Optimal. Leaf size=136 \[ -\frac{a^2 (b c-a d)^3}{b^6 (a+b x)}+\frac{d^2 x^3 (3 b c-2 a d)}{3 b^3}+\frac{3 d x^2 (b c-a d)^2}{2 b^4}+\frac{x (b c-4 a d) (b c-a d)^2}{b^5}-\frac{a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6}+\frac{d^3 x^4}{4 b^2} \]
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Rubi [A] time = 0.124279, antiderivative size = 136, 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} \[ -\frac{a^2 (b c-a d)^3}{b^6 (a+b x)}+\frac{d^2 x^3 (3 b c-2 a d)}{3 b^3}+\frac{3 d x^2 (b c-a d)^2}{2 b^4}+\frac{x (b c-4 a d) (b c-a d)^2}{b^5}-\frac{a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6}+\frac{d^3 x^4}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int \frac{x^2 (c+d x)^3}{(a+b x)^2} \, dx &=\int \left (\frac{(b c-4 a d) (b c-a d)^2}{b^5}+\frac{3 d (b c-a d)^2 x}{b^4}+\frac{d^2 (3 b c-2 a d) x^2}{b^3}+\frac{d^3 x^3}{b^2}-\frac{a^2 (-b c+a d)^3}{b^5 (a+b x)^2}+\frac{a (-b c+a d)^2 (-2 b c+5 a d)}{b^5 (a+b x)}\right ) \, dx\\ &=\frac{(b c-4 a d) (b c-a d)^2 x}{b^5}+\frac{3 d (b c-a d)^2 x^2}{2 b^4}+\frac{d^2 (3 b c-2 a d) x^3}{3 b^3}+\frac{d^3 x^4}{4 b^2}-\frac{a^2 (b c-a d)^3}{b^6 (a+b x)}-\frac{a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0497328, size = 130, normalized size = 0.96 \[ \frac{\frac{12 a^2 (a d-b c)^3}{a+b x}+4 b^3 d^2 x^3 (3 b c-2 a d)+18 b^2 d x^2 (b c-a d)^2+12 b x (b c-4 a d) (b c-a d)^2+12 a (b c-a d)^2 (5 a d-2 b c) \log (a+b x)+3 b^4 d^3 x^4}{12 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 260, normalized size = 1.9 \begin{align*}{\frac{{d}^{3}{x}^{4}}{4\,{b}^{2}}}-{\frac{2\,{x}^{3}a{d}^{3}}{3\,{b}^{3}}}+{\frac{c{x}^{3}{d}^{2}}{{b}^{2}}}+{\frac{3\,{a}^{2}{x}^{2}{d}^{3}}{2\,{b}^{4}}}-3\,{\frac{a{x}^{2}c{d}^{2}}{{b}^{3}}}+{\frac{3\,{x}^{2}{c}^{2}d}{2\,{b}^{2}}}-4\,{\frac{{a}^{3}{d}^{3}x}{{b}^{5}}}+9\,{\frac{{a}^{2}c{d}^{2}x}{{b}^{4}}}-6\,{\frac{a{c}^{2}dx}{{b}^{3}}}+{\frac{{c}^{3}x}{{b}^{2}}}+{\frac{{a}^{5}{d}^{3}}{{b}^{6} \left ( bx+a \right ) }}-3\,{\frac{{a}^{4}c{d}^{2}}{{b}^{5} \left ( bx+a \right ) }}+3\,{\frac{{a}^{3}{c}^{2}d}{{b}^{4} \left ( bx+a \right ) }}-{\frac{{a}^{2}{c}^{3}}{{b}^{3} \left ( bx+a \right ) }}+5\,{\frac{{a}^{4}\ln \left ( bx+a \right ){d}^{3}}{{b}^{6}}}-12\,{\frac{{a}^{3}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{5}}}+9\,{\frac{{a}^{2}\ln \left ( bx+a \right ){c}^{2}d}{{b}^{4}}}-2\,{\frac{a\ln \left ( bx+a \right ){c}^{3}}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02642, size = 297, normalized size = 2.18 \begin{align*} -\frac{a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}}{b^{7} x + a b^{6}} + \frac{3 \, b^{3} d^{3} x^{4} + 4 \,{\left (3 \, b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{3} + 18 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 12 \,{\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 )} x}{12 \, b^{5}} - \frac{{\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.23935, size = 653, normalized size = 4.8 \begin{align*} \frac{3 \, b^{5} d^{3} x^{5} - 12 \, a^{2} b^{3} c^{3} + 36 \, a^{3} b^{2} c^{2} d - 36 \, a^{4} b c d^{2} + 12 \, a^{5} d^{3} +{\left (12 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{4} + 2 \,{\left (9 \, b^{5} c^{2} d - 12 \, a b^{4} c d^{2} + 5 \, a^{2} b^{3} d^{3}\right )} x^{3} + 6 \,{\left (2 \, b^{5} c^{3} - 9 \, a b^{4} c^{2} d + 12 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{2} + 12 \,{\left (a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 4 \, a^{4} b d^{3}\right )} x - 12 \,{\left (2 \, a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 12 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} +{\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{7} x + a b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.13529, size = 199, normalized size = 1.46 \begin{align*} \frac{a \left (a d - b c\right )^{2} \left (5 a d - 2 b c\right ) \log{\left (a + b x \right )}}{b^{6}} + \frac{a^{5} d^{3} - 3 a^{4} b c d^{2} + 3 a^{3} b^{2} c^{2} d - a^{2} b^{3} c^{3}}{a b^{6} + b^{7} x} + \frac{d^{3} x^{4}}{4 b^{2}} - \frac{x^{3} \left (2 a d^{3} - 3 b c d^{2}\right )}{3 b^{3}} + \frac{x^{2} \left (3 a^{2} d^{3} - 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{2 b^{4}} - \frac{x \left (4 a^{3} d^{3} - 9 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - b^{3} c^{3}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20933, size = 386, normalized size = 2.84 \begin{align*} \frac{{\left (3 \, d^{3} + \frac{4 \,{\left (3 \, b^{2} c d^{2} - 5 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac{6 \,{\left (3 \, b^{4} c^{2} d - 12 \, a b^{3} c d^{2} + 10 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{12 \,{\left (b^{6} c^{3} - 9 \, a b^{5} c^{2} d + 18 \, a^{2} b^{4} c d^{2} - 10 \, a^{3} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )}{\left (b x + a\right )}^{4}}{12 \, b^{6}} + \frac{{\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{6}} - \frac{\frac{a^{2} b^{7} c^{3}}{b x + a} - \frac{3 \, a^{3} b^{6} c^{2} d}{b x + a} + \frac{3 \, a^{4} b^{5} c d^{2}}{b x + a} - \frac{a^{5} b^{4} d^{3}}{b x + a}}{b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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