Optimal. Leaf size=52 \[ \frac{\sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{c}{a}} x\right )\right |-1\right )}{\sqrt [4]{\frac{c}{a}} \sqrt{c x^4-a}} \]
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Rubi [A] time = 0.0463449, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {1200, 1199, 424} \[ \frac{\sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{c}{a}} x\right )\right |-1\right )}{\sqrt [4]{\frac{c}{a}} \sqrt{c x^4-a}} \]
Antiderivative was successfully verified.
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Rule 1200
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{1+\sqrt{\frac{c}{a}} x^2}{\sqrt{-a+c x^4}} \, dx &=\frac{\sqrt{1-\frac{c x^4}{a}} \int \frac{1+\sqrt{\frac{c}{a}} x^2}{\sqrt{1-\frac{c x^4}{a}}} \, dx}{\sqrt{-a+c x^4}}\\ &=\frac{\sqrt{1-\frac{c x^4}{a}} \int \frac{\sqrt{1+\sqrt{\frac{c}{a}} x^2}}{\sqrt{1-\sqrt{\frac{c}{a}} x^2}} \, dx}{\sqrt{-a+c x^4}}\\ &=\frac{\sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{c}{a}} x\right )\right |-1\right )}{\sqrt [4]{\frac{c}{a}} \sqrt{-a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0305537, size = 85, normalized size = 1.63 \[ \frac{\sqrt{1-\frac{c x^4}{a}} \left (x^3 \sqrt{\frac{c}{a}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{c x^4}{a}\right )+3 x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{c x^4}{a}\right )\right )}{3 \sqrt{c x^4-a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.071, size = 165, normalized size = 3.2 \begin{align*}{\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}}+{\sqrt{{\frac{c}{a}}}\sqrt{a}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}{\frac{1}{\sqrt{c}}}} \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{x^{2} \sqrt{\frac{c}{a}} + 1}{\sqrt{c 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{x^{2} \sqrt{\frac{c}{a}} + 1}{\sqrt{c x^{4} - a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.75264, size = 76, normalized size = 1.46 \begin{align*} - \frac{i x^{3} \sqrt{\frac{c}{a}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4}}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} - \frac{i 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{c x^{4}}{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{x^{2} \sqrt{\frac{c}{a}} + 1}{\sqrt{c x^{4} - a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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