Optimal. Leaf size=103 \[ -\frac{c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3 x}+\frac{c^2 (b c-3 a d)}{2 a^2 x^2}-\frac{\log (x) (b c-a d)^3}{a^4}+\frac{(b c-a d)^3 \log (a+b x)}{a^4}-\frac{c^3}{3 a x^3} \]
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Rubi [A] time = 0.0654395, antiderivative size = 103, 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} \[ -\frac{c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3 x}+\frac{c^2 (b c-3 a d)}{2 a^2 x^2}-\frac{\log (x) (b c-a d)^3}{a^4}+\frac{(b c-a d)^3 \log (a+b x)}{a^4}-\frac{c^3}{3 a x^3} \]
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)} \, dx &=\int \left (\frac{c^3}{a x^4}+\frac{c^2 (-b c+3 a d)}{a^2 x^3}+\frac{c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x^2}+\frac{(-b c+a d)^3}{a^4 x}-\frac{b (-b c+a d)^3}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac{c^3}{3 a x^3}+\frac{c^2 (b c-3 a d)}{2 a^2 x^2}-\frac{c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x}-\frac{(b c-a d)^3 \log (x)}{a^4}+\frac{(b c-a d)^3 \log (a+b x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0626361, size = 93, normalized size = 0.9 \[ -\frac{\frac{a c \left (a^2 \left (2 c^2+9 c d x+18 d^2 x^2\right )-3 a b c x (c+6 d x)+6 b^2 c^2 x^2\right )}{x^3}+6 \log (x) (b c-a d)^3-6 (b c-a d)^3 \log (a+b x)}{6 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 188, normalized size = 1.8 \begin{align*} -{\frac{{c}^{3}}{3\,a{x}^{3}}}+{\frac{\ln \left ( x \right ){d}^{3}}{a}}-3\,{\frac{\ln \left ( x \right ) cb{d}^{2}}{{a}^{2}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}{c}^{2}d}{{a}^{3}}}-{\frac{\ln \left ( x \right ){b}^{3}{c}^{3}}{{a}^{4}}}-3\,{\frac{c{d}^{2}}{ax}}+3\,{\frac{{c}^{2}bd}{{a}^{2}x}}-{\frac{{c}^{3}{b}^{2}}{{a}^{3}x}}-{\frac{3\,{c}^{2}d}{2\,a{x}^{2}}}+{\frac{{c}^{3}b}{2\,{a}^{2}{x}^{2}}}-{\frac{\ln \left ( bx+a \right ){d}^{3}}{a}}+3\,{\frac{\ln \left ( bx+a \right ) cb{d}^{2}}{{a}^{2}}}-3\,{\frac{\ln \left ( bx+a \right ){b}^{2}{c}^{2}d}{{a}^{3}}}+{\frac{\ln \left ( bx+a \right ){b}^{3}{c}^{3}}{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03198, size = 211, normalized size = 2.05 \begin{align*} \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{4}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x\right )}{a^{4}} - \frac{2 \, a^{2} c^{3} + 6 \,{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{2} - 3 \,{\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x}{6 \, a^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51539, size = 338, normalized size = 3.28 \begin{align*} -\frac{2 \, a^{3} c^{3} - 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (b x + a\right ) + 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (x\right ) + 6 \,{\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 3 \,{\left (a^{2} b c^{3} - 3 \, a^{3} c^{2} d\right )} x}{6 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.8072, size = 289, normalized size = 2.81 \begin{align*} - \frac{2 a^{2} c^{3} + x^{2} \left (18 a^{2} c d^{2} - 18 a b c^{2} d + 6 b^{2} c^{3}\right ) + x \left (9 a^{2} c^{2} d - 3 a b c^{3}\right )}{6 a^{3} x^{3}} + \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3} - a \left (a d - b c\right )^{3}}{2 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 6 a b^{3} c^{2} d - 2 b^{4} c^{3}} \right )}}{a^{4}} - \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3} + a \left (a d - b c\right )^{3}}{2 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 6 a b^{3} c^{2} d - 2 b^{4} c^{3}} \right )}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14995, size = 228, normalized size = 2.21 \begin{align*} -\frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{2 \, a^{3} c^{3} + 6 \,{\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 3 \,{\left (a^{2} b c^{3} - 3 \, a^{3} c^{2} d\right )} x}{6 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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