Optimal. Leaf size=195 \[ \frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}-\frac{b^4}{a^2 (a+b x) (b c-a d)^3}+\frac{b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{1}{a^2 c^3 x}-\frac{2 d^3 (2 b c-a d)}{c^3 (c+d x) (b c-a d)^3}-\frac{d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]
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Rubi [A] time = 0.231376, antiderivative size = 195, 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} \[ \frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}-\frac{b^4}{a^2 (a+b x) (b c-a d)^3}+\frac{b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{1}{a^2 c^3 x}-\frac{2 d^3 (2 b c-a d)}{c^3 (c+d x) (b c-a d)^3}-\frac{d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int \frac{1}{x^2 (a+b x)^2 (c+d x)^3} \, dx &=\int \left (\frac{1}{a^2 c^3 x^2}+\frac{-2 b c-3 a d}{a^3 c^4 x}-\frac{b^5}{a^2 (-b c+a d)^3 (a+b x)^2}-\frac{b^5 (-2 b c+5 a d)}{a^3 (-b c+a d)^4 (a+b x)}+\frac{d^4}{c^2 (b c-a d)^2 (c+d x)^3}+\frac{2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)^2}+\frac{d^4 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )}{c^4 (b c-a d)^4 (c+d x)}\right ) \, dx\\ &=-\frac{1}{a^2 c^3 x}-\frac{b^4}{a^2 (b c-a d)^3 (a+b x)}-\frac{d^3}{2 c^2 (b c-a d)^2 (c+d x)^2}-\frac{2 d^3 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)}-\frac{(2 b c+3 a d) \log (x)}{a^3 c^4}+\frac{b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}\\ \end{align*}
Mathematica [A] time = 0.263727, size = 193, normalized size = 0.99 \[ \frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}+\frac{b^4}{a^2 (a+b x) (a d-b c)^3}+\frac{b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{1}{a^2 c^3 x}+\frac{2 d^3 (a d-2 b c)}{c^3 (c+d x) (b c-a d)^3}-\frac{d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 266, normalized size = 1.4 \begin{align*} -{\frac{{d}^{3}}{2\,{c}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}+3\,{\frac{{d}^{5}\ln \left ( dx+c \right ){a}^{2}}{{c}^{4} \left ( ad-bc \right ) ^{4}}}-10\,{\frac{{d}^{4}\ln \left ( dx+c \right ) ab}{{c}^{3} \left ( ad-bc \right ) ^{4}}}+10\,{\frac{{d}^{3}\ln \left ( dx+c \right ){b}^{2}}{{c}^{2} \left ( ad-bc \right ) ^{4}}}-2\,{\frac{{d}^{4}a}{{c}^{3} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}+4\,{\frac{{d}^{3}b}{{c}^{2} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-{\frac{1}{{a}^{2}{c}^{3}x}}-3\,{\frac{\ln \left ( x \right ) d}{{a}^{2}{c}^{4}}}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}{c}^{3}}}+{\frac{{b}^{4}}{ \left ( ad-bc \right ) ^{3}{a}^{2} \left ( bx+a \right ) }}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{4}{a}^{2}}}+2\,{\frac{{b}^{5}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{4}{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 3.22616, size = 863, normalized size = 4.43 \begin{align*} \frac{{\left (2 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x + a\right )}{a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}} + \frac{{\left (10 \, b^{2} c^{2} d^{3} - 10 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left (d x + c\right )}{b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}} - \frac{2 \, a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 6 \, a^{3} b c^{3} d^{2} - 2 \, a^{4} c^{2} d^{3} + 2 \,{\left (2 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 7 \, a^{2} b^{2} c d^{4} - 3 \, a^{3} b d^{5}\right )} x^{3} +{\left (8 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + 15 \, a^{2} b^{2} c^{2} d^{3} + 5 \, a^{3} b c d^{4} - 6 \, a^{4} d^{5}\right )} x^{2} +{\left (4 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d - 6 \, a^{2} b^{2} c^{3} d^{2} + 19 \, a^{3} b c^{2} d^{3} - 9 \, a^{4} c d^{4}\right )} x}{2 \,{\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{4} +{\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{3} +{\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{2} +{\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x\right )}} - \frac{{\left (2 \, b c + 3 \, a d\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 [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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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 [B] time = 1.20558, size = 698, normalized size = 3.58 \begin{align*} -\frac{b^{9}}{{\left (a^{2} b^{8} c^{3} - 3 \, a^{3} b^{7} c^{2} d + 3 \, a^{4} b^{6} c d^{2} - a^{5} b^{5} d^{3}\right )}{\left (b x + a\right )}} + \frac{{\left (10 \, b^{3} c^{2} d^{3} - 10 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{8} - 4 \, a b^{4} c^{7} d + 6 \, a^{2} b^{3} c^{6} d^{2} - 4 \, a^{3} b^{2} c^{5} d^{3} + a^{4} b c^{4} d^{4}} - \frac{{\left (2 \, b^{2} c + 3 \, a b d\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{3} b c^{4}} + \frac{2 \, b^{5} c^{5} d^{2} - 8 \, a b^{4} c^{4} d^{3} + 12 \, a^{2} b^{3} c^{3} d^{4} - 17 \, a^{3} b^{2} c^{2} d^{5} + 6 \, a^{4} b c d^{6} + \frac{4 \, b^{7} c^{6} d - 20 \, a b^{6} c^{5} d^{2} + 40 \, a^{2} b^{5} c^{4} d^{3} - 50 \, a^{3} b^{4} c^{3} d^{4} + 43 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6}}{{\left (b x + a\right )} b} + \frac{2 \,{\left (b^{9} c^{7} - 6 \, a b^{8} c^{6} d + 15 \, a^{2} b^{7} c^{5} d^{2} - 20 \, a^{3} b^{6} c^{4} d^{3} + 20 \, a^{4} b^{5} c^{3} d^{4} - 13 \, a^{5} b^{4} c^{2} d^{5} + 3 \, a^{6} b^{3} c d^{6}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{2 \,{\left (b c - a d\right )}^{4} a^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2} c^{4}{\left (\frac{a}{b x + a} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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