Optimal. Leaf size=128 \[ \frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}-\frac{\sqrt{a-b x^4}}{a x}-\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}} \]
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Rubi [A] time = 0.0768875, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{\sqrt{a-b x^4}}{a x}+\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}-\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}} \]
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^2 \sqrt{a-b x^4}} \, dx &=-\frac{\sqrt{a-b x^4}}{a x}-\frac{b \int \frac{x^2}{\sqrt{a-b x^4}} \, dx}{a}\\ &=-\frac{\sqrt{a-b x^4}}{a x}+\frac{\sqrt{b} \int \frac{1}{\sqrt{a-b x^4}} \, dx}{\sqrt{a}}-\frac{\sqrt{b} \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a-b x^4}} \, dx}{\sqrt{a}}\\ &=-\frac{\sqrt{a-b x^4}}{a x}+\frac{\left (\sqrt{b} \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{\sqrt{a} \sqrt{a-b x^4}}-\frac{\left (\sqrt{b} \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}{\sqrt{a} \sqrt{a-b x^4}}\\ &=-\frac{\sqrt{a-b x^4}}{a x}+\frac{\sqrt [4]{b} \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 )}{\sqrt [4]{a} \sqrt{a-b x^4}}-\frac{\left (\sqrt{b} \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}{\sqrt{a} \sqrt{a-b x^4}}\\ &=-\frac{\sqrt{a-b x^4}}{a x}-\frac{\sqrt [4]{b} \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 )}{\sqrt [4]{a} \sqrt{a-b x^4}}+\frac{\sqrt [4]{b} \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 )}{\sqrt [4]{a} \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0092366, size = 50, normalized size = 0.39 \[ -\frac{\sqrt{1-\frac{b x^4}{a}} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{b x^4}{a}\right )}{x \sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 106, normalized size = 0.8 \begin{align*} -{\frac{1}{ax}\sqrt{-b{x}^{4}+a}}+{\sqrt{b}\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 ){\frac{1}{\sqrt{a}}}{\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^{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{\sqrt{-b x^{4} + a}}{b x^{6} - a x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.04219, size = 41, normalized size = 0.32 \begin{align*} \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} x \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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