Optimal. Leaf size=59 \[ \frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2} \]
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Rubi [A] time = 0.0340354, antiderivative size = 59, 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, 50, 63, 208} \[ \frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{x} \, dx &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\frac{1}{x^4}\right )\right )\\ &=-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{1}{4} a \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x^4}\right )\\ &=-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{1}{4} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^4}\right )\\ &=-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^4}}\right )}{2 b}\\ &=-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2}+\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0173859, size = 52, normalized size = 0.88 \[ -\frac{b \sqrt{a+\frac{b}{x^4}} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{a x^4}{b}\right )}{6 x^4 \sqrt{\frac{a x^4}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 79, normalized size = 1.3 \begin{align*}{\frac{1}{6} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{a}^{3/2}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{6}-4\,a{x}^{4}\sqrt{a{x}^{4}+b}-b\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\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 = 1.56579, size = 335, normalized size = 5.68 \begin{align*} \left [\frac{3 \, a^{\frac{3}{2}} x^{4} \log \left (-2 \, a x^{4} - 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right ) - 2 \,{\left (4 \, a x^{4} + b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{12 \, x^{4}}, -\frac{3 \, \sqrt{-a} a x^{4} \arctan \left (\frac{\sqrt{-a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) +{\left (4 \, a x^{4} + b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{6 \, x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.3766, size = 80, normalized size = 1.36 \begin{align*} - \frac{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{4}}}}{3} - \frac{a^{\frac{3}{2}} \log{\left (\frac{b}{a x^{4}} \right )}}{4} + \frac{a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b}{a x^{4}}} + 1 \right )}}{2} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{4}}}}{6 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12247, size = 68, normalized size = 1.15 \begin{align*} -\frac{a^{2} \arctan \left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} - \frac{1}{6} \,{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{a + \frac{b}{x^{4}}} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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