Optimal. Leaf size=124 \[ \frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt{a-c x^4}}+\frac{a^{3/4} e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{a-c x^4}} \]
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Rubi [A] time = 0.0888635, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1201, 224, 221, 1200, 1199, 424} \[ \frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{a-c x^4}}+\frac{a^{3/4} e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{a-c x^4}} \]
Antiderivative was successfully verified.
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Rule 1201
Rule 224
Rule 221
Rule 1200
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{d+e x^2}{\sqrt{a-c x^4}} \, dx &=\frac{\left (\sqrt{a} e\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a-c x^4}} \, dx}{\sqrt{c}}+\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{1}{\sqrt{a-c x^4}} \, dx\\ &=\frac{\left (\sqrt{a} e \sqrt{1-\frac{c x^4}{a}}\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{1-\frac{c x^4}{a}}} \, dx}{\sqrt{c} \sqrt{a-c x^4}}+\frac{\left (\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \sqrt{1-\frac{c x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{c x^4}{a}}} \, dx}{\sqrt{a-c x^4}}\\ &=\frac{\sqrt [4]{a} \left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{a-c x^4}}+\frac{\left (\sqrt{a} e \sqrt{1-\frac{c x^4}{a}}\right ) \int \frac{\sqrt{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}}{\sqrt{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}} \, dx}{\sqrt{c} \sqrt{a-c x^4}}\\ &=\frac{a^{3/4} e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{a-c x^4}}+\frac{\sqrt [4]{a} \left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{a-c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0312186, size = 77, normalized size = 0.62 \[ \frac{\sqrt{1-\frac{c x^4}{a}} \left (3 d x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{c x^4}{a}\right )+e x^3 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{c x^4}{a}\right )\right )}{3 \sqrt{a-c x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 154, normalized size = 1.2 \begin{align*} -{e\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}}}}+{d\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}}}} \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{e x^{2} + d}{\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{\sqrt{-c x^{4} + a}{\left (e x^{2} + d\right )}}{c x^{4} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.70771, size = 82, normalized size = 0.66 \begin{align*} \frac{d 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} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} + \frac{e x^{3} \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} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{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{e x^{2} + d}{\sqrt{-c x^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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