Optimal. Leaf size=18 \[ \frac{1}{3 b \left (a+\frac{b}{x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.0061118, antiderivative size = 18, normalized size of antiderivative = 1., 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 = {261} \[ \frac{1}{3 b \left (a+\frac{b}{x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^3} \, dx &=\frac{1}{3 b \left (a+\frac{b}{x^2}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0091765, size = 28, normalized size = 1.56 \[ \frac{a x^2+b}{3 b x^2 \left (a+\frac{b}{x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 29, normalized size = 1.6 \begin{align*}{\frac{a{x}^{2}+b}{3\,b{x}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02597, size = 19, normalized size = 1.06 \begin{align*} \frac{1}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52651, size = 85, normalized size = 4.72 \begin{align*} \frac{x^{4} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.93812, size = 48, normalized size = 2.67 \begin{align*} \begin{cases} \frac{1}{3 a b \sqrt{a + \frac{b}{x^{2}}} + \frac{3 b^{2} \sqrt{a + \frac{b}{x^{2}}}}{x^{2}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{5}{2}} x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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