Optimal. Leaf size=74 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{5/2}}-\frac{3 b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a^2}+\frac{x^4 \sqrt{a+\frac{b}{x^2}}}{4 a} \]
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Rubi [A] time = 0.0366002, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{5/2}}-\frac{3 b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a^2}+\frac{x^4 \sqrt{a+\frac{b}{x^2}}}{4 a} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{a+\frac{b}{x^2}}} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{\sqrt{a+\frac{b}{x^2}} x^4}{4 a}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{8 a}\\ &=-\frac{3 b \sqrt{a+\frac{b}{x^2}} x^2}{8 a^2}+\frac{\sqrt{a+\frac{b}{x^2}} x^4}{4 a}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{16 a^2}\\ &=-\frac{3 b \sqrt{a+\frac{b}{x^2}} x^2}{8 a^2}+\frac{\sqrt{a+\frac{b}{x^2}} x^4}{4 a}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^2}}\right )}{8 a^2}\\ &=-\frac{3 b \sqrt{a+\frac{b}{x^2}} x^2}{8 a^2}+\frac{\sqrt{a+\frac{b}{x^2}} x^4}{4 a}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0340435, size = 87, normalized size = 1.18 \[ \frac{\sqrt{a} x \left (2 a^2 x^4-a b x^2-3 b^2\right )+3 b^2 \sqrt{a x^2+b} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b}}\right )}{8 a^{5/2} x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 87, normalized size = 1.2 \begin{align*}{\frac{1}{8\,x}\sqrt{a{x}^{2}+b} \left ( 2\,{x}^{3}\sqrt{a{x}^{2}+b}{a}^{5/2}-3\,{a}^{3/2}\sqrt{a{x}^{2}+b}xb+3\,\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ) a{b}^{2} \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}{a}^{-{\frac{7}{2}}}} \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 = 1.60108, size = 359, normalized size = 4.85 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) + 2 \,{\left (2 \, a^{2} x^{4} - 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, a^{3}}, -\frac{3 \, \sqrt{-a} b^{2} \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) -{\left (2 \, a^{2} x^{4} - 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, a^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.38876, size = 95, normalized size = 1.28 \begin{align*} \frac{x^{5}}{4 \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{\sqrt{b} x^{3}}{8 a \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{3 b^{\frac{3}{2}} x}{8 a^{2} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20663, size = 134, normalized size = 1.81 \begin{align*} -\frac{1}{8} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} - \frac{5 \, a \sqrt{\frac{a x^{2} + b}{x^{2}}} - \frac{3 \,{\left (a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{x^{2}}}{{\left (a - \frac{a x^{2} + b}{x^{2}}\right )}^{2} a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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