Optimal. Leaf size=146 \[ -\frac{20 a^{7/4} b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{21 \sqrt{a+\frac{b}{x^4}}}+\frac{1}{3} x^3 \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{21 x}-\frac{20 a b \sqrt{a+\frac{b}{x^4}}}{21 x} \]
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Rubi [A] time = 0.0696952, antiderivative size = 146, 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, 277, 195, 220} \[ -\frac{20 a^{7/4} b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 \sqrt{a+\frac{b}{x^4}}}+\frac{1}{3} x^3 \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{21 x}-\frac{20 a b \sqrt{a+\frac{b}{x^4}}}{21 x} \]
Antiderivative was successfully verified.
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Rule 335
Rule 277
Rule 195
Rule 220
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^4}\right )^{5/2} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{\left (a+b x^4\right )^{5/2}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \left (a+\frac{b}{x^4}\right )^{5/2} x^3-\frac{1}{3} (10 b) \operatorname{Subst}\left (\int \left (a+b x^4\right )^{3/2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{21 x}+\frac{1}{3} \left (a+\frac{b}{x^4}\right )^{5/2} x^3-\frac{1}{7} (20 a b) \operatorname{Subst}\left (\int \sqrt{a+b x^4} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{20 a b \sqrt{a+\frac{b}{x^4}}}{21 x}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{21 x}+\frac{1}{3} \left (a+\frac{b}{x^4}\right )^{5/2} x^3-\frac{1}{21} \left (40 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{20 a b \sqrt{a+\frac{b}{x^4}}}{21 x}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{21 x}+\frac{1}{3} \left (a+\frac{b}{x^4}\right )^{5/2} x^3-\frac{20 a^{7/4} 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 )}{21 \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}
Mathematica [C] time = 0.0135696, size = 54, normalized size = 0.37 \[ -\frac{b^2 \sqrt{a+\frac{b}{x^4}} \, _2F_1\left (-\frac{5}{2},-\frac{7}{4};-\frac{3}{4};-\frac{a x^4}{b}\right )}{7 x^5 \sqrt{\frac{a x^4}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 181, normalized size = 1.2 \begin{align*}{\frac{{x}^{3}}{21\, \left ( a{x}^{4}+b \right ) ^{3}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( 7\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{12}{a}^{3}+40\,b{a}^{2}\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}^{7}-9\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{8}{a}^{2}b-19\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{4}a{b}^{2}-3\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{3} \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{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} 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 (\frac{{\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.97254, size = 44, normalized size = 0.3 \begin{align*} - \frac{a^{\frac{5}{2}} x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{3}{4} \\ \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{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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