Optimal. Leaf size=103 \[ \frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac{a (b c-a d)^3}{b^5 (a+b x)}+\frac{3 d x (b c-a d)^2}{b^4}+\frac{(b c-4 a d) (b c-a d)^2 \log (a+b x)}{b^5}+\frac{d^3 x^3}{3 b^2} \]
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Rubi [A] time = 0.1028, antiderivative size = 103, 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} \[ \frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac{a (b c-a d)^3}{b^5 (a+b x)}+\frac{3 d x (b c-a d)^2}{b^4}+\frac{(b c-4 a d) (b c-a d)^2 \log (a+b x)}{b^5}+\frac{d^3 x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x (c+d x)^3}{(a+b x)^2} \, dx &=\int \left (\frac{3 d (b c-a d)^2}{b^4}+\frac{d^2 (3 b c-2 a d) x}{b^3}+\frac{d^3 x^2}{b^2}+\frac{a (-b c+a d)^3}{b^4 (a+b x)^2}+\frac{(b c-4 a d) (b c-a d)^2}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{3 d (b c-a d)^2 x}{b^4}+\frac{d^2 (3 b c-2 a d) x^2}{2 b^3}+\frac{d^3 x^3}{3 b^2}+\frac{a (b c-a d)^3}{b^5 (a+b x)}+\frac{(b c-4 a d) (b c-a d)^2 \log (a+b x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0711383, size = 100, normalized size = 0.97 \[ \frac{3 b^2 d^2 x^2 (3 b c-2 a d)-\frac{6 a (a d-b c)^3}{a+b x}+18 b d x (b c-a d)^2+6 (b c-4 a d) (b c-a d)^2 \log (a+b x)+2 b^3 d^3 x^3}{6 b^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 205, normalized size = 2. \begin{align*}{\frac{{d}^{3}{x}^{3}}{3\,{b}^{2}}}-{\frac{{d}^{3}{x}^{2}a}{{b}^{3}}}+{\frac{3\,{d}^{2}{x}^{2}c}{2\,{b}^{2}}}+3\,{\frac{{a}^{2}{d}^{3}x}{{b}^{4}}}-6\,{\frac{ac{d}^{2}x}{{b}^{3}}}+3\,{\frac{{c}^{2}dx}{{b}^{2}}}-{\frac{{a}^{4}{d}^{3}}{{b}^{5} \left ( bx+a \right ) }}+3\,{\frac{{a}^{3}c{d}^{2}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{{a}^{2}{c}^{2}d}{{b}^{3} \left ( bx+a \right ) }}+{\frac{a{c}^{3}}{{b}^{2} \left ( bx+a \right ) }}-4\,{\frac{\ln \left ( bx+a \right ){a}^{3}{d}^{3}}{{b}^{5}}}+9\,{\frac{{a}^{2}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{4}}}-6\,{\frac{\ln \left ( bx+a \right ) a{c}^{2}d}{{b}^{3}}}+{\frac{\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 = 1.32664, size = 224, normalized size = 2.17 \begin{align*} \frac{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}}{b^{6} x + a b^{5}} + \frac{2 \, b^{2} d^{3} x^{3} + 3 \,{\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{2} + 18 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x}{6 \, b^{4}} + \frac{{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} 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 [B] time = 2.1975, size = 506, normalized size = 4.91 \begin{align*} \frac{2 \, b^{4} d^{3} x^{4} + 6 \, a b^{3} c^{3} - 18 \, a^{2} b^{2} c^{2} d + 18 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} +{\left (9 \, b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{3} + 3 \,{\left (6 \, b^{4} c^{2} d - 9 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x^{2} + 18 \,{\left (a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x + 6 \,{\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{6} x + a b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.02518, size = 146, normalized size = 1.42 \begin{align*} - \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 b^{5} + b^{6} x} + \frac{d^{3} x^{3}}{3 b^{2}} - \frac{x^{2} \left (2 a d^{3} - 3 b c d^{2}\right )}{2 b^{3}} + \frac{x \left (3 a^{2} d^{3} - 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{4}} - \frac{\left (a d - b c\right )^{2} \left (4 a d - b c\right ) \log{\left (a + b x \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16212, size = 312, normalized size = 3.03 \begin{align*} \frac{\frac{{\left (2 \, d^{3} + \frac{3 \,{\left (3 \, b^{2} c d^{2} - 4 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac{18 \,{\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )}{\left (b x + a\right )}^{3}}{b^{4}} - \frac{6 \,{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} + \frac{6 \,{\left (\frac{a b^{6} c^{3}}{b x + a} - \frac{3 \, a^{2} b^{5} c^{2} d}{b x + a} + \frac{3 \, a^{3} b^{4} c d^{2}}{b x + a} - \frac{a^{4} b^{3} d^{3}}{b x + a}\right )}}{b^{7}}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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