Optimal. Leaf size=30 \[ \frac{a^2 x^4}{4}+\frac{2}{3} a b x^3+\frac{b^2 x^2}{2} \]
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Rubi [A] time = 0.012121, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 43} \[ \frac{a^2 x^4}{4}+\frac{2}{3} a b x^3+\frac{b^2 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^2 x^3 \, dx &=\int x (b+a x)^2 \, dx\\ &=\int \left (b^2 x+2 a b x^2+a^2 x^3\right ) \, dx\\ &=\frac{b^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{a^2 x^4}{4}\\ \end{align*}
Mathematica [A] time = 0.0012466, size = 30, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{2}{3} a b x^3+\frac{b^2 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 0.8 \begin{align*}{\frac{{b}^{2}{x}^{2}}{2}}+{\frac{2\,{x}^{3}ab}{3}}+{\frac{{a}^{2}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946022, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{4} \, a^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, b^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38786, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{4} \, a^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, b^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.058638, size = 26, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17525, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{4} \, a^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, b^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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