Optimal. Leaf size=107 \[ \frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}-\frac{b^3 \log (a+b x)}{a^3 (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}+\frac{d^3 \log (c+d x)}{c^3 (b c-a d)}-\frac{1}{2 a c x^2} \]
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Rubi [A] time = 0.0801743, antiderivative size = 107, 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} \[ \frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}-\frac{b^3 \log (a+b x)}{a^3 (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}+\frac{d^3 \log (c+d x)}{c^3 (b c-a d)}-\frac{1}{2 a c x^2} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin{align*} \int \frac{1}{x^3 (a+b x) (c+d x)} \, dx &=\int \left (\frac{1}{a c x^3}+\frac{-b c-a d}{a^2 c^2 x^2}+\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac{b^4}{a^3 (-b c+a d) (a+b x)}+\frac{d^4}{c^3 (b c-a d) (c+d x)}\right ) \, dx\\ &=-\frac{1}{2 a c x^2}+\frac{b c+a d}{a^2 c^2 x}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \log (x)}{a^3 c^3}-\frac{b^3 \log (a+b x)}{a^3 (b c-a d)}+\frac{d^3 \log (c+d x)}{c^3 (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0489417, size = 106, normalized size = 0.99 \[ \frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}+\frac{b^3 \log (a+b x)}{a^3 (a d-b c)}+\frac{a d+b c}{a^2 c^2 x}+\frac{d^3 \log (c+d x)}{c^3 (b c-a d)}-\frac{1}{2 a c x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 117, normalized size = 1.1 \begin{align*} -{\frac{{d}^{3}\ln \left ( dx+c \right ) }{{c}^{3} \left ( ad-bc \right ) }}-{\frac{1}{2\,ac{x}^{2}}}+{\frac{d}{a{c}^{2}x}}+{\frac{b}{{a}^{2}cx}}+{\frac{\ln \left ( x \right ){d}^{2}}{a{c}^{3}}}+{\frac{b\ln \left ( x \right ) d}{{a}^{2}{c}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}c}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ){a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03271, size = 143, normalized size = 1.34 \begin{align*} -\frac{b^{3} \log \left (b x + a\right )}{a^{3} b c - a^{4} d} + \frac{d^{3} \log \left (d x + c\right )}{b c^{4} - a c^{3} d} + \frac{{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} \log \left (x\right )}{a^{3} c^{3}} - \frac{a c - 2 \,{\left (b c + a d\right )} x}{2 \, a^{2} c^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 21.3442, size = 247, normalized size = 2.31 \begin{align*} -\frac{2 \, b^{3} c^{3} x^{2} \log \left (b x + a\right ) - 2 \, a^{3} d^{3} x^{2} \log \left (d x + c\right ) + a^{2} b c^{3} - a^{3} c^{2} d - 2 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{2} \log \left (x\right ) - 2 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x}{2 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{2}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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