Optimal. Leaf size=55 \[ \frac{a^2}{3 b^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{2 a}{b^3 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^3} \]
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Rubi [A] time = 0.0293698, antiderivative size = 55, 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} \[ \frac{a^2}{3 b^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{2 a}{b^3 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^3} \]
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^7} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{5/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{5/2}}-\frac{2 a}{b^2 (a+b x)^{3/2}}+\frac{1}{b^2 \sqrt{a+b x}}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{a^2}{3 b^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{2 a}{b^3 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0188686, size = 48, normalized size = 0.87 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (8 a^2 x^4+12 a b x^2+3 b^2\right )}{3 b^3 \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 50, normalized size = 0.9 \begin{align*} -{\frac{ \left ( a{x}^{2}+b \right ) \left ( 8\,{a}^{2}{x}^{4}+12\,ab{x}^{2}+3\,{b}^{2} \right ) }{3\,{b}^{3}{x}^{6}} \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 = 1.00109, size = 63, normalized size = 1.15 \begin{align*} -\frac{\sqrt{a + \frac{b}{x^{2}}}}{b^{3}} - \frac{2 \, a}{\sqrt{a + \frac{b}{x^{2}}} b^{3}} + \frac{a^{2}}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52117, size = 128, normalized size = 2.33 \begin{align*} -\frac{{\left (8 \, a^{2} x^{4} + 12 \, a b x^{2} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.2919, size = 153, normalized size = 2.78 \begin{align*} \begin{cases} - \frac{8 a^{2} x^{4}}{3 a b^{3} x^{4} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{4} x^{2} \sqrt{a + \frac{b}{x^{2}}}} - \frac{12 a b x^{2}}{3 a b^{3} x^{4} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{4} x^{2} \sqrt{a + \frac{b}{x^{2}}}} - \frac{3 b^{2}}{3 a b^{3} x^{4} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{4} x^{2} \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{6 a^{\frac{5}{2}} x^{6}} & \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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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