Optimal. Leaf size=31 \[ \frac{a^2 \log (a+b x)}{b^3}-\frac{a x}{b^2}+\frac{x^2}{2 b} \]
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Rubi [A] time = 0.015678, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{a^2 \log (a+b x)}{b^3}-\frac{a x}{b^2}+\frac{x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{a+b x} \, dx &=\int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx\\ &=-\frac{a x}{b^2}+\frac{x^2}{2 b}+\frac{a^2 \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0032206, size = 31, normalized size = 1. \[ \frac{a^2 \log (a+b x)}{b^3}-\frac{a x}{b^2}+\frac{x^2}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 30, normalized size = 1. \begin{align*} -{\frac{ax}{{b}^{2}}}+{\frac{{x}^{2}}{2\,b}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94316, size = 39, normalized size = 1.26 \begin{align*} \frac{a^{2} \log \left (b x + a\right )}{b^{3}} + \frac{b x^{2} - 2 \, a x}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01677, size = 68, normalized size = 2.19 \begin{align*} \frac{b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.267409, size = 26, normalized size = 0.84 \begin{align*} \frac{a^{2} \log{\left (a + b x \right )}}{b^{3}} - \frac{a x}{b^{2}} + \frac{x^{2}}{2 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07159, size = 41, normalized size = 1.32 \begin{align*} \frac{a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac{b x^{2} - 2 \, a x}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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