Optimal. Leaf size=110 \[ \frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+\frac{b}{x^4}}}+\frac{x^3 \sqrt{a+\frac{b}{x^4}}}{3 a} \]
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Rubi [A] time = 0.0451209, antiderivative size = 110, 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, 325, 220} \[ \frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+\frac{b}{x^4}}}+\frac{x^3 \sqrt{a+\frac{b}{x^4}}}{3 a} \]
Antiderivative was successfully verified.
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Rule 335
Rule 325
Rule 220
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{a+\frac{b}{x^4}}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{a+\frac{b}{x^4}} x^3}{3 a}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{3 a}\\ &=\frac{\sqrt{a+\frac{b}{x^4}} x^3}{3 a}+\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 )}{6 a^{5/4} \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}
Mathematica [C] time = 0.0200952, size = 64, normalized size = 0.58 \[ \frac{-b \sqrt{\frac{a x^4}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{a x^4}{b}\right )+a x^4+b}{3 a x \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.009, size = 124, normalized size = 1.1 \begin{align*}{\frac{1}{3\,a{x}^{2}} \left ( \sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{x}^{5}a-b\sqrt{-{ \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{ \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}xb \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\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 \frac{x^{2}}{\sqrt{a + \frac{b}{x^{4}}}}\,{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 (\frac{x^{6} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a x^{4} + b}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.38849, size = 42, normalized size = 0.38 \begin{align*} - \frac{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 \sqrt{a} \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 \frac{x^{2}}{\sqrt{a + \frac{b}{x^{4}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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