Optimal. Leaf size=76 \[ -\frac{a^3}{3 b^4 \left (a+\frac{b}{x^2}\right )^{3/2}}+\frac{3 a^2}{b^4 \sqrt{a+\frac{b}{x^2}}}+\frac{3 a \sqrt{a+\frac{b}{x^2}}}{b^4}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^4} \]
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Rubi [A] time = 0.0404859, antiderivative size = 76, 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} \[ -\frac{a^3}{3 b^4 \left (a+\frac{b}{x^2}\right )^{3/2}}+\frac{3 a^2}{b^4 \sqrt{a+\frac{b}{x^2}}}+\frac{3 a \sqrt{a+\frac{b}{x^2}}}{b^4}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^4} \]
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^9} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{5/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{5/2}}+\frac{3 a^2}{b^3 (a+b x)^{3/2}}-\frac{3 a}{b^3 \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b^3}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{a^3}{3 b^4 \left (a+\frac{b}{x^2}\right )^{3/2}}+\frac{3 a^2}{b^4 \sqrt{a+\frac{b}{x^2}}}+\frac{3 a \sqrt{a+\frac{b}{x^2}}}{b^4}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^4}\\ \end{align*}
Mathematica [A] time = 0.0127014, size = 62, normalized size = 0.82 \[ \frac{24 a^2 b x^4+16 a^3 x^6+6 a b^2 x^2-b^3}{3 b^4 x^4 \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 61, normalized size = 0.8 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ( 16\,{a}^{3}{x}^{6}+24\,{a}^{2}b{x}^{4}+6\,a{b}^{2}{x}^{2}-{b}^{3} \right ) }{3\,{x}^{8}{b}^{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 = 0.979099, size = 86, normalized size = 1.13 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}{3 \, b^{4}} + \frac{3 \, \sqrt{a + \frac{b}{x^{2}}} a}{b^{4}} + \frac{3 \, a^{2}}{\sqrt{a + \frac{b}{x^{2}}} b^{4}} - \frac{a^{3}}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59622, size = 153, normalized size = 2.01 \begin{align*} \frac{{\left (16 \, a^{3} x^{6} + 24 \, a^{2} b x^{4} + 6 \, a b^{2} x^{2} - b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{2} b^{4} x^{6} + 2 \, a b^{5} x^{4} + b^{6} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.9793, size = 201, normalized size = 2.64 \begin{align*} \begin{cases} \frac{16 a^{3} x^{6}}{3 a b^{4} x^{6} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{5} x^{4} \sqrt{a + \frac{b}{x^{2}}}} + \frac{24 a^{2} b x^{4}}{3 a b^{4} x^{6} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{5} x^{4} \sqrt{a + \frac{b}{x^{2}}}} + \frac{6 a b^{2} x^{2}}{3 a b^{4} x^{6} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{5} x^{4} \sqrt{a + \frac{b}{x^{2}}}} - \frac{b^{3}}{3 a b^{4} x^{6} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{5} x^{4} \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{8 a^{\frac{5}{2}} x^{8}} & \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^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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