Optimal. Leaf size=35 \[ \frac{1}{b^2 \sqrt{a+\frac{b}{x^2}}}-\frac{a}{3 b^2 \left (a+\frac{b}{x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.0206982, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{1}{b^2 \sqrt{a+\frac{b}{x^2}}}-\frac{a}{3 b^2 \left (a+\frac{b}{x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^5} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{5/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{5/2}}+\frac{1}{b (a+b x)^{3/2}}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{a}{3 b^2 \left (a+\frac{b}{x^2}\right )^{3/2}}+\frac{1}{b^2 \sqrt{a+\frac{b}{x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0126012, size = 37, normalized size = 1.06 \[ \frac{2 a x^2+3 b}{3 b^2 \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 39, normalized size = 1.1 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ( 2\,a{x}^{2}+3\,b \right ) }{3\,{b}^{2}{x}^{4}} \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 = 1.00135, size = 39, normalized size = 1.11 \begin{align*} \frac{1}{\sqrt{a + \frac{b}{x^{2}}} b^{2}} - \frac{a}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53224, size = 109, normalized size = 3.11 \begin{align*} \frac{{\left (2 \, a x^{4} + 3 \, b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.47328, size = 94, normalized size = 2.69 \begin{align*} \begin{cases} \frac{2 a x^{2}}{3 a b^{2} x^{2} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{3} \sqrt{a + \frac{b}{x^{2}}}} + \frac{3 b}{3 a b^{2} x^{2} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{3} \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{4 a^{\frac{5}{2}} x^{4}} & \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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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