Optimal. Leaf size=156 \[ \frac{(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}+\frac{(c d-b e)^3 \log (b+c x) \left (-b c (B d-2 A e)+2 A c^2 d-3 b^2 B e\right )}{b^3 c^4}+\frac{d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}-\frac{A d^4}{b^2 x}+\frac{e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac{B e^4 x^2}{2 c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.225319, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}+\frac{(c d-b e)^3 \log (b+c x) \left (-b c (B d-2 A e)+2 A c^2 d-3 b^2 B e\right )}{b^3 c^4}+\frac{d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}-\frac{A d^4}{b^2 x}+\frac{e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac{B e^4 x^2}{2 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{e^3 (4 B c d-2 b B e+A c e)}{c^3}+\frac{A d^4}{b^2 x^2}+\frac{d^3 (b B d-2 A c d+4 A b e)}{b^3 x}+\frac{B e^4 x}{c^2}-\frac{(b B-A c) (-c d+b e)^4}{b^2 c^3 (b+c x)^2}+\frac{(c d-b e)^3 \left (2 A c^2 d-3 b^2 B e-b c (B d-2 A e)\right )}{b^3 c^3 (b+c x)}\right ) \, dx\\ &=-\frac{A d^4}{b^2 x}+\frac{e^3 (4 B c d-2 b B e+A c e) x}{c^3}+\frac{B e^4 x^2}{2 c^2}+\frac{(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}+\frac{d^3 (b B d-2 A c d+4 A b e) \log (x)}{b^3}+\frac{(c d-b e)^3 \left (2 A c^2 d-3 b^2 B e-b c (B d-2 A e)\right ) \log (b+c x)}{b^3 c^4}\\ \end{align*}
Mathematica [A] time = 0.096569, size = 155, normalized size = 0.99 \[ \frac{(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}+\frac{(b e-c d)^3 \log (b+c x) \left (b c (B d-2 A e)-2 A c^2 d+3 b^2 B e\right )}{b^3 c^4}+\frac{d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}-\frac{A d^4}{b^2 x}+\frac{e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac{B e^4 x^2}{2 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.016, size = 403, normalized size = 2.6 \begin{align*} -{\frac{A{d}^{4}}{{b}^{2}x}}+{\frac{B{e}^{4}{x}^{2}}{2\,{c}^{2}}}-8\,{\frac{b\ln \left ( cx+b \right ) Bd{e}^{3}}{{c}^{3}}}+4\,{\frac{Adb{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}-4\,{\frac{{b}^{2}Bd{e}^{3}}{{c}^{3} \left ( cx+b \right ) }}+6\,{\frac{bB{d}^{2}{e}^{2}}{{c}^{2} \left ( cx+b \right ) }}-{\frac{\ln \left ( cx+b \right ) B{d}^{4}}{{b}^{2}}}+{\frac{{e}^{4}Ax}{{c}^{2}}}+{\frac{{d}^{4}\ln \left ( x \right ) B}{{b}^{2}}}+{\frac{B{d}^{4}}{b \left ( cx+b \right ) }}+4\,{\frac{{d}^{3}\ln \left ( x \right ) Ae}{{b}^{2}}}-2\,{\frac{{d}^{4}\ln \left ( x \right ) Ac}{{b}^{3}}}-2\,{\frac{B{e}^{4}bx}{{c}^{3}}}+4\,{\frac{{e}^{3}Bdx}{{c}^{2}}}-4\,{\frac{B{d}^{3}e}{c \left ( cx+b \right ) }}-2\,{\frac{b\ln \left ( cx+b \right ) A{e}^{4}}{{c}^{3}}}+4\,{\frac{\ln \left ( cx+b \right ) Ad{e}^{3}}{{c}^{2}}}-4\,{\frac{\ln \left ( cx+b \right ) A{d}^{3}e}{{b}^{2}}}+2\,{\frac{c\ln \left ( cx+b \right ) A{d}^{4}}{{b}^{3}}}+3\,{\frac{{b}^{2}\ln \left ( cx+b \right ) B{e}^{4}}{{c}^{4}}}+6\,{\frac{\ln \left ( cx+b \right ) B{d}^{2}{e}^{2}}{{c}^{2}}}-{\frac{A{b}^{2}{e}^{4}}{{c}^{3} \left ( cx+b \right ) }}-{\frac{A{d}^{4}c}{{b}^{2} \left ( cx+b \right ) }}+{\frac{B{e}^{4}{b}^{3}}{{c}^{4} \left ( cx+b \right ) }}-6\,{\frac{A{d}^{2}{e}^{2}}{c \left ( cx+b \right ) }}+4\,{\frac{A{d}^{3}e}{b \left ( cx+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.15994, size = 419, normalized size = 2.69 \begin{align*} -\frac{A b c^{4} d^{4} -{\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} - 4 \,{\left (B b^{2} c^{3} - A b c^{4}\right )} d^{3} e + 6 \,{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d^{2} e^{2} - 4 \,{\left (B b^{4} c - A b^{3} c^{2}\right )} d e^{3} +{\left (B b^{5} - A b^{4} c\right )} e^{4}\right )} x}{b^{2} c^{5} x^{2} + b^{3} c^{4} x} + \frac{{\left (4 \, A b d^{3} e +{\left (B b - 2 \, A c\right )} d^{4}\right )} \log \left (x\right )}{b^{3}} + \frac{B c e^{4} x^{2} + 2 \,{\left (4 \, B c d e^{3} -{\left (2 \, B b - A c\right )} e^{4}\right )} x}{2 \, c^{3}} - \frac{{\left (4 \, A b c^{4} d^{3} e - 6 \, B b^{3} c^{2} d^{2} e^{2} +{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} + 4 \,{\left (2 \, B b^{4} c - A b^{3} c^{2}\right )} d e^{3} -{\left (3 \, B b^{5} - 2 \, A b^{4} c\right )} e^{4}\right )} \log \left (c x + b\right )}{b^{3} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.74277, size = 1025, normalized size = 6.57 \begin{align*} \frac{B b^{3} c^{3} e^{4} x^{4} - 2 \, A b^{2} c^{4} d^{4} +{\left (8 \, B b^{3} c^{3} d e^{3} -{\left (3 \, B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} e^{4}\right )} x^{3} + 2 \,{\left (4 \, B b^{4} c^{2} d e^{3} -{\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} e^{4}\right )} x^{2} + 2 \,{\left ({\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} - 4 \,{\left (B b^{3} c^{3} - A b^{2} c^{4}\right )} d^{3} e + 6 \,{\left (B b^{4} c^{2} - A b^{3} c^{3}\right )} d^{2} e^{2} - 4 \,{\left (B b^{5} c - A b^{4} c^{2}\right )} d e^{3} +{\left (B b^{6} - A b^{5} c\right )} e^{4}\right )} x - 2 \,{\left ({\left (4 \, A b c^{5} d^{3} e - 6 \, B b^{3} c^{3} d^{2} e^{2} +{\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4} + 4 \,{\left (2 \, B b^{4} c^{2} - A b^{3} c^{3}\right )} d e^{3} -{\left (3 \, B b^{5} c - 2 \, A b^{4} c^{2}\right )} e^{4}\right )} x^{2} +{\left (4 \, A b^{2} c^{4} d^{3} e - 6 \, B b^{4} c^{2} d^{2} e^{2} +{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} + 4 \,{\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} d e^{3} -{\left (3 \, B b^{6} - 2 \, A b^{5} c\right )} e^{4}\right )} x\right )} \log \left (c x + b\right ) + 2 \,{\left ({\left (4 \, A b c^{5} d^{3} e +{\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4}\right )} x^{2} +{\left (4 \, A b^{2} c^{4} d^{3} e +{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4}\right )} x\right )} \log \left (x\right )}{2 \,{\left (b^{3} c^{5} x^{2} + b^{4} c^{4} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 20.2068, size = 641, normalized size = 4.11 \begin{align*} \frac{B e^{4} x^{2}}{2 c^{2}} + \frac{- A b c^{4} d^{4} + x \left (- A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 6 A b^{2} c^{3} d^{2} e^{2} + 4 A b c^{4} d^{3} e - 2 A c^{5} d^{4} + B b^{5} e^{4} - 4 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 4 B b^{2} c^{3} d^{3} e + B b c^{4} d^{4}\right )}{b^{3} c^{4} x + b^{2} c^{5} x^{2}} - \frac{x \left (- A c e^{4} + 2 B b e^{4} - 4 B c d e^{3}\right )}{c^{3}} + \frac{d^{3} \left (4 A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{- 4 A b^{2} c^{3} d^{3} e + 2 A b c^{4} d^{4} - B b^{2} c^{3} d^{4} + b c^{3} d^{3} \left (4 A b e - 2 A c d + B b d\right )}{- 2 A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 8 A b c^{4} d^{3} e + 4 A c^{5} d^{4} + 3 B b^{5} e^{4} - 8 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 2 B b c^{4} d^{4}} \right )}}{b^{3}} + \frac{\left (b e - c d\right )^{3} \left (- 2 A b c e - 2 A c^{2} d + 3 B b^{2} e + B b c d\right ) \log{\left (x + \frac{- 4 A b^{2} c^{3} d^{3} e + 2 A b c^{4} d^{4} - B b^{2} c^{3} d^{4} + \frac{b \left (b e - c d\right )^{3} \left (- 2 A b c e - 2 A c^{2} d + 3 B b^{2} e + B b c d\right )}{c}}{- 2 A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 8 A b c^{4} d^{3} e + 4 A c^{5} d^{4} + 3 B b^{5} e^{4} - 8 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 2 B b c^{4} d^{4}} \right )}}{b^{3} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.37316, size = 425, normalized size = 2.72 \begin{align*} \frac{{\left (B b d^{4} - 2 \, A c d^{4} + 4 \, A b d^{3} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac{B c^{2} x^{2} e^{4} + 8 \, B c^{2} d x e^{3} - 4 \, B b c x e^{4} + 2 \, A c^{2} x e^{4}}{2 \, c^{4}} - \frac{{\left (B b c^{4} d^{4} - 2 \, A c^{5} d^{4} + 4 \, A b c^{4} d^{3} e - 6 \, B b^{3} c^{2} d^{2} e^{2} + 8 \, B b^{4} c d e^{3} - 4 \, A b^{3} c^{2} d e^{3} - 3 \, B b^{5} e^{4} + 2 \, A b^{4} c e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{4}} - \frac{A b c^{4} d^{4} -{\left (B b c^{4} d^{4} - 2 \, A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 4 \, A b c^{4} d^{3} e + 6 \, B b^{3} c^{2} d^{2} e^{2} - 6 \, A b^{2} c^{3} d^{2} e^{2} - 4 \, B b^{4} c d e^{3} + 4 \, A b^{3} c^{2} d e^{3} + B b^{5} e^{4} - A b^{4} c e^{4}\right )} x}{{\left (c x + b\right )} b^{2} c^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]