Optimal. Leaf size=53 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt{a-b x^4}} \]
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Rubi [A] time = 0.0113811, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {224, 221} \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
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Rule 224
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a-b x^4}} \, dx &=\frac{\sqrt{1-\frac{b x^4}{a}} \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{\sqrt{a-b x^4}}\\ &=\frac{\sqrt [4]{a} \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]{b} \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0341167, size = 72, normalized size = 1.36 \[ -\frac{i \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}}\right ),-1\right )}{\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 64, normalized size = 1.2 \begin{align*}{\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\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}}\,{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^{4} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.848854, size = 37, normalized size = 0.7 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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