Optimal. Leaf size=132 \[ \frac{c^2 (2 b c-3 a d)}{2 a^3 x^2}-\frac{3 c (b c-a d)^2}{a^4 x}-\frac{(b c-a d)^3}{a^4 (a+b x)}-\frac{\log (x) (b c-a d)^2 (4 b c-a d)}{a^5}+\frac{(b c-a d)^2 (4 b c-a d) \log (a+b x)}{a^5}-\frac{c^3}{3 a^2 x^3} \]
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Rubi [A] time = 0.105787, antiderivative size = 132, 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} \[ \frac{c^2 (2 b c-3 a d)}{2 a^3 x^2}-\frac{3 c (b c-a d)^2}{a^4 x}-\frac{(b c-a d)^3}{a^4 (a+b x)}-\frac{\log (x) (b c-a d)^2 (4 b c-a d)}{a^5}+\frac{(b c-a d)^2 (4 b c-a d) \log (a+b x)}{a^5}-\frac{c^3}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int \frac{(c+d x)^3}{x^4 (a+b x)^2} \, dx &=\int \left (\frac{c^3}{a^2 x^4}+\frac{c^2 (-2 b c+3 a d)}{a^3 x^3}+\frac{3 c (-b c+a d)^2}{a^4 x^2}+\frac{(-4 b c+a d) (-b c+a d)^2}{a^5 x}-\frac{b (-b c+a d)^3}{a^4 (a+b x)^2}-\frac{b (-4 b c+a d) (-b c+a d)^2}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{c^3}{3 a^2 x^3}+\frac{c^2 (2 b c-3 a d)}{2 a^3 x^2}-\frac{3 c (b c-a d)^2}{a^4 x}-\frac{(b c-a d)^3}{a^4 (a+b x)}-\frac{(b c-a d)^2 (4 b c-a d) \log (x)}{a^5}+\frac{(b c-a d)^2 (4 b c-a d) \log (a+b x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.064494, size = 126, normalized size = 0.95 \[ -\frac{\frac{3 a^2 c^2 (3 a d-2 b c)}{x^2}+\frac{2 a^3 c^3}{x^3}+\frac{18 a c (b c-a d)^2}{x}-\frac{6 a (a d-b c)^3}{a+b x}+6 \log (x) (b c-a d)^2 (4 b c-a d)-6 (b c-a d)^2 (4 b c-a d) \log (a+b x)}{6 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 256, normalized size = 1.9 \begin{align*} -{\frac{{c}^{3}}{3\,{a}^{2}{x}^{3}}}+{\frac{\ln \left ( x \right ){d}^{3}}{{a}^{2}}}-6\,{\frac{\ln \left ( x \right ) cb{d}^{2}}{{a}^{3}}}+9\,{\frac{\ln \left ( x \right ){b}^{2}{c}^{2}d}{{a}^{4}}}-4\,{\frac{\ln \left ( x \right ){b}^{3}{c}^{3}}{{a}^{5}}}-{\frac{3\,{c}^{2}d}{2\,{a}^{2}{x}^{2}}}+{\frac{{c}^{3}b}{{a}^{3}{x}^{2}}}-3\,{\frac{c{d}^{2}}{{a}^{2}x}}+6\,{\frac{{c}^{2}db}{{a}^{3}x}}-3\,{\frac{{c}^{3}{b}^{2}}{{a}^{4}x}}-{\frac{\ln \left ( bx+a \right ){d}^{3}}{{a}^{2}}}+6\,{\frac{\ln \left ( bx+a \right ) cb{d}^{2}}{{a}^{3}}}-9\,{\frac{\ln \left ( bx+a \right ){b}^{2}{c}^{2}d}{{a}^{4}}}+4\,{\frac{\ln \left ( bx+a \right ){b}^{3}{c}^{3}}{{a}^{5}}}+{\frac{{d}^{3}}{a \left ( bx+a \right ) }}-3\,{\frac{c{d}^{2}b}{{a}^{2} \left ( bx+a \right ) }}+3\,{\frac{{c}^{2}d{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}-{\frac{{b}^{3}{c}^{3}}{{a}^{4} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08011, size = 296, normalized size = 2.24 \begin{align*} -\frac{2 \, a^{3} c^{3} + 6 \,{\left (4 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} + 3 \,{\left (4 \, a b^{2} c^{3} - 9 \, a^{2} b c^{2} d + 6 \, a^{3} c d^{2}\right )} x^{2} -{\left (4 \, a^{2} b c^{3} - 9 \, a^{3} c^{2} d\right )} x}{6 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac{{\left (4 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{5}} - \frac{{\left (4 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.28526, size = 643, normalized size = 4.87 \begin{align*} -\frac{2 \, a^{4} c^{3} + 6 \,{\left (4 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3} + 3 \,{\left (4 \, a^{2} b^{2} c^{3} - 9 \, a^{3} b c^{2} d + 6 \, a^{4} c d^{2}\right )} x^{2} -{\left (4 \, a^{3} b c^{3} - 9 \, a^{4} c^{2} d\right )} x - 6 \,{\left ({\left (4 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} +{\left (4 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) + 6 \,{\left ({\left (4 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} +{\left (4 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.12692, size = 386, normalized size = 2.92 \begin{align*} \frac{- 2 a^{3} c^{3} + x^{3} \left (6 a^{3} d^{3} - 36 a^{2} b c d^{2} + 54 a b^{2} c^{2} d - 24 b^{3} c^{3}\right ) + x^{2} \left (- 18 a^{3} c d^{2} + 27 a^{2} b c^{2} d - 12 a b^{2} c^{3}\right ) + x \left (- 9 a^{3} c^{2} d + 4 a^{2} b c^{3}\right )}{6 a^{5} x^{3} + 6 a^{4} b x^{4}} + \frac{\left (a d - 4 b c\right ) \left (a d - b c\right )^{2} \log{\left (x + \frac{a^{4} d^{3} - 6 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - 4 a b^{3} c^{3} - a \left (a d - 4 b c\right ) \left (a d - b c\right )^{2}}{2 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 18 a b^{3} c^{2} d - 8 b^{4} c^{3}} \right )}}{a^{5}} - \frac{\left (a d - 4 b c\right ) \left (a d - b c\right )^{2} \log{\left (x + \frac{a^{4} d^{3} - 6 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - 4 a b^{3} c^{3} + a \left (a d - 4 b c\right ) \left (a d - b c\right )^{2}}{2 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 18 a b^{3} c^{2} d - 8 b^{4} c^{3}} \right )}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14967, size = 378, normalized size = 2.86 \begin{align*} -\frac{{\left (4 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{5} b} - \frac{\frac{b^{7} c^{3}}{b x + a} - \frac{3 \, a b^{6} c^{2} d}{b x + a} + \frac{3 \, a^{2} b^{5} c d^{2}}{b x + a} - \frac{a^{3} b^{4} d^{3}}{b x + a}}{a^{4} b^{4}} + \frac{26 \, b^{3} c^{3} - 45 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - \frac{3 \,{\left (20 \, a b^{4} c^{3} - 33 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2}\right )}}{{\left (b x + a\right )} b} + \frac{18 \,{\left (2 \, a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + a^{4} b^{3} c d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{6 \, a^{5}{\left (\frac{a}{b x + a} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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