Optimal. Leaf size=158 \[ \frac{3 b^{5/4} \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{5 a^{5/4} \sqrt{a-b x^4}}-\frac{3 b^{5/4} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt{a-b x^4}}-\frac{3 b \sqrt{a-b x^4}}{5 a^2 x}-\frac{\sqrt{a-b x^4}}{5 a x^5} \]
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Rubi [A] time = 0.091197, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {325, 307, 224, 221, 1200, 1199, 424} \[ \frac{3 b^{5/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt{a-b x^4}}-\frac{3 b^{5/4} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt{a-b x^4}}-\frac{3 b \sqrt{a-b x^4}}{5 a^2 x}-\frac{\sqrt{a-b x^4}}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 325
Rule 307
Rule 224
Rule 221
Rule 1200
Rule 1199
Rule 424
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{\left (3 b^{3/2} \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{5 a^{3/2} \sqrt{a-b x^4}}-\frac{\left (3 b^{3/2} \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{5 a^{3/2} \sqrt{a-b x^4}}\\ &=-\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^{5/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt{a-b x^4}}-\frac{\left (3 b^{3/2} \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{\sqrt{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}}{\sqrt{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}} \, dx}{5 a^{3/2} \sqrt{a-b x^4}}\\ &=-\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^{5/4} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt{a-b x^4}}+\frac{3 b^{5/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0094924, size = 52, normalized size = 0.33 \[ -\frac{\sqrt{1-\frac{b x^4}{a}} \, _2F_1\left (-\frac{5}{4},\frac{1}{2};-\frac{1}{4};\frac{b x^4}{a}\right )}{5 x^5 \sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 126, normalized size = 0.8 \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}{5}{b}^{{\frac{3}{2}}}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{\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 [A] time = 1.43258, size = 39, normalized size = 0.25 \begin{align*} - \frac{i \Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 \sqrt{b} x^{7} \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 \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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