + 15 B c d e^{2}\right ) + x \left (6 A c d e^{2} + 12 B a e^{3} + 6 B c d^{2} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.24666, size = 126, normalized size = 1.17 \begin{align*} -\frac{{\left (20 \, B c x^{3} e^{3} + 15 \, B c d x^{2} e^{2} + 6 \, B c d^{2} x e + B c d^{3} + 15 \, A c x^{2} e^{3} + 6 \, A c d x e^{2} + A c d^{2} e + 12 \, B a x e^{3} + 2 \, B a d e^{2} + 10 \, A a e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]