Optimal. Leaf size=156 \[ \frac{\left (10 a^2 d^2-8 a b c d+b^2 c^2\right ) (b c-a d) \log (a+b x)}{b^6}-\frac{a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac{3 d^2 x^2 (b c-a d)}{2 b^4}+\frac{a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac{3 d x (b c-2 a d) (b c-a d)}{b^5}+\frac{d^3 x^3}{3 b^3} \]
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Rubi [A] time = 0.15293, antiderivative size = 156, 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} \[ \frac{\left (10 a^2 d^2-8 a b c d+b^2 c^2\right ) (b c-a d) \log (a+b x)}{b^6}-\frac{a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac{3 d^2 x^2 (b c-a d)}{2 b^4}+\frac{a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac{3 d x (b c-2 a d) (b c-a d)}{b^5}+\frac{d^3 x^3}{3 b^3} \]
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)^3} \, dx &=\int \left (\frac{3 d (b c-2 a d) (b c-a d)}{b^5}+\frac{3 d^2 (b c-a d) x}{b^4}+\frac{d^3 x^2}{b^3}-\frac{a^2 (-b c+a d)^3}{b^5 (a+b x)^3}+\frac{a (-b c+a d)^2 (-2 b c+5 a d)}{b^5 (a+b x)^2}+\frac{(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right )}{b^5 (a+b x)}\right ) \, dx\\ &=\frac{3 d (b c-2 a d) (b c-a d) x}{b^5}+\frac{3 d^2 (b c-a d) x^2}{2 b^4}+\frac{d^3 x^3}{3 b^3}-\frac{a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac{a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac{(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0843345, size = 160, normalized size = 1.03 \[ \frac{18 b d x \left (2 a^2 d^2-3 a b c d+b^2 c^2\right )+6 \left (18 a^2 b c d^2-10 a^3 d^3-9 a b^2 c^2 d+b^3 c^3\right ) \log (a+b x)+\frac{3 a^2 (a d-b c)^3}{(a+b x)^2}+9 b^2 d^2 x^2 (b c-a d)-\frac{6 a (b c-a d)^2 (5 a d-2 b c)}{a+b x}+2 b^3 d^3 x^3}{6 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 280, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}{x}^{3}}{3\,{b}^{3}}}-{\frac{3\,{d}^{3}{x}^{2}a}{2\,{b}^{4}}}+{\frac{3\,{d}^{2}{x}^{2}c}{2\,{b}^{3}}}+6\,{\frac{{a}^{2}{d}^{3}x}{{b}^{5}}}-9\,{\frac{ac{d}^{2}x}{{b}^{4}}}+3\,{\frac{{c}^{2}dx}{{b}^{3}}}-5\,{\frac{{a}^{4}{d}^{3}}{{b}^{6} \left ( bx+a \right ) }}+12\,{\frac{{a}^{3}c{d}^{2}}{{b}^{5} \left ( bx+a \right ) }}-9\,{\frac{{a}^{2}{c}^{2}d}{{b}^{4} \left ( bx+a \right ) }}+2\,{\frac{a{c}^{3}}{{b}^{3} \left ( bx+a \right ) }}+{\frac{{a}^{5}{d}^{3}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{a}^{4}c{d}^{2}}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{3\,{a}^{3}{c}^{2}d}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{2}{c}^{3}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}-10\,{\frac{\ln \left ( bx+a \right ){a}^{3}{d}^{3}}{{b}^{6}}}+18\,{\frac{{a}^{2}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{5}}}-9\,{\frac{\ln \left ( bx+a \right ) a{c}^{2}d}{{b}^{4}}}+{\frac{\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 = 1.20467, size = 306, normalized size = 1.96 \begin{align*} \frac{3 \, a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 21 \, a^{4} b c d^{2} - 9 \, a^{5} d^{3} + 2 \,{\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}{2 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac{2 \, b^{2} d^{3} x^{3} + 9 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 18 \,{\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x}{6 \, b^{5}} + \frac{{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} 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 = 2.99063, size = 745, normalized size = 4.78 \begin{align*} \frac{2 \, b^{5} d^{3} x^{5} + 9 \, a^{2} b^{3} c^{3} - 45 \, a^{3} b^{2} c^{2} d + 63 \, a^{4} b c d^{2} - 27 \, a^{5} d^{3} +{\left (9 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{4} + 2 \,{\left (9 \, b^{5} c^{2} d - 18 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{3} + 9 \,{\left (4 \, a b^{4} c^{2} d - 11 \, a^{2} b^{3} c d^{2} + 7 \, a^{3} b^{2} d^{3}\right )} x^{2} + 6 \,{\left (2 \, a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )} x + 6 \,{\left (a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 18 \, a^{4} b c d^{2} - 10 \, a^{5} d^{3} +{\left (b^{5} c^{3} - 9 \, a b^{4} c^{2} d + 18 \, a^{2} b^{3} c d^{2} - 10 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 18 \, a^{3} b^{2} c d^{2} - 10 \, a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.98321, size = 230, normalized size = 1.47 \begin{align*} - \frac{9 a^{5} d^{3} - 21 a^{4} b c d^{2} + 15 a^{3} b^{2} c^{2} d - 3 a^{2} b^{3} c^{3} + x \left (10 a^{4} b d^{3} - 24 a^{3} b^{2} c d^{2} + 18 a^{2} b^{3} c^{2} d - 4 a b^{4} c^{3}\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} + \frac{d^{3} x^{3}}{3 b^{3}} - \frac{x^{2} \left (3 a d^{3} - 3 b c d^{2}\right )}{2 b^{4}} + \frac{x \left (6 a^{2} d^{3} - 9 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{5}} - \frac{\left (a d - b c\right ) \left (10 a^{2} d^{2} - 8 a b c d + b^{2} c^{2}\right ) \log{\left (a + b x \right )}}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16637, size = 300, normalized size = 1.92 \begin{align*} \frac{{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{3 \, a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 21 \, a^{4} b c d^{2} - 9 \, a^{5} d^{3} + 2 \,{\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}{2 \,{\left (b x + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} d^{3} x^{3} + 9 \, b^{6} c d^{2} x^{2} - 9 \, a b^{5} d^{3} x^{2} + 18 \, b^{6} c^{2} d x - 54 \, a b^{5} c d^{2} x + 36 \, a^{2} b^{4} d^{3} x}{6 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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