Optimal. Leaf size=34 \[ -\frac{3 a^2 b}{x}+a^3 x-\frac{a b^2}{x^3}-\frac{b^3}{5 x^5} \]
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Rubi [A] time = 0.0153526, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {193, 270} \[ -\frac{3 a^2 b}{x}+a^3 x-\frac{a b^2}{x^3}-\frac{b^3}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 193
Rule 270
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right )^3 \, dx &=\int \frac{\left (b+a x^2\right )^3}{x^6} \, dx\\ &=\int \left (a^3+\frac{b^3}{x^6}+\frac{3 a b^2}{x^4}+\frac{3 a^2 b}{x^2}\right ) \, dx\\ &=-\frac{b^3}{5 x^5}-\frac{a b^2}{x^3}-\frac{3 a^2 b}{x}+a^3 x\\ \end{align*}
Mathematica [A] time = 0.0046173, size = 34, normalized size = 1. \[ -\frac{3 a^2 b}{x}+a^3 x-\frac{a b^2}{x^3}-\frac{b^3}{5 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 33, normalized size = 1. \begin{align*} -{\frac{{b}^{3}}{5\,{x}^{5}}}-{\frac{{b}^{2}a}{{x}^{3}}}-3\,{\frac{{a}^{2}b}{x}}+{a}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968912, size = 43, normalized size = 1.26 \begin{align*} a^{3} x - \frac{3 \, a^{2} b}{x} - \frac{a b^{2}}{x^{3}} - \frac{b^{3}}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40245, size = 76, normalized size = 2.24 \begin{align*} \frac{5 \, a^{3} x^{6} - 15 \, a^{2} b x^{4} - 5 \, a b^{2} x^{2} - b^{3}}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.336675, size = 32, normalized size = 0.94 \begin{align*} a^{3} x - \frac{15 a^{2} b x^{4} + 5 a b^{2} x^{2} + b^{3}}{5 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16378, size = 45, normalized size = 1.32 \begin{align*} a^{3} x - \frac{15 \, a^{2} b x^{4} + 5 \, a b^{2} x^{2} + b^{3}}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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