Optimal. Leaf size=261 \[ -\frac{3 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{10 a^{7/4} \sqrt{a+b x^4}}-\frac{3 b^{3/2} x \sqrt{a+b x^4}}{5 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{7/4} \sqrt{a+b x^4}}+\frac{3 b \sqrt{a+b x^4}}{5 a^2 x}-\frac{\sqrt{a+b x^4}}{5 a x^5} \]
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Rubi [A] time = 0.0921261, antiderivative size = 261, 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 = {325, 305, 220, 1196} \[ -\frac{3 b^{3/2} x \sqrt{a+b x^4}}{5 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{7/4} \sqrt{a+b x^4}}+\frac{3 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{7/4} \sqrt{a+b x^4}}+\frac{3 b \sqrt{a+b x^4}}{5 a^2 x}-\frac{\sqrt{a+b x^4}}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 325
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{x^6 \sqrt{a+b x^4}} \, dx &=-\frac{\sqrt{a+b x^4}}{5 a x^5}-\frac{(3 b) \int \frac{1}{x^2 \sqrt{a+b x^4}} \, dx}{5 a}\\ &=-\frac{\sqrt{a+b x^4}}{5 a x^5}+\frac{3 b \sqrt{a+b x^4}}{5 a^2 x}-\frac{\left (3 b^2\right ) \int \frac{x^2}{\sqrt{a+b x^4}} \, dx}{5 a^2}\\ &=-\frac{\sqrt{a+b x^4}}{5 a x^5}+\frac{3 b \sqrt{a+b x^4}}{5 a^2 x}-\frac{\left (3 b^{3/2}\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{5 a^{3/2}}+\frac{\left (3 b^{3/2}\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{5 a^{3/2}}\\ &=-\frac{\sqrt{a+b x^4}}{5 a x^5}+\frac{3 b \sqrt{a+b x^4}}{5 a^2 x}-\frac{3 b^{3/2} x \sqrt{a+b x^4}}{5 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{7/4} \sqrt{a+b x^4}}-\frac{3 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{7/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0087659, size = 51, normalized size = 0.2 \[ -\frac{\sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (-\frac{5}{4},\frac{1}{2};-\frac{1}{4};-\frac{b x^4}{a}\right )}{5 x^5 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 133, normalized size = 0.5 \begin{align*} -{\frac{1}{5\,a{x}^{5}}\sqrt{b{x}^{4}+a}}+{\frac{3\,b}{5\,{a}^{2}x}\sqrt{b{x}^{4}+a}}-{{\frac{3\,i}{5}}{b}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \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}{\sqrt{b x^{4} + a} x^{6}}\,{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{\sqrt{b x^{4} + a}}{b x^{10} + a x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.39197, size = 44, normalized size = 0.17 \begin{align*} \frac{\Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x^{5} \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{1}{\sqrt{b x^{4} + a} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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