Optimal. Leaf size=164 \[ \frac{c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}-\frac{a^4}{b^2 (a+b x) (b c-a d)^3}-\frac{a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac{2 c^3 (b c-2 a d)}{d^3 (c+d x) (b c-a d)^3}-\frac{c^4}{2 d^3 (c+d x)^2 (b c-a d)^2} \]
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Rubi [A] time = 0.178563, antiderivative size = 164, 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} \[ \frac{c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}-\frac{a^4}{b^2 (a+b x) (b c-a d)^3}-\frac{a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac{2 c^3 (b c-2 a d)}{d^3 (c+d x) (b c-a d)^3}-\frac{c^4}{2 d^3 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int \frac{x^4}{(a+b x)^2 (c+d x)^3} \, dx &=\int \left (\frac{a^4}{b (b c-a d)^3 (a+b x)^2}+\frac{a^3 (-4 b c+a d)}{b (b c-a d)^4 (a+b x)}+\frac{c^4}{d^2 (-b c+a d)^2 (c+d x)^3}+\frac{2 c^3 (b c-2 a d)}{d^2 (-b c+a d)^3 (c+d x)^2}+\frac{c^2 \left (b^2 c^2-4 a b c d+6 a^2 d^2\right )}{d^2 (-b c+a d)^4 (c+d x)}\right ) \, dx\\ &=-\frac{a^4}{b^2 (b c-a d)^3 (a+b x)}-\frac{c^4}{2 d^3 (b c-a d)^2 (c+d x)^2}+\frac{2 c^3 (b c-2 a d)}{d^3 (b c-a d)^3 (c+d x)}-\frac{a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac{c^2 \left (b^2 c^2-4 a b c d+6 a^2 d^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}\\ \end{align*}
Mathematica [A] time = 0.22432, size = 162, normalized size = 0.99 \[ \frac{c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}-\frac{a^4}{b^2 (a+b x) (b c-a d)^3}+\frac{a^3 (a d-4 b c) \log (a+b x)}{b^2 (b c-a d)^4}-\frac{2 c^3 (b c-2 a d)}{d^3 (c+d x) (a d-b c)^3}-\frac{c^4}{2 d^3 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 231, normalized size = 1.4 \begin{align*} -{\frac{{c}^{4}}{2\,{d}^{3} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}+6\,{\frac{{c}^{2}\ln \left ( dx+c \right ){a}^{2}}{ \left ( ad-bc \right ) ^{4}d}}-4\,{\frac{{c}^{3}\ln \left ( dx+c \right ) ab}{ \left ( ad-bc \right ) ^{4}{d}^{2}}}+{\frac{{c}^{4}\ln \left ( dx+c \right ){b}^{2}}{ \left ( ad-bc \right ) ^{4}{d}^{3}}}+4\,{\frac{{c}^{3}a}{{d}^{2} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-2\,{\frac{{c}^{4}b}{{d}^{3} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}+{\frac{{a}^{4}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{4}{b}^{2}}}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{4}b}}+{\frac{{a}^{4}}{{b}^{2} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.48842, size = 699, normalized size = 4.26 \begin{align*} -\frac{{\left (4 \, a^{3} b c - a^{4} d\right )} \log \left (b x + a\right )}{b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}} + \frac{{\left (b^{2} c^{4} - 4 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}} + \frac{3 \, a b^{3} c^{5} - 7 \, a^{2} b^{2} c^{4} d - 2 \, a^{4} c^{2} d^{3} + 2 \,{\left (2 \, b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} - a^{4} d^{5}\right )} x^{2} +{\left (3 \, b^{4} c^{5} - 3 \, a b^{3} c^{4} d - 8 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{4} c d^{4}\right )} x}{2 \,{\left (a b^{5} c^{5} d^{3} - 3 \, a^{2} b^{4} c^{4} d^{4} + 3 \, a^{3} b^{3} c^{3} d^{5} - a^{4} b^{2} c^{2} d^{6} +{\left (b^{6} c^{3} d^{5} - 3 \, a b^{5} c^{2} d^{6} + 3 \, a^{2} b^{4} c d^{7} - a^{3} b^{3} d^{8}\right )} x^{3} +{\left (2 \, b^{6} c^{4} d^{4} - 5 \, a b^{5} c^{3} d^{5} + 3 \, a^{2} b^{4} c^{2} d^{6} + a^{3} b^{3} c d^{7} - a^{4} b^{2} d^{8}\right )} x^{2} +{\left (b^{6} c^{5} d^{3} - a b^{5} c^{4} d^{4} - 3 \, a^{2} b^{4} c^{3} d^{5} + 5 \, a^{3} b^{3} c^{2} d^{6} - 2 \, a^{4} b^{2} c d^{7}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.46779, size = 1562, normalized size = 9.52 \begin{align*} \frac{3 \, a b^{4} c^{6} - 10 \, a^{2} b^{3} c^{5} d + 7 \, a^{3} b^{2} c^{4} d^{2} - 2 \, a^{4} b c^{3} d^{3} + 2 \, a^{5} c^{2} d^{4} + 2 \,{\left (2 \, b^{5} c^{5} d - 6 \, a b^{4} c^{4} d^{2} + 4 \, a^{2} b^{3} c^{3} d^{3} - a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} +{\left (3 \, b^{5} c^{6} - 6 \, a b^{4} c^{5} d - 5 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 4 \, a^{4} b c^{2} d^{4} + 4 \, a^{5} c d^{5}\right )} x - 2 \,{\left (4 \, a^{4} b c^{3} d^{3} - a^{5} c^{2} d^{4} +{\left (4 \, a^{3} b^{2} c d^{5} - a^{4} b d^{6}\right )} x^{3} +{\left (8 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - a^{5} d^{6}\right )} x^{2} +{\left (4 \, a^{3} b^{2} c^{3} d^{3} + 7 \, a^{4} b c^{2} d^{4} - 2 \, a^{5} c d^{5}\right )} x\right )} \log \left (b x + a\right ) + 2 \,{\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} +{\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4}\right )} x^{3} +{\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} + 6 \, a^{3} b^{2} c^{2} d^{4}\right )} x^{2} +{\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 12 \, a^{3} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{6} c^{6} d^{3} - 4 \, a^{2} b^{5} c^{5} d^{4} + 6 \, a^{3} b^{4} c^{4} d^{5} - 4 \, a^{4} b^{3} c^{3} d^{6} + a^{5} b^{2} c^{2} d^{7} +{\left (b^{7} c^{4} d^{5} - 4 \, a b^{6} c^{3} d^{6} + 6 \, a^{2} b^{5} c^{2} d^{7} - 4 \, a^{3} b^{4} c d^{8} + a^{4} b^{3} d^{9}\right )} x^{3} +{\left (2 \, b^{7} c^{5} d^{4} - 7 \, a b^{6} c^{4} d^{5} + 8 \, a^{2} b^{5} c^{3} d^{6} - 2 \, a^{3} b^{4} c^{2} d^{7} - 2 \, a^{4} b^{3} c d^{8} + a^{5} b^{2} d^{9}\right )} x^{2} +{\left (b^{7} c^{6} d^{3} - 2 \, a b^{6} c^{5} d^{4} - 2 \, a^{2} b^{5} c^{4} d^{5} + 8 \, a^{3} b^{4} c^{3} d^{6} - 7 \, a^{4} b^{3} c^{2} d^{7} + 2 \, a^{5} b^{2} c d^{8}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.22527, size = 1083, normalized size = 6.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.263, size = 419, normalized size = 2.55 \begin{align*} -\frac{a^{4} b^{3}}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )}{\left (b x + a\right )}} + \frac{{\left (b^{3} c^{4} - 4 \, a b^{2} c^{3} d + 6 \, a^{2} b c^{2} d^{2}\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{3} - 4 \, a b^{4} c^{3} d^{4} + 6 \, a^{2} b^{3} c^{2} d^{5} - 4 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}} - \frac{\log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{2} d^{3}} - \frac{3 \, b^{2} c^{4} d^{2} - 8 \, a b c^{3} d^{3} + \frac{2 \,{\left (b^{4} c^{5} d - 5 \, a b^{3} c^{4} d^{2} + 4 \, a^{2} b^{2} c^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{2 \,{\left (b c - a d\right )}^{4}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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