Optimal. Leaf size=272 \[ -\frac{4 a^{9/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{3 \sqrt{a+\frac{b}{x^4}}}-\frac{8 a^2 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{3 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}+\frac{8 a^{9/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+\frac{b}{x^4}}}+x \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{9 x^3}-\frac{4 a b \sqrt{a+\frac{b}{x^4}}}{3 x^3} \]
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Rubi [A] time = 0.153864, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {242, 277, 279, 305, 220, 1196} \[ -\frac{8 a^2 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{3 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{4 a^{9/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+\frac{b}{x^4}}}+\frac{8 a^{9/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+\frac{b}{x^4}}}+x \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{9 x^3}-\frac{4 a b \sqrt{a+\frac{b}{x^4}}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 242
Rule 277
Rule 279
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^4}\right )^{5/2} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (a+b x^4\right )^{5/2}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\left (a+\frac{b}{x^4}\right )^{5/2} x-(10 b) \operatorname{Subst}\left (\int x^2 \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}}{9 x^3}+\left (a+\frac{b}{x^4}\right )^{5/2} x-\frac{1}{3} (20 a b) \operatorname{Subst}\left (\int x^2 \sqrt{a+b x^4} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 a b \sqrt{a+\frac{b}{x^4}}}{3 x^3}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{9 x^3}+\left (a+\frac{b}{x^4}\right )^{5/2} x-\frac{1}{3} \left (8 a^2 b\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 a b \sqrt{a+\frac{b}{x^4}}}{3 x^3}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{9 x^3}+\left (a+\frac{b}{x^4}\right )^{5/2} x-\frac{1}{3} \left (8 a^{5/2} \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )+\frac{1}{3} \left (8 a^{5/2} \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 a b \sqrt{a+\frac{b}{x^4}}}{3 x^3}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{9 x^3}-\frac{8 a^2 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{3 \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) x}+\left (a+\frac{b}{x^4}\right )^{5/2} x+\frac{8 a^{9/4} \sqrt [4]{b} \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 ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+\frac{b}{x^4}}}-\frac{4 a^{9/4} \sqrt [4]{b} \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{a+\frac{b}{x^4}}}\\ \end{align*}
Mathematica [C] time = 0.0117494, size = 54, normalized size = 0.2 \[ -\frac{b^2 \sqrt{a+\frac{b}{x^4}} \, _2F_1\left (-\frac{5}{2},-\frac{9}{4};-\frac{5}{4};-\frac{a x^4}{b}\right )}{9 x^7 \sqrt{\frac{a x^4}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 250, normalized size = 0.9 \begin{align*} -{\frac{x}{9\, \left ( a{x}^{4}+b \right ) ^{3}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( -24\,i{a}^{{\frac{5}{2}}}\sqrt{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}}}}}{x}^{9}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +24\,i{a}^{{\frac{5}{2}}}\sqrt{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}}}}}{x}^{9}{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +15\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{12}{a}^{3}+19\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{8}{a}^{2}b+5\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{4}a{b}^{2}+\sqrt{{i\sqrt{a}{\frac{1}{\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}}\,{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^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.44757, size = 42, normalized size = 0.15 \begin{align*} - \frac{a^{\frac{5}{2}} x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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