Optimal. Leaf size=152 \[ \frac{5 b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{8 a^{13/4} \sqrt{a+\frac{b}{x^4}}}+\frac{5 x^3 \sqrt{a+\frac{b}{x^4}}}{4 a^3}-\frac{3 x^3}{4 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x^3}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]
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Rubi [A] time = 0.0772644, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {335, 290, 325, 220} \[ \frac{5 b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{8 a^{13/4} \sqrt{a+\frac{b}{x^4}}}+\frac{5 x^3 \sqrt{a+\frac{b}{x^4}}}{4 a^3}-\frac{3 x^3}{4 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x^3}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]
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 )^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^4\right )^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{x^3}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^4\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{x^3}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{3 x^3}{4 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{15 \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{4 a^2}\\ &=-\frac{x^3}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{3 x^3}{4 a^2 \sqrt{a+\frac{b}{x^4}}}+\frac{5 \sqrt{a+\frac{b}{x^4}} x^3}{4 a^3}+\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{4 a^3}\\ &=-\frac{x^3}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{3 x^3}{4 a^2 \sqrt{a+\frac{b}{x^4}}}+\frac{5 \sqrt{a+\frac{b}{x^4}} x^3}{4 a^3}+\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 )}{8 a^{13/4} \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}
Mathematica [C] time = 0.0403472, size = 94, normalized size = 0.62 \[ \frac{4 a^2 x^8-15 b \left (a x^4+b\right ) \sqrt{\frac{a x^4}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{a x^4}{b}\right )+21 a b x^4+15 b^2}{12 a^3 x \sqrt{a+\frac{b}{x^4}} \left (a x^4+b\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 304, normalized size = 2. \begin{align*}{\frac{1}{12\,{a}^{3}{x}^{10}} \left ( 4\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{13}{a}^{3}-15\,\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 ){x}^{8}{a}^{2}b+25\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{9}{a}^{2}b-30\,\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 ){x}^{4}a{b}^{2}+36\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{5}a{b}^{2}-15\,\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 ){b}^{3}+15\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}x{b}^{3} \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{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{5}{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^{14} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a^{3} x^{12} + 3 \, a^{2} b x^{8} + 3 \, a b^{2} x^{4} + b^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.68666, size = 42, normalized size = 0.28 \begin{align*} - \frac{x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{5}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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