Optimal. Leaf size=71 \[ \frac{d x^2 (2 b c-a d)}{2 b^2}+\frac{x (b c-a d)^2}{b^3}-\frac{a (b c-a d)^2 \log (a+b x)}{b^4}+\frac{d^2 x^3}{3 b} \]
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Rubi [A] time = 0.0530015, antiderivative size = 71, 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} \[ \frac{d x^2 (2 b c-a d)}{2 b^2}+\frac{x (b c-a d)^2}{b^3}-\frac{a (b c-a d)^2 \log (a+b x)}{b^4}+\frac{d^2 x^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x (c+d x)^2}{a+b x} \, dx &=\int \left (\frac{(b c-a d)^2}{b^3}+\frac{d (2 b c-a d) x}{b^2}+\frac{d^2 x^2}{b}-\frac{a (-b c+a d)^2}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{(b c-a d)^2 x}{b^3}+\frac{d (2 b c-a d) x^2}{2 b^2}+\frac{d^2 x^3}{3 b}-\frac{a (b c-a d)^2 \log (a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.027347, size = 74, normalized size = 1.04 \[ \frac{b x \left (6 a^2 d^2-3 a b d (4 c+d x)+2 b^2 \left (3 c^2+3 c d x+d^2 x^2\right )\right )-6 a (b c-a d)^2 \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 110, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{3}}{3\,b}}-{\frac{a{x}^{2}{d}^{2}}{2\,{b}^{2}}}+{\frac{c{x}^{2}d}{b}}+{\frac{{a}^{2}{d}^{2}x}{{b}^{3}}}-2\,{\frac{acdx}{{b}^{2}}}+{\frac{{c}^{2}x}{b}}-{\frac{{a}^{3}\ln \left ( bx+a \right ){d}^{2}}{{b}^{4}}}+2\,{\frac{{a}^{2}\ln \left ( bx+a \right ) cd}{{b}^{3}}}-{\frac{a\ln \left ( bx+a \right ){c}^{2}}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02429, size = 131, normalized size = 1.85 \begin{align*} \frac{2 \, b^{2} d^{2} x^{3} + 3 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{2} + 6 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{6 \, b^{3}} - \frac{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left (b x + a\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94013, size = 205, normalized size = 2.89 \begin{align*} \frac{2 \, b^{3} d^{2} x^{3} + 3 \,{\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2} + 6 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x - 6 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.499274, size = 76, normalized size = 1.07 \begin{align*} - \frac{a \left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{b^{4}} + \frac{d^{2} x^{3}}{3 b} - \frac{x^{2} \left (a d^{2} - 2 b c d\right )}{2 b^{2}} + \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26777, size = 134, normalized size = 1.89 \begin{align*} \frac{2 \, b^{2} d^{2} x^{3} + 6 \, b^{2} c d x^{2} - 3 \, a b d^{2} x^{2} + 6 \, b^{2} c^{2} x - 12 \, a b c d x + 6 \, a^{2} d^{2} x}{6 \, b^{3}} - \frac{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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