Optimal. Leaf size=124 \[ \frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{a-c x^4}}+\frac{a^{3/4} e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{a-c x^4}} \]
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Rubi [A] time = 0.220254, 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 \[ \frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{a-c x^4}}+\frac{a^{3/4} e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{a-c x^4}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^2)/Sqrt[a - c*x^4],x]
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Rubi in Sympy [A] time = 36.2828, size = 112, normalized size = 0.9 \[ \frac{a^{\frac{3}{4}} e \sqrt{1 - \frac{c x^{4}}{a}} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{c^{\frac{3}{4}} \sqrt{a - c x^{4}}} - \frac{\sqrt [4]{a} \sqrt{1 - \frac{c x^{4}}{a}} \left (\sqrt{a} e - \sqrt{c} d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{c^{\frac{3}{4}} \sqrt{a - c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d)/(-c*x**4+a)**(1/2),x)
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Mathematica [C] time = 0.126764, size = 127, normalized size = 1.02 \[ \frac{i \sqrt{1-\frac{c x^4}{a}} \left (\left (\sqrt{c} d-\sqrt{a} e\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{a} e E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\sqrt{a} \left (-\frac{\sqrt{c}}{\sqrt{a}}\right )^{3/2} \sqrt{a-c x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^2)/Sqrt[a - c*x^4],x]
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Maple [A] time = 0.006, size = 154, normalized size = 1.2 \[{d\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+a}}}}-{e\sqrt{a}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+a}}}{\frac{1}{\sqrt{c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)/(-c*x^4+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{2} + d}{\sqrt{-c x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)/sqrt(-c*x^4 + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e x^{2} + d}{\sqrt{-c x^{4} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)/sqrt(-c*x^4 + a),x, algorithm="fricas")
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Sympy [A] time = 4.11344, size = 82, normalized size = 0.66 \[ \frac{d x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} + \frac{e x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)/(-c*x**4+a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{2} + d}{\sqrt{-c x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)/sqrt(-c*x^4 + a),x, algorithm="giac")
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