Optimal. Leaf size=146 \[ \frac{e^2 \log (a+b x) (B d-A e)}{(b d-a e)^4}-\frac{e^2 (B d-A e) \log (d+e x)}{(b d-a e)^4}+\frac{e (B d-A e)}{(a+b x) (b d-a e)^3}-\frac{B d-A e}{2 (a+b x)^2 (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 (b d-a e)} \]
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Rubi [A] time = 0.127773, antiderivative size = 146, 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, 77} \[ \frac{e^2 \log (a+b x) (B d-A e)}{(b d-a e)^4}-\frac{e^2 (B d-A e) \log (d+e x)}{(b d-a e)^4}+\frac{e (B d-A e)}{(a+b x) (b d-a e)^3}-\frac{B d-A e}{2 (a+b x)^2 (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{A+B x}{(a+b x)^4 (d+e x)} \, dx\\ &=\int \left (\frac{A b-a B}{(b d-a e) (a+b x)^4}+\frac{b (B d-A e)}{(b d-a e)^2 (a+b x)^3}+\frac{b e (-B d+A e)}{(b d-a e)^3 (a+b x)^2}-\frac{b e^2 (-B d+A e)}{(b d-a e)^4 (a+b x)}+\frac{e^3 (-B d+A e)}{(b d-a e)^4 (d+e x)}\right ) \, dx\\ &=-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3}-\frac{B d-A e}{2 (b d-a e)^2 (a+b x)^2}+\frac{e (B d-A e)}{(b d-a e)^3 (a+b x)}+\frac{e^2 (B d-A e) \log (a+b x)}{(b d-a e)^4}-\frac{e^2 (B d-A e) \log (d+e x)}{(b d-a e)^4}\\ \end{align*}
Mathematica [A] time = 0.0964573, size = 136, normalized size = 0.93 \[ \frac{6 e^2 \log (a+b x) (B d-A e)+\frac{2 (a B-A b) (b d-a e)^3}{b (a+b x)^3}+\frac{3 (b d-a e)^2 (A e-B d)}{(a+b x)^2}+\frac{6 e (a e-b d) (A e-B d)}{a+b x}+6 e^2 (A e-B d) \log (d+e x)}{6 (b d-a e)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 220, normalized size = 1.5 \begin{align*}{\frac{{e}^{3}\ln \left ( ex+d \right ) A}{ \left ( ae-bd \right ) ^{4}}}-{\frac{{e}^{2}\ln \left ( ex+d \right ) Bd}{ \left ( ae-bd \right ) ^{4}}}+{\frac{A}{ \left ( 3\,ae-3\,bd \right ) \left ( bx+a \right ) ^{3}}}-{\frac{aB}{ \left ( 3\,ae-3\,bd \right ) b \left ( bx+a \right ) ^{3}}}+{\frac{Ae}{2\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{2}}}-{\frac{Bd}{2\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{2}}}+{\frac{A{e}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}-{\frac{Bde}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}-{\frac{{e}^{3}\ln \left ( bx+a \right ) A}{ \left ( ae-bd \right ) ^{4}}}+{\frac{{e}^{2}\ln \left ( bx+a \right ) Bd}{ \left ( ae-bd \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15105, size = 610, normalized size = 4.18 \begin{align*} \frac{{\left (B d e^{2} - A e^{3}\right )} \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{{\left (B d e^{2} - A e^{3}\right )} \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{{\left (B a b^{2} + 2 \, A b^{3}\right )} d^{2} -{\left (5 \, B a^{2} b + 7 \, A a b^{2}\right )} d e -{\left (2 \, B a^{3} - 11 \, A a^{2} b\right )} e^{2} - 6 \,{\left (B b^{3} d e - A b^{3} e^{2}\right )} x^{2} + 3 \,{\left (B b^{3} d^{2} + 5 \, A a b^{2} e^{2} -{\left (5 \, B a b^{2} + A b^{3}\right )} d e\right )} x}{6 \,{\left (a^{3} b^{4} d^{3} - 3 \, a^{4} b^{3} d^{2} e + 3 \, a^{5} b^{2} d e^{2} - a^{6} b e^{3} +{\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} x^{3} + 3 \,{\left (a b^{6} d^{3} - 3 \, a^{2} b^{5} d^{2} e + 3 \, a^{3} b^{4} d e^{2} - a^{4} b^{3} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{5} d^{3} - 3 \, a^{3} b^{4} d^{2} e + 3 \, a^{4} b^{3} d e^{2} - a^{5} b^{2} e^{3}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68311, size = 1284, normalized size = 8.79 \begin{align*} -\frac{{\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} - 3 \,{\left (2 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d^{2} e + 3 \,{\left (B a^{3} b + 6 \, A a^{2} b^{2}\right )} d e^{2} +{\left (2 \, B a^{4} - 11 \, A a^{3} b\right )} e^{3} - 6 \,{\left (B b^{4} d^{2} e + A a b^{3} e^{3} -{\left (B a b^{3} + A b^{4}\right )} d e^{2}\right )} x^{2} + 3 \,{\left (B b^{4} d^{3} - 5 \, A a^{2} b^{2} e^{3} -{\left (6 \, B a b^{3} + A b^{4}\right )} d^{2} e +{\left (5 \, B a^{2} b^{2} + 6 \, A a b^{3}\right )} d e^{2}\right )} x - 6 \,{\left (B a^{3} b d e^{2} - A a^{3} b e^{3} +{\left (B b^{4} d e^{2} - A b^{4} e^{3}\right )} x^{3} + 3 \,{\left (B a b^{3} d e^{2} - A a b^{3} e^{3}\right )} x^{2} + 3 \,{\left (B a^{2} b^{2} d e^{2} - A a^{2} b^{2} e^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left (B a^{3} b d e^{2} - A a^{3} b e^{3} +{\left (B b^{4} d e^{2} - A b^{4} e^{3}\right )} x^{3} + 3 \,{\left (B a b^{3} d e^{2} - A a b^{3} e^{3}\right )} x^{2} + 3 \,{\left (B a^{2} b^{2} d e^{2} - A a^{2} b^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\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} +{\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^{3} + 3 \,{\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^{2} + 3 \,{\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\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.69624, size = 818, normalized size = 5.6 \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 = 1.15323, size = 491, normalized size = 3.36 \begin{align*} \frac{{\left (B b d e^{2} - A b e^{3}\right )} \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{{\left (B d e^{3} - A e^{4}\right )} \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{B a b^{3} d^{3} + 2 \, A b^{4} d^{3} - 6 \, B a^{2} b^{2} d^{2} e - 9 \, A a b^{3} d^{2} e + 3 \, B a^{3} b d e^{2} + 18 \, A a^{2} b^{2} d e^{2} + 2 \, B a^{4} e^{3} - 11 \, A a^{3} b e^{3} - 6 \,{\left (B b^{4} d^{2} e - B a b^{3} d e^{2} - A b^{4} d e^{2} + A a b^{3} e^{3}\right )} x^{2} + 3 \,{\left (B b^{4} d^{3} - 6 \, B a b^{3} d^{2} e - A b^{4} d^{2} e + 5 \, B a^{2} b^{2} d e^{2} + 6 \, A a b^{3} d e^{2} - 5 \, A a^{2} b^{2} e^{3}\right )} x}{6 \,{\left (b d - a e\right )}^{4}{\left (b x + a\right )}^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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