Optimal. Leaf size=28 \[ -\frac{a^2}{x}-\frac{2 a b}{3 x^3}-\frac{b^2}{5 x^5} \]
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Rubi [A] time = 0.0121368, antiderivative size = 28, 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, 270} \[ -\frac{a^2}{x}-\frac{2 a b}{3 x^3}-\frac{b^2}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 263
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right )^2}{x^2} \, dx &=\int \frac{\left (b+a x^2\right )^2}{x^6} \, dx\\ &=\int \left (\frac{b^2}{x^6}+\frac{2 a b}{x^4}+\frac{a^2}{x^2}\right ) \, dx\\ &=-\frac{b^2}{5 x^5}-\frac{2 a b}{3 x^3}-\frac{a^2}{x}\\ \end{align*}
Mathematica [A] time = 0.0007557, size = 28, normalized size = 1. \[ -\frac{a^2}{x}-\frac{2 a b}{3 x^3}-\frac{b^2}{5 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 25, normalized size = 0.9 \begin{align*} -{\frac{{b}^{2}}{5\,{x}^{5}}}-{\frac{2\,ab}{3\,{x}^{3}}}-{\frac{{a}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00412, size = 35, normalized size = 1.25 \begin{align*} -\frac{15 \, a^{2} x^{4} + 10 \, a b x^{2} + 3 \, b^{2}}{15 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46131, size = 61, normalized size = 2.18 \begin{align*} -\frac{15 \, a^{2} x^{4} + 10 \, a b x^{2} + 3 \, b^{2}}{15 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.31196, size = 27, normalized size = 0.96 \begin{align*} - \frac{15 a^{2} x^{4} + 10 a b x^{2} + 3 b^{2}}{15 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14852, size = 35, normalized size = 1.25 \begin{align*} -\frac{15 \, a^{2} x^{4} + 10 \, a b x^{2} + 3 \, b^{2}}{15 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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