3.171 \(\int \frac{1+\sqrt{\frac{c}{a}} x^2}{\sqrt{-a+c x^4}} \, dx\)

Optimal. Leaf size=52 \[ \frac{\sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{c}{a}} x\right )\right |-1\right )}{\sqrt [4]{\frac{c}{a}} \sqrt{c x^4-a}} \]

[Out]

(Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c/a)^(1/4)*x], -1])/((c/a)^(1/4)*Sqrt[-a
+ c*x^4])

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Rubi [A]  time = 0.134496, 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 \[ \frac{\sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{c}{a}} x\right )\right |-1\right )}{\sqrt [4]{\frac{c}{a}} \sqrt{c x^4-a}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + Sqrt[c/a]*x^2)/Sqrt[-a + c*x^4],x]

[Out]

(Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c/a)^(1/4)*x], -1])/((c/a)^(1/4)*Sqrt[-a
+ c*x^4])

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Rubi in Sympy [A]  time = 18.4757, size = 41, normalized size = 0.79 \[ \frac{\sqrt{1 - \frac{c x^{4}}{a}} E\left (\operatorname{asin}{\left (x \sqrt [4]{\frac{c}{a}} \right )}\middle | -1\right )}{\sqrt [4]{\frac{c}{a}} \sqrt{- a + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x**2*(c/a)**(1/2))/(c*x**4-a)**(1/2),x)

[Out]

sqrt(1 - c*x**4/a)*elliptic_e(asin(x*(c/a)**(1/4)), -1)/((c/a)**(1/4)*sqrt(-a +
c*x**4))

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Mathematica [C]  time = 0.149101, size = 142, normalized size = 2.73 \[ \frac{i \sqrt{1-\frac{c x^4}{a}} \left (\left (\sqrt{c}-\sqrt{a} \sqrt{\frac{c}{a}}\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{a} \sqrt{\frac{c}{a}} E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\sqrt{a} \left (-\frac{\sqrt{c}}{\sqrt{a}}\right )^{3/2} \sqrt{c x^4-a}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Sqrt[c/a]*x^2)/Sqrt[-a + c*x^4],x]

[Out]

(I*Sqrt[1 - (c*x^4)/a]*(Sqrt[a]*Sqrt[c/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqr
t[a])]*x], -1] + (Sqrt[c] - Sqrt[a]*Sqrt[c/a])*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c
]/Sqrt[a])]*x], -1]))/(Sqrt[a]*(-(Sqrt[c]/Sqrt[a]))^(3/2)*Sqrt[-a + c*x^4])

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Maple [B]  time = 0.042, size = 165, normalized size = 3.2 \[{1\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}}+{1\sqrt{{\frac{c}{a}}}\sqrt{a}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x^2*(c/a)^(1/2))/(c*x^4-a)^(1/2),x)

[Out]

1/(-1/a^(1/2)*c^(1/2))^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1-1/a^(1/2)*c^(1/2
)*x^2)^(1/2)/(c*x^4-a)^(1/2)*EllipticF(x*(-1/a^(1/2)*c^(1/2))^(1/2),I)+(c/a)^(1/
2)*a^(1/2)/(-1/a^(1/2)*c^(1/2))^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1-1/a^(1/
2)*c^(1/2)*x^2)^(1/2)/(c*x^4-a)^(1/2)/c^(1/2)*(EllipticF(x*(-1/a^(1/2)*c^(1/2))^
(1/2),I)-EllipticE(x*(-1/a^(1/2)*c^(1/2))^(1/2),I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \sqrt{\frac{c}{a}} + 1}{\sqrt{c x^{4} - a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2*sqrt(c/a) + 1)/sqrt(c*x^4 - a),x, algorithm="maxima")

[Out]

integrate((x^2*sqrt(c/a) + 1)/sqrt(c*x^4 - a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2} \sqrt{\frac{c}{a}} + 1}{\sqrt{c x^{4} - a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2*sqrt(c/a) + 1)/sqrt(c*x^4 - a),x, algorithm="fricas")

[Out]

integral((x^2*sqrt(c/a) + 1)/sqrt(c*x^4 - a), x)

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Sympy [A]  time = 4.39726, size = 76, normalized size = 1.46 \[ - \frac{i x^{3} \sqrt{\frac{c}{a}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4}}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} - \frac{i x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4}}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x**2*(c/a)**(1/2))/(c*x**4-a)**(1/2),x)

[Out]

-I*x**3*sqrt(c/a)*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x**4/a)/(4*sqrt(a)*gamm
a(7/4)) - I*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4/a)/(4*sqrt(a)*gamma(5/
4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \sqrt{\frac{c}{a}} + 1}{\sqrt{c x^{4} - a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2*sqrt(c/a) + 1)/sqrt(c*x^4 - a),x, algorithm="giac")

[Out]

integrate((x^2*sqrt(c/a) + 1)/sqrt(c*x^4 - a), x)