Optimal. Leaf size=134 \[ \frac{d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}-\frac{b^3 \log (a+b x)}{a (b c-a d)^3}-\frac{d (2 b c-a d)}{c^2 (c+d x) (b c-a d)^2}-\frac{d}{2 c (c+d x)^2 (b c-a d)}+\frac{\log (x)}{a c^3} \]
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Rubi [A] time = 0.106698, antiderivative size = 134, 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 = {72} \[ \frac{d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}-\frac{b^3 \log (a+b x)}{a (b c-a d)^3}-\frac{d (2 b c-a d)}{c^2 (c+d x) (b c-a d)^2}-\frac{d}{2 c (c+d x)^2 (b c-a d)}+\frac{\log (x)}{a c^3} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin{align*} \int \frac{1}{x (a+b x) (c+d x)^3} \, dx &=\int \left (\frac{1}{a c^3 x}+\frac{b^4}{a (-b c+a d)^3 (a+b x)}+\frac{d^2}{c (b c-a d) (c+d x)^3}+\frac{d^2 (2 b c-a d)}{c^2 (b c-a d)^2 (c+d x)^2}+\frac{d^2 \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{c^3 (b c-a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac{d}{2 c (b c-a d) (c+d x)^2}-\frac{d (2 b c-a d)}{c^2 (b c-a d)^2 (c+d x)}+\frac{\log (x)}{a c^3}-\frac{b^3 \log (a+b x)}{a (b c-a d)^3}+\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 0.249302, size = 116, normalized size = 0.87 \[ \frac{\frac{d \left (\frac{c (b c-a d) (b c (5 c+4 d x)-a d (3 c+2 d x))}{(c+d x)^2}-2 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)\right )}{c^3}+\frac{2 b^3 \log (a+b x)}{a}}{2 (a d-b c)^3}+\frac{\log (x)}{a c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 184, normalized size = 1.4 \begin{align*}{\frac{d}{2\,c \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}+{\frac{a{d}^{2}}{{c}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-2\,{\frac{bd}{c \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-{\frac{{d}^{3}\ln \left ( dx+c \right ){a}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{3}}}+3\,{\frac{{d}^{2}\ln \left ( dx+c \right ) ab}{{c}^{2} \left ( ad-bc \right ) ^{3}}}-3\,{\frac{d\ln \left ( dx+c \right ){b}^{2}}{c \left ( ad-bc \right ) ^{3}}}+{\frac{\ln \left ( x \right ) }{a{c}^{3}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23681, size = 359, normalized size = 2.68 \begin{align*} -\frac{b^{3} \log \left (b x + a\right )}{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}} + \frac{{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (d x + c\right )}{b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}} - \frac{5 \, b c^{2} d - 3 \, a c d^{2} + 2 \,{\left (2 \, b c d^{2} - a d^{3}\right )} x}{2 \,{\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} +{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x\right )}} + \frac{\log \left (x\right )}{a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 63.4492, size = 1008, normalized size = 7.52 \begin{align*} -\frac{5 \, a b^{2} c^{4} d - 8 \, a^{2} b c^{3} d^{2} + 3 \, a^{3} c^{2} d^{3} + 2 \,{\left (2 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x + 2 \,{\left (b^{3} c^{3} d^{2} x^{2} + 2 \, b^{3} c^{4} d x + b^{3} c^{5}\right )} \log \left (b x + a\right ) - 2 \,{\left (3 \, a b^{2} c^{4} d - 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} +{\left (3 \, a b^{2} c^{2} d^{3} - 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \,{\left (3 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x\right )} \log \left (d x + c\right ) - 2 \,{\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} +{\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2} + 2 \,{\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x\right )} \log \left (x\right )}{2 \,{\left (a b^{3} c^{8} - 3 \, a^{2} b^{2} c^{7} d + 3 \, a^{3} b c^{6} d^{2} - a^{4} c^{5} d^{3} +{\left (a b^{3} c^{6} d^{2} - 3 \, a^{2} b^{2} c^{5} d^{3} + 3 \, a^{3} b c^{4} d^{4} - a^{4} c^{3} d^{5}\right )} x^{2} + 2 \,{\left (a b^{3} c^{7} d - 3 \, a^{2} b^{2} c^{6} d^{2} + 3 \, a^{3} b c^{5} d^{3} - a^{4} c^{4} d^{4}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.214, size = 316, normalized size = 2.36 \begin{align*} -\frac{b^{4} \log \left ({\left | b x + a \right |}\right )}{a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}} + \frac{{\left (3 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}} + \frac{\log \left ({\left | x \right |}\right )}{a c^{3}} - \frac{5 \, b^{2} c^{4} d - 8 \, a b c^{3} d^{2} + 3 \, a^{2} c^{2} d^{3} + 2 \,{\left (2 \, b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x}{2 \,{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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