Optimal. Leaf size=104 \[ -\frac{a^2 (b c-a d)^2}{b^5 (a+b x)}+\frac{d x^2 (b c-a d)}{b^3}+\frac{x (b c-3 a d) (b c-a d)}{b^4}-\frac{2 a (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac{d^2 x^3}{3 b^2} \]
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Rubi [A] time = 0.097313, antiderivative size = 104, 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} \[ -\frac{a^2 (b c-a d)^2}{b^5 (a+b x)}+\frac{d x^2 (b c-a d)}{b^3}+\frac{x (b c-3 a d) (b c-a d)}{b^4}-\frac{2 a (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac{d^2 x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int \frac{x^2 (c+d x)^2}{(a+b x)^2} \, dx &=\int \left (\frac{(b c-3 a d) (b c-a d)}{b^4}+\frac{2 d (b c-a d) x}{b^3}+\frac{d^2 x^2}{b^2}+\frac{a^2 (-b c+a d)^2}{b^4 (a+b x)^2}+\frac{2 a (b c-2 a d) (-b c+a d)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{(b c-3 a d) (b c-a d) x}{b^4}+\frac{d (b c-a d) x^2}{b^3}+\frac{d^2 x^3}{3 b^2}-\frac{a^2 (b c-a d)^2}{b^5 (a+b x)}-\frac{2 a (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0744315, size = 114, normalized size = 1.1 \[ \frac{3 b x \left (3 a^2 d^2-4 a b c d+b^2 c^2\right )-6 a \left (2 a^2 d^2-3 a b c d+b^2 c^2\right ) \log (a+b x)-\frac{3 a^2 (b c-a d)^2}{a+b x}+3 b^2 d x^2 (b c-a d)+b^3 d^2 x^3}{3 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 164, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{3}}{3\,{b}^{2}}}-{\frac{a{x}^{2}{d}^{2}}{{b}^{3}}}+{\frac{c{x}^{2}d}{{b}^{2}}}+3\,{\frac{{a}^{2}{d}^{2}x}{{b}^{4}}}-4\,{\frac{acdx}{{b}^{3}}}+{\frac{{c}^{2}x}{{b}^{2}}}-{\frac{{a}^{4}{d}^{2}}{{b}^{5} \left ( bx+a \right ) }}+2\,{\frac{{a}^{3}cd}{{b}^{4} \left ( bx+a \right ) }}-{\frac{{a}^{2}{c}^{2}}{{b}^{3} \left ( bx+a \right ) }}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ){d}^{2}}{{b}^{5}}}+6\,{\frac{{a}^{2}\ln \left ( bx+a \right ) cd}{{b}^{4}}}-2\,{\frac{a\ln \left ( bx+a \right ){c}^{2}}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00557, size = 186, normalized size = 1.79 \begin{align*} -\frac{a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}}{b^{6} x + a b^{5}} + \frac{b^{2} d^{2} x^{3} + 3 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2} + 3 \,{\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x}{3 \, b^{4}} - \frac{2 \,{\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\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 = 2.21397, size = 416, normalized size = 4. \begin{align*} \frac{b^{4} d^{2} x^{4} - 3 \, a^{2} b^{2} c^{2} + 6 \, a^{3} b c d - 3 \, a^{4} d^{2} +{\left (3 \, b^{4} c d - 2 \, a b^{3} d^{2}\right )} x^{3} + 3 \,{\left (b^{4} c^{2} - 3 \, a b^{3} c d + 2 \, a^{2} b^{2} d^{2}\right )} x^{2} + 3 \,{\left (a b^{3} c^{2} - 4 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x - 6 \,{\left (a^{2} b^{2} c^{2} - 3 \, a^{3} b c d + 2 \, a^{4} d^{2} +{\left (a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{3 \,{\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 = 0.878595, size = 122, normalized size = 1.17 \begin{align*} - \frac{2 a \left (a d - b c\right ) \left (2 a d - b c\right ) \log{\left (a + b x \right )}}{b^{5}} - \frac{a^{4} d^{2} - 2 a^{3} b c d + a^{2} b^{2} c^{2}}{a b^{5} + b^{6} x} + \frac{d^{2} x^{3}}{3 b^{2}} - \frac{x^{2} \left (a d^{2} - b c d\right )}{b^{3}} + \frac{x \left (3 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14805, size = 254, normalized size = 2.44 \begin{align*} \frac{{\left (d^{2} + \frac{3 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )}}{{\left (b x + a\right )} b} + \frac{3 \,{\left (b^{4} c^{2} - 6 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )}{\left (b x + a\right )}^{3}}{3 \, b^{5}} + \frac{2 \,{\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{5}} - \frac{\frac{a^{2} b^{5} c^{2}}{b x + a} - \frac{2 \, a^{3} b^{4} c d}{b x + a} + \frac{a^{4} b^{3} d^{2}}{b x + a}}{b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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