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

Optimal. Leaf size=54 \[ \frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{c x^4-a}} \]

[Out]

(a^(3/4)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4
)*Sqrt[-a + c*x^4])

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Rubi [A]  time = 0.135692, antiderivative size = 54, 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 \[ \frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{c x^4-a}} \]

Antiderivative was successfully verified.

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

[Out]

(a^(3/4)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4
)*Sqrt[-a + c*x^4])

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Rubi in Sympy [A]  time = 25.5978, size = 48, normalized size = 0.89 \[ \frac{a^{\frac{3}{4}} \sqrt{1 - \frac{c x^{4}}{a}} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{\sqrt [4]{c} \sqrt{- a + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/4)*sqrt(1 - c*x**4/a)*elliptic_e(asin(c**(1/4)*x/a**(1/4)), -1)/(c**(1/4)*
sqrt(-a + c*x**4))

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Mathematica [C]  time = 0.0976819, size = 78, normalized size = 1.44 \[ \frac{i \sqrt{c} \sqrt{1-\frac{c x^4}{a}} E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\left (-\frac{\sqrt{c}}{\sqrt{a}}\right )^{3/2} \sqrt{c x^4-a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.067, size = 158, normalized size = 2.9 \[{1\sqrt{a}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}}+{1\sqrt{a}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)*EllipticF(x*(-1/a^(1/2)*c^(1/2))^(1/2),I)+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)*(EllipticF(x*(-1/a^(1/2)*c^(1/2))^(1/2),I)-Ellip
ticE(x*(-1/a^(1/2)*c^(1/2))^(1/2),I))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.29037, size = 70, normalized size = 1.3 \[ - \frac{i x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4}}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} - \frac{i \sqrt{c} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4}}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError