Optimal. Leaf size=130 \[ \frac{e^3}{(a+b x) (b d-a e)^4}-\frac{e^2}{2 (a+b x)^2 (b d-a e)^3}+\frac{e^4 \log (a+b x)}{(b d-a e)^5}-\frac{e^4 \log (d+e x)}{(b d-a e)^5}+\frac{e}{3 (a+b x)^3 (b d-a e)^2}-\frac{1}{4 (a+b x)^4 (b d-a e)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0850072, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 44} \[ \frac{e^3}{(a+b x) (b d-a e)^4}-\frac{e^2}{2 (a+b x)^2 (b d-a e)^3}+\frac{e^4 \log (a+b x)}{(b d-a e)^5}-\frac{e^4 \log (d+e x)}{(b d-a e)^5}+\frac{e}{3 (a+b x)^3 (b d-a e)^2}-\frac{1}{4 (a+b x)^4 (b d-a e)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^5 (d+e x)} \, dx\\ &=\int \left (\frac{b}{(b d-a e) (a+b x)^5}-\frac{b e}{(b d-a e)^2 (a+b x)^4}+\frac{b e^2}{(b d-a e)^3 (a+b x)^3}-\frac{b e^3}{(b d-a e)^4 (a+b x)^2}+\frac{b e^4}{(b d-a e)^5 (a+b x)}-\frac{e^5}{(b d-a e)^5 (d+e x)}\right ) \, dx\\ &=-\frac{1}{4 (b d-a e) (a+b x)^4}+\frac{e}{3 (b d-a e)^2 (a+b x)^3}-\frac{e^2}{2 (b d-a e)^3 (a+b x)^2}+\frac{e^3}{(b d-a e)^4 (a+b x)}+\frac{e^4 \log (a+b x)}{(b d-a e)^5}-\frac{e^4 \log (d+e x)}{(b d-a e)^5}\\ \end{align*}
Mathematica [A] time = 0.0509937, size = 130, normalized size = 1. \[ \frac{e^3}{(a+b x) (b d-a e)^4}-\frac{e^2}{2 (a+b x)^2 (b d-a e)^3}+\frac{e^4 \log (a+b x)}{(b d-a e)^5}-\frac{e^4 \log (d+e x)}{(b d-a e)^5}+\frac{e}{3 (a+b x)^3 (b d-a e)^2}+\frac{1}{4 (a+b x)^4 (a e-b d)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 125, normalized size = 1. \begin{align*}{\frac{{e}^{4}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{5}}}+{\frac{1}{ \left ( 4\,ae-4\,bd \right ) \left ( bx+a \right ) ^{4}}}+{\frac{e}{3\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{3}}}+{\frac{{e}^{2}}{2\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{2}}}+{\frac{{e}^{3}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}-{\frac{{e}^{4}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.13727, size = 753, normalized size = 5.79 \begin{align*} \frac{e^{4} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{e^{4} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac{12 \, b^{3} e^{3} x^{3} - 3 \, b^{3} d^{3} + 13 \, a b^{2} d^{2} e - 23 \, a^{2} b d e^{2} + 25 \, a^{3} e^{3} - 6 \,{\left (b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{3} d^{2} e - 5 \, a b^{2} d e^{2} + 13 \, a^{2} b e^{3}\right )} x}{12 \,{\left (a^{4} b^{4} d^{4} - 4 \, a^{5} b^{3} d^{3} e + 6 \, a^{6} b^{2} d^{2} e^{2} - 4 \, a^{7} b d e^{3} + a^{8} e^{4} +{\left (b^{8} d^{4} - 4 \, a b^{7} d^{3} e + 6 \, a^{2} b^{6} d^{2} e^{2} - 4 \, a^{3} b^{5} d e^{3} + a^{4} b^{4} e^{4}\right )} x^{4} + 4 \,{\left (a b^{7} d^{4} - 4 \, a^{2} b^{6} d^{3} e + 6 \, a^{3} b^{5} d^{2} e^{2} - 4 \, a^{4} b^{4} d e^{3} + a^{5} b^{3} e^{4}\right )} x^{3} + 6 \,{\left (a^{2} b^{6} d^{4} - 4 \, a^{3} b^{5} d^{3} e + 6 \, a^{4} b^{4} d^{2} e^{2} - 4 \, a^{5} b^{3} d e^{3} + a^{6} b^{2} e^{4}\right )} x^{2} + 4 \,{\left (a^{3} b^{5} d^{4} - 4 \, a^{4} b^{4} d^{3} e + 6 \, a^{5} b^{3} d^{2} e^{2} - 4 \, a^{6} b^{2} d e^{3} + a^{7} b e^{4}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.56557, size = 1320, normalized size = 10.15 \begin{align*} -\frac{3 \, b^{4} d^{4} - 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} - 48 \, a^{3} b d e^{3} + 25 \, a^{4} e^{4} - 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 18 \, a^{2} b^{2} d e^{3} - 13 \, a^{3} b e^{4}\right )} x - 12 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (a^{4} b^{5} d^{5} - 5 \, a^{5} b^{4} d^{4} e + 10 \, a^{6} b^{3} d^{3} e^{2} - 10 \, a^{7} b^{2} d^{2} e^{3} + 5 \, a^{8} b d e^{4} - a^{9} e^{5} +{\left (b^{9} d^{5} - 5 \, a b^{8} d^{4} e + 10 \, a^{2} b^{7} d^{3} e^{2} - 10 \, a^{3} b^{6} d^{2} e^{3} + 5 \, a^{4} b^{5} d e^{4} - a^{5} b^{4} e^{5}\right )} x^{4} + 4 \,{\left (a b^{8} d^{5} - 5 \, a^{2} b^{7} d^{4} e + 10 \, a^{3} b^{6} d^{3} e^{2} - 10 \, a^{4} b^{5} d^{2} e^{3} + 5 \, a^{5} b^{4} d e^{4} - a^{6} b^{3} e^{5}\right )} x^{3} + 6 \,{\left (a^{2} b^{7} d^{5} - 5 \, a^{3} b^{6} d^{4} e + 10 \, a^{4} b^{5} d^{3} e^{2} - 10 \, a^{5} b^{4} d^{2} e^{3} + 5 \, a^{6} b^{3} d e^{4} - a^{7} b^{2} e^{5}\right )} x^{2} + 4 \,{\left (a^{3} b^{6} d^{5} - 5 \, a^{4} b^{5} d^{4} e + 10 \, a^{5} b^{4} d^{3} e^{2} - 10 \, a^{6} b^{3} d^{2} e^{3} + 5 \, a^{7} b^{2} d e^{4} - a^{8} b e^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 2.73769, size = 802, normalized size = 6.17 \begin{align*} \frac{e^{4} \log{\left (x + \frac{- \frac{a^{6} e^{10}}{\left (a e - b d\right )^{5}} + \frac{6 a^{5} b d e^{9}}{\left (a e - b d\right )^{5}} - \frac{15 a^{4} b^{2} d^{2} e^{8}}{\left (a e - b d\right )^{5}} + \frac{20 a^{3} b^{3} d^{3} e^{7}}{\left (a e - b d\right )^{5}} - \frac{15 a^{2} b^{4} d^{4} e^{6}}{\left (a e - b d\right )^{5}} + \frac{6 a b^{5} d^{5} e^{5}}{\left (a e - b d\right )^{5}} + a e^{5} - \frac{b^{6} d^{6} e^{4}}{\left (a e - b d\right )^{5}} + b d e^{4}}{2 b e^{5}} \right )}}{\left (a e - b d\right )^{5}} - \frac{e^{4} \log{\left (x + \frac{\frac{a^{6} e^{10}}{\left (a e - b d\right )^{5}} - \frac{6 a^{5} b d e^{9}}{\left (a e - b d\right )^{5}} + \frac{15 a^{4} b^{2} d^{2} e^{8}}{\left (a e - b d\right )^{5}} - \frac{20 a^{3} b^{3} d^{3} e^{7}}{\left (a e - b d\right )^{5}} + \frac{15 a^{2} b^{4} d^{4} e^{6}}{\left (a e - b d\right )^{5}} - \frac{6 a b^{5} d^{5} e^{5}}{\left (a e - b d\right )^{5}} + a e^{5} + \frac{b^{6} d^{6} e^{4}}{\left (a e - b d\right )^{5}} + b d e^{4}}{2 b e^{5}} \right )}}{\left (a e - b d\right )^{5}} + \frac{25 a^{3} e^{3} - 23 a^{2} b d e^{2} + 13 a b^{2} d^{2} e - 3 b^{3} d^{3} + 12 b^{3} e^{3} x^{3} + x^{2} \left (42 a b^{2} e^{3} - 6 b^{3} d e^{2}\right ) + x \left (52 a^{2} b e^{3} - 20 a b^{2} d e^{2} + 4 b^{3} d^{2} e\right )}{12 a^{8} e^{4} - 48 a^{7} b d e^{3} + 72 a^{6} b^{2} d^{2} e^{2} - 48 a^{5} b^{3} d^{3} e + 12 a^{4} b^{4} d^{4} + x^{4} \left (12 a^{4} b^{4} e^{4} - 48 a^{3} b^{5} d e^{3} + 72 a^{2} b^{6} d^{2} e^{2} - 48 a b^{7} d^{3} e + 12 b^{8} d^{4}\right ) + x^{3} \left (48 a^{5} b^{3} e^{4} - 192 a^{4} b^{4} d e^{3} + 288 a^{3} b^{5} d^{2} e^{2} - 192 a^{2} b^{6} d^{3} e + 48 a b^{7} d^{4}\right ) + x^{2} \left (72 a^{6} b^{2} e^{4} - 288 a^{5} b^{3} d e^{3} + 432 a^{4} b^{4} d^{2} e^{2} - 288 a^{3} b^{5} d^{3} e + 72 a^{2} b^{6} d^{4}\right ) + x \left (48 a^{7} b e^{4} - 192 a^{6} b^{2} d e^{3} + 288 a^{5} b^{3} d^{2} e^{2} - 192 a^{4} b^{4} d^{3} e + 48 a^{3} b^{5} d^{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.1941, size = 435, normalized size = 3.35 \begin{align*} \frac{b e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} - \frac{e^{5} \log \left ({\left | x e + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac{3 \, b^{4} d^{4} - 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} - 48 \, a^{3} b d e^{3} + 25 \, a^{4} e^{4} - 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 18 \, a^{2} b^{2} d e^{3} - 13 \, a^{3} b e^{4}\right )} x}{12 \,{\left (b d - a e\right )}^{5}{\left (b x + a\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]