Optimal. Leaf size=73 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt{c x^4-a}} \]
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Rubi [A] time = 0.120486, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt{c x^4-a}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x^2)*Sqrt[-a + c*x^4]),x]
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Rubi in Sympy [A] time = 15.5972, size = 63, normalized size = 0.86 \[ \frac{\sqrt [4]{a} \sqrt{1 - \frac{c x^{4}}{a}} \Pi \left (- \frac{\sqrt{a} e}{\sqrt{c} d}; \operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{\sqrt [4]{c} d \sqrt{- a + c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x**2+d)/(c*x**4-a)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0704225, size = 92, normalized size = 1.26 \[ -\frac{i \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{d \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \sqrt{c x^4-a}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x^2)*Sqrt[-a + c*x^4]),x]
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Maple [A] time = 0.023, size = 99, normalized size = 1.4 \[{\frac{1}{d}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},{\frac{e}{d}\sqrt{a}{\frac{1}{\sqrt{c}}}},{1\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x^2+d)/(c*x^4-a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} - a}{\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 - a)*(e*x^2 + d)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{c x^{4} - a}{\left (e x^{2} + d\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 - a)*(e*x^2 + d)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- a + c x^{4}} \left (d + e x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x**2+d)/(c*x**4-a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} - a}{\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 - a)*(e*x^2 + d)),x, algorithm="giac")
[Out]