Optimal. Leaf size=71 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{16 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2} \]
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Rubi [A] time = 0.0520998, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {335, 275, 195, 217, 206} \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{16 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 335
Rule 275
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{x^3} \, dx &=-\operatorname{Subst}\left (\int x \left (a+b x^4\right )^{3/2} \, dx,x,\frac{1}{x}\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}-\frac{1}{8} (3 a) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{3 a \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}-\frac{1}{16} \left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{3 a \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}-\frac{1}{16} \left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x^4}} x^2}\right )\\ &=-\frac{3 a \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}-\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x^4}} x^2}\right )}{16 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.036271, size = 85, normalized size = 1.2 \[ -\frac{\sqrt{a+\frac{b}{x^4}} \left (3 a^2 x^8 \sqrt{\frac{a x^4}{b}+1} \tanh ^{-1}\left (\sqrt{\frac{a x^4}{b}+1}\right )+5 a^2 x^8+7 a b x^4+2 b^2\right )}{16 x^6 \left (a x^4+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 93, normalized size = 1.3 \begin{align*} -{\frac{1}{16\,{x}^{2}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{a}^{2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){x}^{8}+5\,a\sqrt{a{x}^{4}+b}{x}^{4}\sqrt{b}+2\,{b}^{3/2}\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{b}}}} \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 = 1.51737, size = 363, normalized size = 5.11 \begin{align*} \left [\frac{3 \, a^{2} \sqrt{b} x^{6} \log \left (\frac{a x^{4} - 2 \, \sqrt{b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ) - 2 \,{\left (5 \, a b x^{4} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{32 \, b x^{6}}, \frac{3 \, a^{2} \sqrt{-b} x^{6} \arctan \left (\frac{\sqrt{-b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{b}\right ) -{\left (5 \, a b x^{4} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{16 \, b x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.45369, size = 75, normalized size = 1.06 \begin{align*} - \frac{5 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{4}}}}{16 x^{2}} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{4}}}}{8 x^{6}} - \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{16 \sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.122, size = 82, normalized size = 1.15 \begin{align*} \frac{1}{16} \, a^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{5 \,{\left (a x^{4} + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x^{4} + b} b}{a^{2} x^{8}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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