Optimal. Leaf size=129 \[ \frac{a^3}{b (a+b x) (b c-a d)^3}+\frac{3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac{3 a^2 c \log (c+d x)}{(b c-a d)^4}-\frac{c^2 (b c-3 a d)}{d^2 (c+d x) (b c-a d)^3}+\frac{c^3}{2 d^2 (c+d x)^2 (b c-a d)^2} \]
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Rubi [A] time = 0.131083, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {88} \[ \frac{a^3}{b (a+b x) (b c-a d)^3}+\frac{3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac{3 a^2 c \log (c+d x)}{(b c-a d)^4}-\frac{c^2 (b c-3 a d)}{d^2 (c+d x) (b c-a d)^3}+\frac{c^3}{2 d^2 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int \frac{x^3}{(a+b x)^2 (c+d x)^3} \, dx &=\int \left (-\frac{a^3}{(b c-a d)^3 (a+b x)^2}+\frac{3 a^2 b c}{(b c-a d)^4 (a+b x)}-\frac{c^3}{d (-b c+a d)^2 (c+d x)^3}-\frac{c^2 (b c-3 a d)}{d (-b c+a d)^3 (c+d x)^2}-\frac{3 a^2 c d}{(-b c+a d)^4 (c+d x)}\right ) \, dx\\ &=\frac{a^3}{b (b c-a d)^3 (a+b x)}+\frac{c^3}{2 d^2 (b c-a d)^2 (c+d x)^2}-\frac{c^2 (b c-3 a d)}{d^2 (b c-a d)^3 (c+d x)}+\frac{3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac{3 a^2 c \log (c+d x)}{(b c-a d)^4}\\ \end{align*}
Mathematica [A] time = 0.162406, size = 130, normalized size = 1.01 \[ \frac{a^3}{b (a+b x) (b c-a d)^3}+\frac{3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac{3 a^2 c \log (c+d x)}{(b c-a d)^4}+\frac{b c^3-3 a c^2 d}{d^2 (c+d x) (a d-b c)^3}+\frac{c^3}{2 d^2 (c+d x)^2 (a d-b c)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 147, normalized size = 1.1 \begin{align*} -3\,{\frac{{c}^{2}a}{ \left ( ad-bc \right ) ^{3}d \left ( dx+c \right ) }}+{\frac{{c}^{3}b}{ \left ( ad-bc \right ) ^{3}{d}^{2} \left ( dx+c \right ) }}+{\frac{{c}^{3}}{2\,{d}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}-3\,{\frac{c{a}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{4}}}-{\frac{{a}^{3}}{ \left ( ad-bc \right ) ^{3}b \left ( bx+a \right ) }}+3\,{\frac{c{a}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14602, size = 625, normalized size = 4.84 \begin{align*} \frac{3 \, a^{2} c \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac{3 \, a^{2} c \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac{a b^{2} c^{4} - 5 \, a^{2} b c^{3} d - 2 \, a^{3} c^{2} d^{2} + 2 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - a^{3} d^{4}\right )} x^{2} +{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} - 4 \, a^{3} c d^{3}\right )} x}{2 \,{\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5} +{\left (b^{5} c^{3} d^{4} - 3 \, a b^{4} c^{2} d^{5} + 3 \, a^{2} b^{3} c d^{6} - a^{3} b^{2} d^{7}\right )} x^{3} +{\left (2 \, b^{5} c^{4} d^{3} - 5 \, a b^{4} c^{3} d^{4} + 3 \, a^{2} b^{3} c^{2} d^{5} + a^{3} b^{2} c d^{6} - a^{4} b d^{7}\right )} x^{2} +{\left (b^{5} c^{5} d^{2} - a b^{4} c^{4} d^{3} - 3 \, a^{2} b^{3} c^{3} d^{4} + 5 \, a^{3} b^{2} c^{2} d^{5} - 2 \, a^{4} b c d^{6}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.78596, size = 1220, normalized size = 9.46 \begin{align*} -\frac{a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} + 2 \, a^{4} c^{2} d^{3} + 2 \,{\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} - a^{3} b c d^{4} + a^{4} d^{5}\right )} x^{2} +{\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3} + 4 \, a^{4} c d^{4}\right )} x - 6 \,{\left (a^{2} b^{2} c d^{4} x^{3} + a^{3} b c^{3} d^{2} +{\left (2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4}\right )} x^{2} +{\left (a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left (a^{2} b^{2} c d^{4} x^{3} + a^{3} b c^{3} d^{2} +{\left (2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4}\right )} x^{2} +{\left (a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{5} c^{6} d^{2} - 4 \, a^{2} b^{4} c^{5} d^{3} + 6 \, a^{3} b^{3} c^{4} d^{4} - 4 \, a^{4} b^{2} c^{3} d^{5} + a^{5} b c^{2} d^{6} +{\left (b^{6} c^{4} d^{4} - 4 \, a b^{5} c^{3} d^{5} + 6 \, a^{2} b^{4} c^{2} d^{6} - 4 \, a^{3} b^{3} c d^{7} + a^{4} b^{2} d^{8}\right )} x^{3} +{\left (2 \, b^{6} c^{5} d^{3} - 7 \, a b^{5} c^{4} d^{4} + 8 \, a^{2} b^{4} c^{3} d^{5} - 2 \, a^{3} b^{3} c^{2} d^{6} - 2 \, a^{4} b^{2} c d^{7} + a^{5} b d^{8}\right )} x^{2} +{\left (b^{6} c^{6} d^{2} - 2 \, a b^{5} c^{5} d^{3} - 2 \, a^{2} b^{4} c^{4} d^{4} + 8 \, a^{3} b^{3} c^{3} d^{5} - 7 \, a^{4} b^{2} c^{2} d^{6} + 2 \, a^{5} b c d^{7}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.00997, size = 717, normalized size = 5.56 \begin{align*} - \frac{3 a^{2} c \log{\left (x + \frac{- \frac{3 a^{7} c d^{5}}{\left (a d - b c\right )^{4}} + \frac{15 a^{6} b c^{2} d^{4}}{\left (a d - b c\right )^{4}} - \frac{30 a^{5} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{4}} + \frac{30 a^{4} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{4}} - \frac{15 a^{3} b^{4} c^{5} d}{\left (a d - b c\right )^{4}} + 3 a^{3} c d + \frac{3 a^{2} b^{5} c^{6}}{\left (a d - b c\right )^{4}} + 3 a^{2} b c^{2}}{6 a^{2} b c d} \right )}}{\left (a d - b c\right )^{4}} + \frac{3 a^{2} c \log{\left (x + \frac{\frac{3 a^{7} c d^{5}}{\left (a d - b c\right )^{4}} - \frac{15 a^{6} b c^{2} d^{4}}{\left (a d - b c\right )^{4}} + \frac{30 a^{5} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{4}} - \frac{30 a^{4} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + \frac{15 a^{3} b^{4} c^{5} d}{\left (a d - b c\right )^{4}} + 3 a^{3} c d - \frac{3 a^{2} b^{5} c^{6}}{\left (a d - b c\right )^{4}} + 3 a^{2} b c^{2}}{6 a^{2} b c d} \right )}}{\left (a d - b c\right )^{4}} - \frac{2 a^{3} c^{2} d^{2} + 5 a^{2} b c^{3} d - a b^{2} c^{4} + x^{2} \left (2 a^{3} d^{4} + 6 a b^{2} c^{2} d^{2} - 2 b^{3} c^{3} d\right ) + x \left (4 a^{3} c d^{3} + 6 a^{2} b c^{2} d^{2} + 3 a b^{2} c^{3} d - b^{3} c^{4}\right )}{2 a^{4} b c^{2} d^{5} - 6 a^{3} b^{2} c^{3} d^{4} + 6 a^{2} b^{3} c^{4} d^{3} - 2 a b^{4} c^{5} d^{2} + x^{3} \left (2 a^{3} b^{2} d^{7} - 6 a^{2} b^{3} c d^{6} + 6 a b^{4} c^{2} d^{5} - 2 b^{5} c^{3} d^{4}\right ) + x^{2} \left (2 a^{4} b d^{7} - 2 a^{3} b^{2} c d^{6} - 6 a^{2} b^{3} c^{2} d^{5} + 10 a b^{4} c^{3} d^{4} - 4 b^{5} c^{4} d^{3}\right ) + x \left (4 a^{4} b c d^{6} - 10 a^{3} b^{2} c^{2} d^{5} + 6 a^{2} b^{3} c^{3} d^{4} + 2 a b^{4} c^{4} d^{3} - 2 b^{5} c^{5} d^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2065, size = 311, normalized size = 2.41 \begin{align*} -\frac{3 \, a^{2} b c \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac{a^{3} b^{2}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )}{\left (b x + a\right )}} + \frac{b^{2} c^{3} - 6 \, a b c^{2} d - \frac{6 \,{\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )}}{{\left (b x + a\right )} b}}{2 \,{\left (b c - a d\right )}^{4}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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