Optimal. Leaf size=68 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 \sqrt{a}}+\frac{1}{4} x^4 \left (a+\frac{b}{x^2}\right )^{3/2}+\frac{3}{8} b x^2 \sqrt{a+\frac{b}{x^2}} \]
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Rubi [A] time = 0.0346493, antiderivative size = 68, 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, 47, 63, 208} \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 \sqrt{a}}+\frac{1}{4} x^4 \left (a+\frac{b}{x^2}\right )^{3/2}+\frac{3}{8} b x^2 \sqrt{a+\frac{b}{x^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right )^{3/2} x^3 \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^3} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{1}{4} \left (a+\frac{b}{x^2}\right )^{3/2} x^4-\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{3}{8} b \sqrt{a+\frac{b}{x^2}} x^2+\frac{1}{4} \left (a+\frac{b}{x^2}\right )^{3/2} x^4-\frac{1}{16} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{3}{8} b \sqrt{a+\frac{b}{x^2}} x^2+\frac{1}{4} \left (a+\frac{b}{x^2}\right )^{3/2} x^4-\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^2}}\right )\\ &=\frac{3}{8} b \sqrt{a+\frac{b}{x^2}} x^2+\frac{1}{4} \left (a+\frac{b}{x^2}\right )^{3/2} x^4+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.110525, size = 66, normalized size = 0.97 \[ \frac{1}{8} x \sqrt{a+\frac{b}{x^2}} \left (\frac{3 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{\frac{a x^2}{b}+1}}+2 a x^3+5 b x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 84, normalized size = 1.2 \begin{align*}{\frac{{x}^{3}}{8} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 2\,x \left ( a{x}^{2}+b \right ) ^{3/2}\sqrt{a}+3\,\sqrt{a}\sqrt{a{x}^{2}+b}xb+3\,\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ){b}^{2} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{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 = 1.60095, size = 354, normalized size = 5.21 \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} + 5 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, a}, -\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} + 5 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.52827, size = 70, normalized size = 1.03 \begin{align*} \frac{a \sqrt{b} x^{3} \sqrt{\frac{a x^{2}}{b} + 1}}{4} + \frac{5 b^{\frac{3}{2}} x \sqrt{\frac{a x^{2}}{b} + 1}}{8} + \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20645, size = 92, normalized size = 1.35 \begin{align*} -\frac{3 \, b^{2} \log \left ({\left | -\sqrt{a} x + \sqrt{a x^{2} + b} \right |}\right ) \mathrm{sgn}\left (x\right )}{8 \, \sqrt{a}} + \frac{3 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{16 \, \sqrt{a}} + \frac{1}{8} \,{\left (2 \, a x^{2} \mathrm{sgn}\left (x\right ) + 5 \, b \mathrm{sgn}\left (x\right )\right )} \sqrt{a x^{2} + b} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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