Optimal. Leaf size=50 \[ \frac{x^4 \sqrt{a+\frac{b}{x^4}}}{4 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{3/2}} \]
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Rubi [A] time = 0.0258321, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{x^4 \sqrt{a+\frac{b}{x^4}}}{4 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{3/2}} \]
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^4}}} \, dx &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x^4}\right )\right )\\ &=\frac{\sqrt{a+\frac{b}{x^4}} x^4}{4 a}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^4}\right )}{8 a}\\ &=\frac{\sqrt{a+\frac{b}{x^4}} x^4}{4 a}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^4}}\right )}{4 a}\\ &=\frac{\sqrt{a+\frac{b}{x^4}} x^4}{4 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0280032, size = 75, normalized size = 1.5 \[ \frac{\sqrt{a} x^2 \left (a x^4+b\right )-b \sqrt{a x^4+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{a x^4+b}}\right )}{4 a^{3/2} x^2 \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 70, normalized size = 1.4 \begin{align*}{\frac{1}{4\,{x}^{2}}\sqrt{a{x}^{4}+b} \left ({x}^{2}\sqrt{a{x}^{4}+b}{a}^{{\frac{3}{2}}}-b\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ) a \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{a}^{-{\frac{5}{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 = 1.57348, size = 297, normalized size = 5.94 \begin{align*} \left [\frac{2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} + \sqrt{a} b \log \left (-2 \, a x^{4} + 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right )}{8 \, a^{2}}, \frac{a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} + \sqrt{-a} b \arctan \left (\frac{\sqrt{-a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right )}{4 \, a^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.47178, size = 46, normalized size = 0.92 \begin{align*} \frac{\sqrt{b} x^{2} \sqrt{\frac{a x^{4}}{b} + 1}}{4 a} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{4 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17596, size = 90, normalized size = 1.8 \begin{align*} \frac{1}{4} \, b{\left (\frac{\arctan \left (\frac{\sqrt{\frac{a x^{4} + b}{x^{4}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{\sqrt{\frac{a x^{4} + b}{x^{4}}}}{{\left (a - \frac{a x^{4} + b}{x^{4}}\right )} a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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