Optimal. Leaf size=110 \[ -\frac{b^2}{(a+b x) (b d-a e)^3}-\frac{3 b^2 e \log (a+b x)}{(b d-a e)^4}+\frac{3 b^2 e \log (d+e x)}{(b d-a e)^4}-\frac{2 b e}{(d+e x) (b d-a e)^3}-\frac{e}{2 (d+e x)^2 (b d-a e)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0786082, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 44} \[ -\frac{b^2}{(a+b x) (b d-a e)^3}-\frac{3 b^2 e \log (a+b x)}{(b d-a e)^4}+\frac{3 b^2 e \log (d+e x)}{(b d-a e)^4}-\frac{2 b e}{(d+e x) (b d-a e)^3}-\frac{e}{2 (d+e x)^2 (b d-a e)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{1}{(a+b x)^2 (d+e x)^3} \, dx\\ &=\int \left (\frac{b^3}{(b d-a e)^3 (a+b x)^2}-\frac{3 b^3 e}{(b d-a e)^4 (a+b x)}+\frac{e^2}{(b d-a e)^2 (d+e x)^3}+\frac{2 b e^2}{(b d-a e)^3 (d+e x)^2}+\frac{3 b^2 e^2}{(b d-a e)^4 (d+e x)}\right ) \, dx\\ &=-\frac{b^2}{(b d-a e)^3 (a+b x)}-\frac{e}{2 (b d-a e)^2 (d+e x)^2}-\frac{2 b e}{(b d-a e)^3 (d+e x)}-\frac{3 b^2 e \log (a+b x)}{(b d-a e)^4}+\frac{3 b^2 e \log (d+e x)}{(b d-a e)^4}\\ \end{align*}
Mathematica [A] time = 0.0995215, size = 97, normalized size = 0.88 \[ -\frac{\frac{2 b^2 (b d-a e)}{a+b x}+6 b^2 e \log (a+b x)+\frac{4 b e (b d-a e)}{d+e x}+\frac{e (b d-a e)^2}{(d+e x)^2}-6 b^2 e \log (d+e x)}{2 (b d-a e)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.055, size = 108, normalized size = 1. \begin{align*} -{\frac{e}{2\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{b}^{2}e\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{4}}}+2\,{\frac{be}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}+{\frac{{b}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}-3\,{\frac{{b}^{2}e\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.16565, size = 521, normalized size = 4.74 \begin{align*} -\frac{3 \, b^{2} e \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac{3 \, b^{2} e \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac{6 \, b^{2} e^{2} x^{2} + 2 \, b^{2} d^{2} + 5 \, a b d e - a^{2} e^{2} + 3 \,{\left (3 \, b^{2} d e + a b e^{2}\right )} x}{2 \,{\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} +{\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} +{\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} +{\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.78047, size = 991, normalized size = 9.01 \begin{align*} -\frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + a b^{2} d^{2} e +{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} +{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2}\right )} x\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} e^{3} x^{3} + a b^{2} d^{2} e +{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} +{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a b^{4} d^{6} - 4 \, a^{2} b^{3} d^{5} e + 6 \, a^{3} b^{2} d^{4} e^{2} - 4 \, a^{4} b d^{3} e^{3} + a^{5} d^{2} e^{4} +{\left (b^{5} d^{4} e^{2} - 4 \, a b^{4} d^{3} e^{3} + 6 \, a^{2} b^{3} d^{2} e^{4} - 4 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{3} +{\left (2 \, b^{5} d^{5} e - 7 \, a b^{4} d^{4} e^{2} + 8 \, a^{2} b^{3} d^{3} e^{3} - 2 \, a^{3} b^{2} d^{2} e^{4} - 2 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x^{2} +{\left (b^{5} d^{6} - 2 \, a b^{4} d^{5} e - 2 \, a^{2} b^{3} d^{4} e^{2} + 8 \, a^{3} b^{2} d^{3} e^{3} - 7 \, a^{4} b d^{2} e^{4} + 2 \, a^{5} d e^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 2.1331, size = 632, normalized size = 5.75 \begin{align*} \frac{3 b^{2} e \log{\left (x + \frac{- \frac{3 a^{5} b^{2} e^{6}}{\left (a e - b d\right )^{4}} + \frac{15 a^{4} b^{3} d e^{5}}{\left (a e - b d\right )^{4}} - \frac{30 a^{3} b^{4} d^{2} e^{4}}{\left (a e - b d\right )^{4}} + \frac{30 a^{2} b^{5} d^{3} e^{3}}{\left (a e - b d\right )^{4}} - \frac{15 a b^{6} d^{4} e^{2}}{\left (a e - b d\right )^{4}} + 3 a b^{2} e^{2} + \frac{3 b^{7} d^{5} e}{\left (a e - b d\right )^{4}} + 3 b^{3} d e}{6 b^{3} e^{2}} \right )}}{\left (a e - b d\right )^{4}} - \frac{3 b^{2} e \log{\left (x + \frac{\frac{3 a^{5} b^{2} e^{6}}{\left (a e - b d\right )^{4}} - \frac{15 a^{4} b^{3} d e^{5}}{\left (a e - b d\right )^{4}} + \frac{30 a^{3} b^{4} d^{2} e^{4}}{\left (a e - b d\right )^{4}} - \frac{30 a^{2} b^{5} d^{3} e^{3}}{\left (a e - b d\right )^{4}} + \frac{15 a b^{6} d^{4} e^{2}}{\left (a e - b d\right )^{4}} + 3 a b^{2} e^{2} - \frac{3 b^{7} d^{5} e}{\left (a e - b d\right )^{4}} + 3 b^{3} d e}{6 b^{3} e^{2}} \right )}}{\left (a e - b d\right )^{4}} + \frac{- a^{2} e^{2} + 5 a b d e + 2 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (3 a b e^{2} + 9 b^{2} d e\right )}{2 a^{4} d^{2} e^{3} - 6 a^{3} b d^{3} e^{2} + 6 a^{2} b^{2} d^{4} e - 2 a b^{3} d^{5} + x^{3} \left (2 a^{3} b e^{5} - 6 a^{2} b^{2} d e^{4} + 6 a b^{3} d^{2} e^{3} - 2 b^{4} d^{3} e^{2}\right ) + x^{2} \left (2 a^{4} e^{5} - 2 a^{3} b d e^{4} - 6 a^{2} b^{2} d^{2} e^{3} + 10 a b^{3} d^{3} e^{2} - 4 b^{4} d^{4} e\right ) + x \left (4 a^{4} d e^{4} - 10 a^{3} b d^{2} e^{3} + 6 a^{2} b^{2} d^{3} e^{2} + 2 a b^{3} d^{4} e - 2 b^{4} d^{5}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.16789, size = 335, normalized size = 3.05 \begin{align*} -\frac{3 \, b^{3} e \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac{3 \, b^{2} e^{2} \log \left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x}{2 \,{\left (b d - a e\right )}^{4}{\left (b x + a\right )}{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]