Optimal. Leaf size=107 \[ \frac{1}{3} x^3 \sqrt{a+\frac{b}{x^4}}-\frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{3 \sqrt [4]{a} \sqrt{a+\frac{b}{x^4}}} \]
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Rubi [A] time = 0.0479689, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {335, 277, 220} \[ \frac{1}{3} x^3 \sqrt{a+\frac{b}{x^4}}-\frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
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Rule 335
Rule 277
Rule 220
Rubi steps
\begin{align*} \int \sqrt{a+\frac{b}{x^4}} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^4}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt{a+\frac{b}{x^4}} x^3-\frac{1}{3} (2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt{a+\frac{b}{x^4}} x^3-\frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}
Mathematica [C] time = 0.164785, size = 93, normalized size = 0.87 \[ \frac{1}{3} x^2 \sqrt{a+\frac{b}{x^4}} \left (x-\frac{2 i b \sqrt{\frac{a x^4}{b}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}\right ),-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.038, size = 130, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}}{3\,a{x}^{4}+3\,b}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( \sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{x}^{5}a+2\,b\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ) +\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}xb \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + \frac{b}{x^{4}}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.35776, size = 44, normalized size = 0.41 \begin{align*} - \frac{\sqrt{a} x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac{1}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + \frac{b}{x^{4}}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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