Optimal. Leaf size=228 \[ \frac{3 \log (x) \left (2 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^4 c^5}-\frac{3 d^4 \left (2 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}+\frac{b^5}{a^3 (a+b x) (b c-a d)^3}-\frac{3 b^5 (b c-2 a d) \log (a+b x)}{a^4 (b c-a d)^4}+\frac{3 a d+2 b c}{a^3 c^4 x}-\frac{1}{2 a^2 c^3 x^2}+\frac{d^4 (5 b c-3 a d)}{c^4 (c+d x) (b c-a d)^3}+\frac{d^4}{2 c^3 (c+d x)^2 (b c-a d)^2} \]
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Rubi [A] time = 0.290479, antiderivative size = 228, 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} \[ \frac{3 \log (x) \left (2 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^4 c^5}-\frac{3 d^4 \left (2 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}+\frac{b^5}{a^3 (a+b x) (b c-a d)^3}-\frac{3 b^5 (b c-2 a d) \log (a+b x)}{a^4 (b c-a d)^4}+\frac{3 a d+2 b c}{a^3 c^4 x}-\frac{1}{2 a^2 c^3 x^2}+\frac{d^4 (5 b c-3 a d)}{c^4 (c+d x) (b c-a d)^3}+\frac{d^4}{2 c^3 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int \frac{1}{x^3 (a+b x)^2 (c+d x)^3} \, dx &=\int \left (\frac{1}{a^2 c^3 x^3}+\frac{-2 b c-3 a d}{a^3 c^4 x^2}+\frac{3 \left (b^2 c^2+2 a b c d+2 a^2 d^2\right )}{a^4 c^5 x}+\frac{b^6}{a^3 (-b c+a d)^3 (a+b x)^2}+\frac{3 b^6 (-b c+2 a d)}{a^4 (-b c+a d)^4 (a+b x)}-\frac{d^5}{c^3 (b c-a d)^2 (c+d x)^3}-\frac{d^5 (5 b c-3 a d)}{c^4 (b c-a d)^3 (c+d x)^2}-\frac{3 d^5 \left (5 b^2 c^2-6 a b c d+2 a^2 d^2\right )}{c^5 (b c-a d)^4 (c+d x)}\right ) \, dx\\ &=-\frac{1}{2 a^2 c^3 x^2}+\frac{2 b c+3 a d}{a^3 c^4 x}+\frac{b^5}{a^3 (b c-a d)^3 (a+b x)}+\frac{d^4}{2 c^3 (b c-a d)^2 (c+d x)^2}+\frac{d^4 (5 b c-3 a d)}{c^4 (b c-a d)^3 (c+d x)}+\frac{3 \left (b^2 c^2+2 a b c d+2 a^2 d^2\right ) \log (x)}{a^4 c^5}-\frac{3 b^5 (b c-2 a d) \log (a+b x)}{a^4 (b c-a d)^4}-\frac{3 d^4 \left (5 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}\\ \end{align*}
Mathematica [A] time = 0.283604, size = 230, normalized size = 1.01 \[ \frac{3 \log (x) \left (2 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^4 c^5}-\frac{3 d^4 \left (2 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}-\frac{b^5}{a^3 (a+b x) (a d-b c)^3}+\frac{3 b^5 (2 a d-b c) \log (a+b x)}{a^4 (b c-a d)^4}+\frac{3 a d+2 b c}{a^3 c^4 x}-\frac{1}{2 a^2 c^3 x^2}+\frac{d^4 (5 b c-3 a d)}{c^4 (c+d x) (b c-a d)^3}+\frac{d^4}{2 c^3 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 307, normalized size = 1.4 \begin{align*}{\frac{{d}^{4}}{2\,{c}^{3} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}+3\,{\frac{{d}^{5}a}{{c}^{4} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-5\,{\frac{{d}^{4}b}{{c}^{3} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-6\,{\frac{{d}^{6}\ln \left ( dx+c \right ){a}^{2}}{{c}^{5} \left ( ad-bc \right ) ^{4}}}+18\,{\frac{{d}^{5}\ln \left ( dx+c \right ) ab}{{c}^{4} \left ( ad-bc \right ) ^{4}}}-15\,{\frac{{d}^{4}\ln \left ( dx+c \right ){b}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{4}}}-{\frac{1}{2\,{a}^{2}{c}^{3}{x}^{2}}}+3\,{\frac{d}{{a}^{2}{c}^{4}x}}+2\,{\frac{b}{{a}^{3}{c}^{3}x}}+6\,{\frac{\ln \left ( x \right ){d}^{2}}{{a}^{2}{c}^{5}}}+6\,{\frac{b\ln \left ( x \right ) d}{{a}^{3}{c}^{4}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{4}{c}^{3}}}-{\frac{{b}^{5}}{ \left ( ad-bc \right ) ^{3}{a}^{3} \left ( bx+a \right ) }}+6\,{\frac{{b}^{5}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{4}{a}^{3}}}-3\,{\frac{{b}^{6}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{4}{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.39517, size = 1017, normalized size = 4.46 \begin{align*} -\frac{3 \,{\left (b^{6} c - 2 \, a b^{5} d\right )} \log \left (b x + a\right )}{a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4}} - \frac{3 \,{\left (5 \, b^{2} c^{2} d^{4} - 6 \, a b c d^{5} + 2 \, a^{2} d^{6}\right )} \log \left (d x + c\right )}{b^{4} c^{9} - 4 \, a b^{3} c^{8} d + 6 \, a^{2} b^{2} c^{7} d^{2} - 4 \, a^{3} b c^{6} d^{3} + a^{4} c^{5} d^{4}} - \frac{a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} - 6 \,{\left (b^{5} c^{4} d^{2} - a b^{4} c^{3} d^{3} - a^{2} b^{3} c^{2} d^{4} + 4 \, a^{3} b^{2} c d^{5} - 2 \, a^{4} b d^{6}\right )} x^{4} - 3 \,{\left (4 \, b^{5} c^{5} d - 3 \, a b^{4} c^{4} d^{2} - 5 \, a^{2} b^{3} c^{3} d^{3} + 10 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - 4 \, a^{5} d^{6}\right )} x^{3} -{\left (6 \, b^{5} c^{6} - 13 \, a^{2} b^{3} c^{4} d^{2} - a^{3} b^{2} c^{3} d^{3} + 32 \, a^{4} b c^{2} d^{4} - 18 \, a^{5} c d^{5}\right )} x^{2} -{\left (3 \, a b^{4} c^{6} - 5 \, a^{2} b^{3} c^{5} d - 3 \, a^{3} b^{2} c^{4} d^{2} + 9 \, a^{4} b c^{3} d^{3} - 4 \, a^{5} c^{2} d^{4}\right )} x}{2 \,{\left ({\left (a^{3} b^{4} c^{7} d^{2} - 3 \, a^{4} b^{3} c^{6} d^{3} + 3 \, a^{5} b^{2} c^{5} d^{4} - a^{6} b c^{4} d^{5}\right )} x^{5} +{\left (2 \, a^{3} b^{4} c^{8} d - 5 \, a^{4} b^{3} c^{7} d^{2} + 3 \, a^{5} b^{2} c^{6} d^{3} + a^{6} b c^{5} d^{4} - a^{7} c^{4} d^{5}\right )} x^{4} +{\left (a^{3} b^{4} c^{9} - a^{4} b^{3} c^{8} d - 3 \, a^{5} b^{2} c^{7} d^{2} + 5 \, a^{6} b c^{6} d^{3} - 2 \, a^{7} c^{5} d^{4}\right )} x^{3} +{\left (a^{4} b^{3} c^{9} - 3 \, a^{5} b^{2} c^{8} d + 3 \, a^{6} b c^{7} d^{2} - a^{7} c^{6} d^{3}\right )} x^{2}\right )}} + \frac{3 \,{\left (b^{2} c^{2} + 2 \, a b c d + 2 \, a^{2} d^{2}\right )} \log \left (x\right )}{a^{4} c^{5}} \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 = 1.23514, size = 1168, normalized size = 5.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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