Optimal. Leaf size=106 \[ \frac{d x^2 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{2 b^3}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{x (b c-a d)^3}{b^4}-\frac{a (b c-a d)^3 \log (a+b x)}{b^5}+\frac{d^3 x^4}{4 b} \]
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Rubi [A] time = 0.0756914, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{d x^2 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{2 b^3}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{x (b c-a d)^3}{b^4}-\frac{a (b c-a d)^3 \log (a+b x)}{b^5}+\frac{d^3 x^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x (c+d x)^3}{a+b x} \, dx &=\int \left (\frac{(b c-a d)^3}{b^4}+\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac{d^2 (3 b c-a d) x^2}{b^2}+\frac{d^3 x^3}{b}+\frac{a (-b c+a d)^3}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{(b c-a d)^3 x}{b^4}+\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^2}{2 b^3}+\frac{d^2 (3 b c-a d) x^3}{3 b^2}+\frac{d^3 x^4}{4 b}-\frac{a (b c-a d)^3 \log (a+b x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0398408, size = 115, normalized size = 1.08 \[ \frac{b x \left (6 a^2 b d^2 (6 c+d x)-12 a^3 d^3-2 a b^2 d \left (18 c^2+9 c d x+2 d^2 x^2\right )+3 b^3 \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )\right )+12 a (a d-b c)^3 \log (a+b x)}{12 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 186, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}{x}^{4}}{4\,b}}-{\frac{{x}^{3}a{d}^{3}}{3\,{b}^{2}}}+{\frac{c{x}^{3}{d}^{2}}{b}}+{\frac{{a}^{2}{x}^{2}{d}^{3}}{2\,{b}^{3}}}-{\frac{3\,a{x}^{2}c{d}^{2}}{2\,{b}^{2}}}+{\frac{3\,{x}^{2}{c}^{2}d}{2\,b}}-{\frac{{a}^{3}{d}^{3}x}{{b}^{4}}}+3\,{\frac{{a}^{2}c{d}^{2}x}{{b}^{3}}}-3\,{\frac{a{c}^{2}dx}{{b}^{2}}}+{\frac{{c}^{3}x}{b}}+{\frac{{a}^{4}\ln \left ( bx+a \right ){d}^{3}}{{b}^{5}}}-3\,{\frac{{a}^{3}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{4}}}+3\,{\frac{{a}^{2}\ln \left ( bx+a \right ){c}^{2}d}{{b}^{3}}}-{\frac{a\ln \left ( bx+a \right ){c}^{3}}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01012, size = 221, normalized size = 2.08 \begin{align*} \frac{3 \, b^{3} d^{3} x^{4} + 4 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} + 6 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 12 \,{\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}{12 \, b^{4}} - \frac{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \log \left (b x + a\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31471, size = 336, normalized size = 3.17 \begin{align*} \frac{3 \, b^{4} d^{3} x^{4} + 4 \,{\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{3} + 6 \,{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{2} + 12 \,{\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 )} x - 12 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.640157, size = 129, normalized size = 1.22 \begin{align*} \frac{a \left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{b^{5}} + \frac{d^{3} x^{4}}{4 b} - \frac{x^{3} \left (a d^{3} - 3 b c d^{2}\right )}{3 b^{2}} + \frac{x^{2} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{2 b^{3}} - \frac{x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22078, size = 230, normalized size = 2.17 \begin{align*} \frac{3 \, b^{3} d^{3} x^{4} + 12 \, b^{3} c d^{2} x^{3} - 4 \, a b^{2} d^{3} x^{3} + 18 \, b^{3} c^{2} d x^{2} - 18 \, a b^{2} c d^{2} x^{2} + 6 \, a^{2} b d^{3} x^{2} + 12 \, b^{3} c^{3} x - 36 \, a b^{2} c^{2} d x + 36 \, a^{2} b c d^{2} x - 12 \, a^{3} d^{3} x}{12 \, b^{4}} - \frac{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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