Optimal. Leaf size=144 \[ \frac{\log (x) \left (3 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^3 c^4}-\frac{b^4 \log (a+b x)}{a^3 (b c-a d)^2}+\frac{2 a d+b c}{a^2 c^3 x}-\frac{d^3}{c^3 (c+d x) (b c-a d)}+\frac{d^3 (4 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^2}-\frac{1}{2 a c^2 x^2} \]
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Rubi [A] time = 0.141431, antiderivative size = 144, 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} \[ \frac{\log (x) \left (3 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^3 c^4}-\frac{b^4 \log (a+b x)}{a^3 (b c-a d)^2}+\frac{2 a d+b c}{a^2 c^3 x}-\frac{d^3}{c^3 (c+d x) (b c-a d)}+\frac{d^3 (4 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^2}-\frac{1}{2 a c^2 x^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int \frac{1}{x^3 (a+b x) (c+d x)^2} \, dx &=\int \left (\frac{1}{a c^2 x^3}+\frac{-b c-2 a d}{a^2 c^3 x^2}+\frac{b^2 c^2+2 a b c d+3 a^2 d^2}{a^3 c^4 x}-\frac{b^5}{a^3 (-b c+a d)^2 (a+b x)}+\frac{d^4}{c^3 (b c-a d) (c+d x)^2}+\frac{d^4 (4 b c-3 a d)}{c^4 (b c-a d)^2 (c+d x)}\right ) \, dx\\ &=-\frac{1}{2 a c^2 x^2}+\frac{b c+2 a d}{a^2 c^3 x}-\frac{d^3}{c^3 (b c-a d) (c+d x)}+\frac{\left (b^2 c^2+2 a b c d+3 a^2 d^2\right ) \log (x)}{a^3 c^4}-\frac{b^4 \log (a+b x)}{a^3 (b c-a d)^2}+\frac{d^3 (4 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.417791, size = 143, normalized size = 0.99 \[ \frac{\log (x) \left (3 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^3 c^4}-\frac{b^4 \log (a+b x)}{a^3 (b c-a d)^2}+\frac{c \left (\frac{2 b c}{a^2 x}+\frac{2 d^3}{(c+d x) (a d-b c)}-\frac{c-4 d x}{a x^2}\right )+\frac{2 d^3 (4 b c-3 a d) \log (c+d x)}{(b c-a d)^2}}{2 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 171, normalized size = 1.2 \begin{align*}{\frac{{d}^{3}}{{c}^{3} \left ( ad-bc \right ) \left ( dx+c \right ) }}-3\,{\frac{{d}^{4}\ln \left ( dx+c \right ) a}{{c}^{4} \left ( ad-bc \right ) ^{2}}}+4\,{\frac{{d}^{3}\ln \left ( dx+c \right ) b}{{c}^{3} \left ( ad-bc \right ) ^{2}}}-{\frac{1}{2\,a{c}^{2}{x}^{2}}}+2\,{\frac{d}{a{c}^{3}x}}+{\frac{b}{{a}^{2}{c}^{2}x}}+3\,{\frac{\ln \left ( x \right ){d}^{2}}{a{c}^{4}}}+2\,{\frac{b\ln \left ( x \right ) d}{{a}^{2}{c}^{3}}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}{c}^{2}}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09693, size = 331, normalized size = 2.3 \begin{align*} -\frac{b^{4} \log \left (b x + a\right )}{a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}} + \frac{{\left (4 \, b c d^{3} - 3 \, a d^{4}\right )} \log \left (d x + c\right )}{b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2}} - \frac{a b c^{3} - a^{2} c^{2} d - 2 \,{\left (b^{2} c^{2} d + a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} -{\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x}{2 \,{\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{3} +{\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{2}\right )}} + \frac{{\left (b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (x\right )}{a^{3} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 47.0734, size = 698, normalized size = 4.85 \begin{align*} -\frac{a^{2} b^{2} c^{5} - 2 \, a^{3} b c^{4} d + a^{4} c^{3} d^{2} - 2 \,{\left (a b^{3} c^{4} d - 4 \, a^{3} b c^{2} d^{3} + 3 \, a^{4} c d^{4}\right )} x^{2} -{\left (2 \, a b^{3} c^{5} - a^{2} b^{2} c^{4} d - 4 \, a^{3} b c^{3} d^{2} + 3 \, a^{4} c^{2} d^{3}\right )} x + 2 \,{\left (b^{4} c^{4} d x^{3} + b^{4} c^{5} x^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left ({\left (4 \, a^{3} b c d^{4} - 3 \, a^{4} d^{5}\right )} x^{3} +{\left (4 \, a^{3} b c^{2} d^{3} - 3 \, a^{4} c d^{4}\right )} x^{2}\right )} \log \left (d x + c\right ) - 2 \,{\left ({\left (b^{4} c^{4} d - 4 \, a^{3} b c d^{4} + 3 \, a^{4} d^{5}\right )} x^{3} +{\left (b^{4} c^{5} - 4 \, a^{3} b c^{2} d^{3} + 3 \, a^{4} c d^{4}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left ({\left (a^{3} b^{2} c^{6} d - 2 \, a^{4} b c^{5} d^{2} + a^{5} c^{4} d^{3}\right )} x^{3} +{\left (a^{3} b^{2} c^{7} - 2 \, a^{4} b c^{6} d + a^{5} c^{5} d^{2}\right )} x^{2}\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 = 1.20544, size = 285, normalized size = 1.98 \begin{align*} -\frac{d^{7}}{{\left (b c^{4} d^{4} - a c^{3} d^{5}\right )}{\left (d x + c\right )}} - \frac{b^{4} d \log \left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{a^{3} b^{2} c^{2} d - 2 \, a^{4} b c d^{2} + a^{5} d^{3}} + \frac{{\left (b^{2} c^{2} d + 2 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} \log \left ({\left | -\frac{c}{d x + c} + 1 \right |}\right )}{a^{3} c^{4} d} + \frac{2 \, a b c d + 5 \, a^{2} d^{2} - \frac{2 \,{\left (a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )}}{{\left (d x + c\right )} d}}{2 \, a^{3} c^{4}{\left (\frac{c}{d x + c} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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