Optimal. Leaf size=132 \[ -\frac{e^3}{(d+e x) (b d-a e)^4}-\frac{3 b e^2}{(a+b x) (b d-a e)^4}-\frac{4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac{4 b e^3 \log (d+e x)}{(b d-a e)^5}+\frac{b e}{(a+b x)^2 (b d-a e)^3}-\frac{b}{3 (a+b x)^3 (b d-a e)^2} \]
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Rubi [A] time = 0.101166, antiderivative size = 132, 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} \[ -\frac{e^3}{(d+e x) (b d-a e)^4}-\frac{3 b e^2}{(a+b x) (b d-a e)^4}-\frac{4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac{4 b e^3 \log (d+e x)}{(b d-a e)^5}+\frac{b e}{(a+b x)^2 (b d-a e)^3}-\frac{b}{3 (a+b x)^3 (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^4 (d+e x)^2} \, dx\\ &=\int \left (\frac{b^2}{(b d-a e)^2 (a+b x)^4}-\frac{2 b^2 e}{(b d-a e)^3 (a+b x)^3}+\frac{3 b^2 e^2}{(b d-a e)^4 (a+b x)^2}-\frac{4 b^2 e^3}{(b d-a e)^5 (a+b x)}+\frac{e^4}{(b d-a e)^4 (d+e x)^2}+\frac{4 b e^4}{(b d-a e)^5 (d+e x)}\right ) \, dx\\ &=-\frac{b}{3 (b d-a e)^2 (a+b x)^3}+\frac{b e}{(b d-a e)^3 (a+b x)^2}-\frac{3 b e^2}{(b d-a e)^4 (a+b x)}-\frac{e^3}{(b d-a e)^4 (d+e x)}-\frac{4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac{4 b e^3 \log (d+e x)}{(b d-a e)^5}\\ \end{align*}
Mathematica [A] time = 0.124786, size = 121, normalized size = 0.92 \[ \frac{\frac{3 e^3 (a e-b d)}{d+e x}-\frac{9 b e^2 (b d-a e)}{a+b x}+\frac{3 b e (b d-a e)^2}{(a+b x)^2}-\frac{b (b d-a e)^3}{(a+b x)^3}-12 b e^3 \log (a+b x)+12 b e^3 \log (d+e x)}{3 (b d-a e)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 132, normalized size = 1. \begin{align*} -{\frac{{e}^{3}}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}-4\,{\frac{{e}^{3}b\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{5}}}-{\frac{b}{3\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{3}}}+4\,{\frac{{e}^{3}b\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{5}}}-3\,{\frac{b{e}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}-{\frac{be}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11037, size = 807, normalized size = 6.11 \begin{align*} -\frac{4 \, b e^{3} \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{4 \, b e^{3} \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} + b^{3} d^{3} - 5 \, a b^{2} d^{2} e + 13 \, a^{2} b d e^{2} + 3 \, a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} + 5 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (b^{3} d^{2} e - 8 \, a b^{2} d e^{2} - 11 \, a^{2} b e^{3}\right )} x}{3 \,{\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} +{\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} +{\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \,{\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} +{\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81873, size = 1501, normalized size = 11.37 \begin{align*} -\frac{b^{4} d^{4} - 6 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 10 \, a^{3} b d e^{3} - 3 \, 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} + 4 \, a b^{3} d e^{3} - 5 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \,{\left (b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} + 11 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} +{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right ) - 12 \,{\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} +{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (a^{3} b^{5} d^{6} - 5 \, a^{4} b^{4} d^{5} e + 10 \, a^{5} b^{3} d^{4} e^{2} - 10 \, a^{6} b^{2} d^{3} e^{3} + 5 \, a^{7} b d^{2} e^{4} - a^{8} d e^{5} +{\left (b^{8} d^{5} e - 5 \, a b^{7} d^{4} e^{2} + 10 \, a^{2} b^{6} d^{3} e^{3} - 10 \, a^{3} b^{5} d^{2} e^{4} + 5 \, a^{4} b^{4} d e^{5} - a^{5} b^{3} e^{6}\right )} x^{4} +{\left (b^{8} d^{6} - 2 \, a b^{7} d^{5} e - 5 \, a^{2} b^{6} d^{4} e^{2} + 20 \, a^{3} b^{5} d^{3} e^{3} - 25 \, a^{4} b^{4} d^{2} e^{4} + 14 \, a^{5} b^{3} d e^{5} - 3 \, a^{6} b^{2} e^{6}\right )} x^{3} + 3 \,{\left (a b^{7} d^{6} - 4 \, a^{2} b^{6} d^{5} e + 5 \, a^{3} b^{5} d^{4} e^{2} - 5 \, a^{5} b^{3} d^{2} e^{4} + 4 \, a^{6} b^{2} d e^{5} - a^{7} b e^{6}\right )} x^{2} +{\left (3 \, a^{2} b^{6} d^{6} - 14 \, a^{3} b^{5} d^{5} e + 25 \, a^{4} b^{4} d^{4} e^{2} - 20 \, a^{5} b^{3} d^{3} e^{3} + 5 \, a^{6} b^{2} d^{2} e^{4} + 2 \, a^{7} b d e^{5} - a^{8} e^{6}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.59816, size = 881, normalized size = 6.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23866, size = 377, normalized size = 2.86 \begin{align*} -\frac{4 \, b e^{4} \log \left ({\left | b - \frac{b d}{x e + d} + \frac{a e}{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{e^{7}}{{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )}{\left (x e + d\right )}} - \frac{13 \, b^{4} e^{3} - \frac{30 \,{\left (b^{4} d e^{4} - a b^{3} e^{5}\right )} e^{\left (-1\right )}}{x e + d} + \frac{18 \,{\left (b^{4} d^{2} e^{5} - 2 \, a b^{3} d e^{6} + a^{2} b^{2} e^{7}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}}{3 \,{\left (b d - a e\right )}^{5}{\left (b - \frac{b d}{x e + d} + \frac{a e}{x e + d}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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