Optimal. Leaf size=131 \[ \frac{5 b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{12 a^{9/4} \sqrt{a+\frac{b}{x^4}}}+\frac{5 x^3 \sqrt{a+\frac{b}{x^4}}}{6 a^2}-\frac{x^3}{2 a \sqrt{a+\frac{b}{x^4}}} \]
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Rubi [A] time = 0.0610674, antiderivative size = 131, 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 = {335, 290, 325, 220} \[ \frac{5 b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt{a+\frac{b}{x^4}}}+\frac{5 x^3 \sqrt{a+\frac{b}{x^4}}}{6 a^2}-\frac{x^3}{2 a \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
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Rule 335
Rule 290
Rule 325
Rule 220
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+\frac{b}{x^4}\right )^{3/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^4\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{x^3}{2 a \sqrt{a+\frac{b}{x^4}}}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{x^3}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{5 \sqrt{a+\frac{b}{x^4}} x^3}{6 a^2}+\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{6 a^2}\\ &=-\frac{x^3}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{5 \sqrt{a+\frac{b}{x^4}} x^3}{6 a^2}+\frac{5 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 )}{12 a^{9/4} \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}
Mathematica [C] time = 0.0198479, size = 67, normalized size = 0.51 \[ \frac{-5 b \sqrt{\frac{a x^4}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{a x^4}{b}\right )+2 a x^4+5 b}{6 a^2 x \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.014, size = 133, normalized size = 1. \begin{align*}{\frac{a{x}^{4}+b}{6\,{a}^{2}{x}^{6}} \left ( 2\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{5}a-5\,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 ) +5\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}xb \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{\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}}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{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 (\frac{x^{10} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a^{2} x^{8} + 2 \, a b x^{4} + b^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.66306, size = 42, normalized size = 0.32 \begin{align*} - \frac{x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{3}{2}} \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}}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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