Optimal. Leaf size=24 \[ -\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3} \]
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Rubi [A] time = 0.0109929, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 270} \[ -\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 270
Rubi steps
\begin{align*} \int \left (\frac{a}{x}+b x\right )^2 \, dx &=\int \frac{\left (a+b x^2\right )^2}{x^2} \, dx\\ &=\int \left (2 a b+\frac{a^2}{x^2}+b^2 x^2\right ) \, dx\\ &=-\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3}\\ \end{align*}
Mathematica [A] time = 0.001072, size = 24, normalized size = 1. \[ -\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, normalized size = 1. \begin{align*} -{\frac{{a}^{2}}{x}}+2\,abx+{\frac{{b}^{2}{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10582, size = 30, normalized size = 1.25 \begin{align*} \frac{1}{3} \, b^{2} x^{3} + 2 \, a b x - \frac{a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.726116, size = 50, normalized size = 2.08 \begin{align*} \frac{b^{2} x^{4} + 6 \, a b x^{2} - 3 \, a^{2}}{3 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.317238, size = 19, normalized size = 0.79 \begin{align*} - \frac{a^{2}}{x} + 2 a b x + \frac{b^{2} x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17389, size = 30, normalized size = 1.25 \begin{align*} \frac{1}{3} \, b^{2} x^{3} + 2 \, a b x - \frac{a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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