Optimal. Leaf size=21 \[ \frac {x^3 \left (a+\frac {b}{x^2}\right )^{3/2}}{3 a} \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {264} \[ \frac {x^3 \left (a+\frac {b}{x^2}\right )^{3/2}}{3 a} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \sqrt {a+\frac {b}{x^2}} x^2 \, dx &=\frac {\left (a+\frac {b}{x^2}\right )^{3/2} x^3}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.24 \[ \frac {x \sqrt {a+\frac {b}{x^2}} \left (a x^2+b\right )}{3 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 27, normalized size = 1.29 \[ \frac {{\left (a x^{3} + b x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 27, normalized size = 1.29 \[ \frac {{\left (a x^{2} + b\right )}^{\frac {3}{2}} \mathrm {sgn}\relax (x)}{3 \, a} - \frac {b^{\frac {3}{2}} \mathrm {sgn}\relax (x)}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 1.29 \[ \frac {\left (a \,x^{2}+b \right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 17, normalized size = 0.81 \[ \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{3}}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 23, normalized size = 1.10 \[ \sqrt {a+\frac {b}{x^2}}\,\left (\frac {x^3}{3}+\frac {b\,x}{3\,a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.85, size = 41, normalized size = 1.95 \[ \frac {\sqrt {b} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3} + \frac {b^{\frac {3}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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