Optimal. Leaf size=133 \[ -\frac{b^3}{(a+b x) (b d-a e)^4}-\frac{3 b^2 e}{(d+e x) (b d-a e)^4}-\frac{4 b^3 e \log (a+b x)}{(b d-a e)^5}+\frac{4 b^3 e \log (d+e x)}{(b d-a e)^5}-\frac{b e}{(d+e x)^2 (b d-a e)^3}-\frac{e}{3 (d+e x)^3 (b d-a e)^2} \]
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Rubi [A] time = 0.107646, antiderivative size = 133, 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} \[ -\frac{b^3}{(a+b x) (b d-a e)^4}-\frac{3 b^2 e}{(d+e x) (b d-a e)^4}-\frac{4 b^3 e \log (a+b x)}{(b d-a e)^5}+\frac{4 b^3 e \log (d+e x)}{(b d-a e)^5}-\frac{b e}{(d+e x)^2 (b d-a e)^3}-\frac{e}{3 (d+e x)^3 (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{1}{(a+b x)^2 (d+e x)^4} \, dx\\ &=\int \left (\frac{b^4}{(b d-a e)^4 (a+b x)^2}-\frac{4 b^4 e}{(b d-a e)^5 (a+b x)}+\frac{e^2}{(b d-a e)^2 (d+e x)^4}+\frac{2 b e^2}{(b d-a e)^3 (d+e x)^3}+\frac{3 b^2 e^2}{(b d-a e)^4 (d+e x)^2}+\frac{4 b^3 e^2}{(b d-a e)^5 (d+e x)}\right ) \, dx\\ &=-\frac{b^3}{(b d-a e)^4 (a+b x)}-\frac{e}{3 (b d-a e)^2 (d+e x)^3}-\frac{b e}{(b d-a e)^3 (d+e x)^2}-\frac{3 b^2 e}{(b d-a e)^4 (d+e x)}-\frac{4 b^3 e \log (a+b x)}{(b d-a e)^5}+\frac{4 b^3 e \log (d+e x)}{(b d-a e)^5}\\ \end{align*}
Mathematica [A] time = 0.124966, size = 120, normalized size = 0.9 \[ \frac{-\frac{3 b^3 (b d-a e)}{a+b x}-\frac{9 b^2 e (b d-a e)}{d+e x}-12 b^3 e \log (a+b x)-\frac{3 b e (b d-a e)^2}{(d+e x)^2}+\frac{e (a e-b d)^3}{(d+e x)^3}+12 b^3 e \log (d+e x)}{3 (b d-a e)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 131, normalized size = 1. \begin{align*} -{\frac{e}{3\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{3}}}-4\,{\frac{e{b}^{3}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{5}}}-3\,{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+{\frac{be}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{3}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}+4\,{\frac{e{b}^{3}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.33082, size = 809, normalized size = 6.08 \begin{align*} -\frac{4 \, b^{3} e \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^{3} e \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 - 5 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (5 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (11 \, b^{3} d^{2} e + 8 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x}{3 \,{\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} +{\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} +{\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \,{\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} +{\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85781, size = 1501, normalized size = 11.29 \begin{align*} -\frac{3 \, b^{4} d^{4} + 10 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 6 \, a^{3} b d e^{3} - a^{4} e^{4} + 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (5 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (11 \, b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + a b^{3} d^{3} e +{\left (3 \, b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3}\right )} x^{2} +{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (b x + a\right ) - 12 \,{\left (b^{4} e^{4} x^{4} + a b^{3} d^{3} e +{\left (3 \, b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3}\right )} x^{2} +{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (a b^{5} d^{8} - 5 \, a^{2} b^{4} d^{7} e + 10 \, a^{3} b^{3} d^{6} e^{2} - 10 \, a^{4} b^{2} d^{5} e^{3} + 5 \, a^{5} b d^{4} e^{4} - a^{6} d^{3} e^{5} +{\left (b^{6} d^{5} e^{3} - 5 \, a b^{5} d^{4} e^{4} + 10 \, a^{2} b^{4} d^{3} e^{5} - 10 \, a^{3} b^{3} d^{2} e^{6} + 5 \, a^{4} b^{2} d e^{7} - a^{5} b e^{8}\right )} x^{4} +{\left (3 \, b^{6} d^{6} e^{2} - 14 \, a b^{5} d^{5} e^{3} + 25 \, a^{2} b^{4} d^{4} e^{4} - 20 \, a^{3} b^{3} d^{3} e^{5} + 5 \, a^{4} b^{2} d^{2} e^{6} + 2 \, a^{5} b d e^{7} - a^{6} e^{8}\right )} x^{3} + 3 \,{\left (b^{6} d^{7} e - 4 \, a b^{5} d^{6} e^{2} + 5 \, a^{2} b^{4} d^{5} e^{3} - 5 \, a^{4} b^{2} d^{3} e^{5} + 4 \, a^{5} b d^{2} e^{6} - a^{6} d e^{7}\right )} x^{2} +{\left (b^{6} d^{8} - 2 \, a b^{5} d^{7} e - 5 \, a^{2} b^{4} d^{6} e^{2} + 20 \, a^{3} b^{3} d^{5} e^{3} - 25 \, a^{4} b^{2} d^{4} e^{4} + 14 \, a^{5} b d^{3} e^{5} - 3 \, a^{6} d^{2} e^{6}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.7665, size = 881, normalized size = 6.62 \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 = 1.12195, size = 455, normalized size = 3.42 \begin{align*} -\frac{4 \, b^{4} e \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{4 \, b^{3} e^{2} \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} + 10 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 6 \, a^{3} b d e^{3} - a^{4} e^{4} + 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (5 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (11 \, b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \,{\left (b d - a e\right )}^{5}{\left (b x + a\right )}{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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