Optimal. Leaf size=80 \[ \frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x^2}\right )^{5/2}-\frac{5}{6} b \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{5}{2} a b \sqrt{a+\frac{b}{x^2}} \]
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Rubi [A] time = 0.0401554, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {266, 47, 50, 63, 208} \[ \frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x^2}\right )^{5/2}-\frac{5}{6} b \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{5}{2} a b \sqrt{a+\frac{b}{x^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right )^{5/2} x \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^2} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{1}{2} \left (a+\frac{b}{x^2}\right )^{5/2} x^2-\frac{1}{4} (5 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{5}{6} b \left (a+\frac{b}{x^2}\right )^{3/2}+\frac{1}{2} \left (a+\frac{b}{x^2}\right )^{5/2} x^2-\frac{1}{4} (5 a b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{5}{2} a b \sqrt{a+\frac{b}{x^2}}-\frac{5}{6} b \left (a+\frac{b}{x^2}\right )^{3/2}+\frac{1}{2} \left (a+\frac{b}{x^2}\right )^{5/2} x^2-\frac{1}{4} \left (5 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{5}{2} a b \sqrt{a+\frac{b}{x^2}}-\frac{5}{6} b \left (a+\frac{b}{x^2}\right )^{3/2}+\frac{1}{2} \left (a+\frac{b}{x^2}\right )^{5/2} x^2-\frac{1}{2} \left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^2}}\right )\\ &=-\frac{5}{2} a b \sqrt{a+\frac{b}{x^2}}-\frac{5}{6} b \left (a+\frac{b}{x^2}\right )^{3/2}+\frac{1}{2} \left (a+\frac{b}{x^2}\right )^{5/2} x^2+\frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0129669, size = 54, normalized size = 0.68 \[ -\frac{b^2 \sqrt{a+\frac{b}{x^2}} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};-\frac{a x^2}{b}\right )}{3 x^2 \sqrt{\frac{a x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 149, normalized size = 1.9 \begin{align*} -{\frac{{x}^{2}}{6\,{b}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( -8\, \left ( a{x}^{2}+b \right ) ^{5/2}{a}^{5/2}{x}^{4}+8\, \left ( a{x}^{2}+b \right ) ^{7/2}{a}^{3/2}{x}^{2}-10\, \left ( a{x}^{2}+b \right ) ^{3/2}{a}^{5/2}{x}^{4}b-15\,\sqrt{a{x}^{2}+b}{a}^{5/2}{x}^{4}{b}^{2}+2\, \left ( a{x}^{2}+b \right ) ^{7/2}b\sqrt{a}-15\,\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ){x}^{3}{a}^{2}{b}^{3} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57919, size = 394, normalized size = 4.92 \begin{align*} \left [\frac{15 \, a^{\frac{3}{2}} b x^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) + 2 \,{\left (3 \, a^{2} x^{4} - 14 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{12 \, x^{2}}, -\frac{15 \, \sqrt{-a} a b x^{2} \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) -{\left (3 \, a^{2} x^{4} - 14 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{6 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.89143, size = 112, normalized size = 1.4 \begin{align*} \frac{a^{\frac{5}{2}} x^{2} \sqrt{1 + \frac{b}{a x^{2}}}}{2} - \frac{7 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x^{2}}}}{3} - \frac{5 a^{\frac{3}{2}} b \log{\left (\frac{b}{a x^{2}} \right )}}{4} + \frac{5 a^{\frac{3}{2}} b \log{\left (\sqrt{1 + \frac{b}{a x^{2}}} + 1 \right )}}{2} - \frac{\sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x^{2}}}}{3 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.746, size = 192, normalized size = 2.4 \begin{align*} \frac{1}{2} \, \sqrt{a x^{2} + b} a^{2} x \mathrm{sgn}\left (x\right ) - \frac{5}{4} \, a^{\frac{3}{2}} b \log \left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2}\right ) \mathrm{sgn}\left (x\right ) + \frac{2 \,{\left (9 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{3}{2}} b^{2} \mathrm{sgn}\left (x\right ) - 12 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} a^{\frac{3}{2}} b^{3} \mathrm{sgn}\left (x\right ) + 7 \, a^{\frac{3}{2}} b^{4} \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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