Optimal. Leaf size=262 \[ \frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{8 a^{7/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{7/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt{a+\frac{b}{x^4}}}{4 a^2 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{1}{4 a^2 x^3 \sqrt{a+\frac{b}{x^4}}}-\frac{1}{6 a x^3 \left (a+\frac{b}{x^4}\right )^{3/2}} \]
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Rubi [A] time = 0.133328, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {335, 290, 305, 220, 1196} \[ \frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{8 a^{7/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{7/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt{a+\frac{b}{x^4}}}{4 a^2 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{1}{4 a^2 x^3 \sqrt{a+\frac{b}{x^4}}}-\frac{1}{6 a x^3 \left (a+\frac{b}{x^4}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 335
Rule 290
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^4}\right )^{5/2} x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^4\right )^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{6 a \left (a+\frac{b}{x^4}\right )^{3/2} x^3}-\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^4\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{1}{6 a \left (a+\frac{b}{x^4}\right )^{3/2} x^3}-\frac{1}{4 a^2 \sqrt{a+\frac{b}{x^4}} x^3}+\frac{\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{4 a^2}\\ &=-\frac{1}{6 a \left (a+\frac{b}{x^4}\right )^{3/2} x^3}-\frac{1}{4 a^2 \sqrt{a+\frac{b}{x^4}} x^3}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{4 a^{3/2} \sqrt{b}}-\frac{\operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{4 a^{3/2} \sqrt{b}}\\ &=-\frac{1}{6 a \left (a+\frac{b}{x^4}\right )^{3/2} x^3}-\frac{1}{4 a^2 \sqrt{a+\frac{b}{x^4}} x^3}+\frac{\sqrt{a+\frac{b}{x^4}}}{4 a^2 \sqrt{b} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) x}-\frac{\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 )}{4 a^{7/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{\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^{7/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}
Mathematica [C] time = 0.033896, size = 77, normalized size = 0.29 \[ \frac{x \left (a x^4+b\right ) \sqrt{\frac{a x^4}{b}+1} \, _2F_1\left (\frac{3}{4},\frac{5}{2};\frac{7}{4};-\frac{a x^4}{b}\right )-b x}{3 a b \sqrt{a+\frac{b}{x^4}} \left (a x^4+b\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 502, normalized size = 1.9 \begin{align*}{\frac{1}{12\,{x}^{10}} \left ( 3\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{a}^{7/2}\sqrt{b}{x}^{11}+3\,i{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \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}^{8}{a}^{3}b+4\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{a}^{5/2}{b}^{3/2}{x}^{7}-3\,i\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 ){x}^{8}{a}^{3}b+6\,i{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \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}^{4}{a}^{2}{b}^{2}+\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{a}^{{\frac{3}{2}}}{b}^{{\frac{5}{2}}}{x}^{3}-6\,i\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 ){x}^{4}{a}^{2}{b}^{2}+3\,i{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \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}}}}}a{b}^{3}-3\,i\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 ) a{b}^{3} \right ){a}^{-{\frac{5}{2}}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{2}}}{b}^{-{\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{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} 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^{8} \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 = 2.8304, size = 39, normalized size = 0.15 \begin{align*} - \frac{\Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{5}{2}} x^{3} \Gamma \left (\frac{7}{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{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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