Optimal. Leaf size=71 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{1}{4} x^4 \sqrt{a+\frac{b}{x^2}}+\frac{b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a} \]
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Rubi [A] time = 0.0357065, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{1}{4} x^4 \sqrt{a+\frac{b}{x^2}}+\frac{b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+\frac{b}{x^2}} x^3 \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{1}{4} \sqrt{a+\frac{b}{x^2}} x^4-\frac{1}{8} b \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{b \sqrt{a+\frac{b}{x^2}} x^2}{8 a}+\frac{1}{4} \sqrt{a+\frac{b}{x^2}} x^4+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{16 a}\\ &=\frac{b \sqrt{a+\frac{b}{x^2}} x^2}{8 a}+\frac{1}{4} \sqrt{a+\frac{b}{x^2}} x^4+\frac{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}\\ &=\frac{b \sqrt{a+\frac{b}{x^2}} x^2}{8 a}+\frac{1}{4} \sqrt{a+\frac{b}{x^2}} x^4-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.039887, size = 88, normalized size = 1.24 \[ x \sqrt{a+\frac{b}{x^2}} \left (\frac{b x}{8 a}+\frac{x^3}{4}\right )-\frac{b^2 x \sqrt{a+\frac{b}{x^2}} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{8 a^{3/2} \sqrt{a x^2+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 82, normalized size = 1.2 \begin{align*}{\frac{x}{8}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( 2\,x \left ( a{x}^{2}+b \right ) ^{3/2}\sqrt{a}-\sqrt{a}\sqrt{a{x}^{2}+b}xb-\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ){b}^{2} \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}{a}^{-{\frac{3}{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 = 1.54799, size = 347, normalized size = 4.89 \begin{align*} \left [\frac{\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} + a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, a^{2}}, \frac{\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} + a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, a^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.43026, size = 92, normalized size = 1.3 \begin{align*} \frac{a x^{5}}{4 \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{3 \sqrt{b} x^{3}}{8 \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{b^{\frac{3}{2}} x}{8 a \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21975, size = 93, normalized size = 1.31 \begin{align*} \frac{1}{8} \, \sqrt{a x^{2} + b}{\left (2 \, x^{2} \mathrm{sgn}\left (x\right ) + \frac{b \mathrm{sgn}\left (x\right )}{a}\right )} x + \frac{b^{2} \log \left ({\left | -\sqrt{a} x + \sqrt{a x^{2} + b} \right |}\right ) \mathrm{sgn}\left (x\right )}{8 \, a^{\frac{3}{2}}} - \frac{b^{2} \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{16 \, a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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