3.639 \(\int \frac{\sqrt{c x}}{\sqrt{3 a-2 a x^2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{\sqrt [4]{6} \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{\sqrt{x} \sqrt{3 a-2 a x^2}} \]

[Out]

-((6^(1/4)*Sqrt[c*x]*Sqrt[3 - 2*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6
]], 2])/(Sqrt[x]*Sqrt[3*a - 2*a*x^2]))

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Rubi [A]  time = 0.101799, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{\sqrt [4]{6} \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{\sqrt{x} \sqrt{3 a-2 a x^2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*x]/Sqrt[3*a - 2*a*x^2],x]

[Out]

-((6^(1/4)*Sqrt[c*x]*Sqrt[3 - 2*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6
]], 2])/(Sqrt[x]*Sqrt[3*a - 2*a*x^2]))

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Rubi in Sympy [A]  time = 44.3939, size = 138, normalized size = 2.06 \[ \frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c} \sqrt{- \frac{2 x^{2}}{3} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{\sqrt{- 2 a x^{2} + 3 a}} - \frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{\sqrt{- 2 a x^{2} + 3 a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(1/2)/(-2*a*x**2+3*a)**(1/2),x)

[Out]

2**(1/4)*3**(3/4)*sqrt(c)*sqrt(-2*x**2/3 + 1)*elliptic_e(asin(2**(1/4)*3**(3/4)*
sqrt(c*x)/(3*sqrt(c))), -1)/sqrt(-2*a*x**2 + 3*a) - 2**(1/4)*3**(3/4)*sqrt(c)*sq
rt(-2*x**2/3 + 1)*elliptic_f(asin(2**(1/4)*3**(3/4)*sqrt(c*x)/(3*sqrt(c))), -1)/
sqrt(-2*a*x**2 + 3*a)

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Mathematica [A]  time = 0.0965715, size = 77, normalized size = 1.15 \[ \frac{\sqrt [4]{6} \sqrt{3-2 x^2} \sqrt{c x} \left (E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )-F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )\right )}{\sqrt{x} \sqrt{a \left (3-2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c*x]/Sqrt[3*a - 2*a*x^2],x]

[Out]

(6^(1/4)*Sqrt[c*x]*Sqrt[3 - 2*x^2]*(EllipticE[ArcSin[(2/3)^(1/4)*Sqrt[x]], -1] -
EllipticF[ArcSin[(2/3)^(1/4)*Sqrt[x]], -1]))/(Sqrt[x]*Sqrt[a*(3 - 2*x^2)])

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Maple [B]  time = 0.02, size = 165, normalized size = 2.5 \[{\frac{\sqrt{3}\sqrt{2}}{12\,ax \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) }\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{-x\sqrt{3}\sqrt{2}} \left ( 2\,{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}},{\frac{\sqrt{2}}{2}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(1/2)/(-2*a*x^2+3*a)^(1/2),x)

[Out]

1/12*(c*x)^(1/2)*(-a*(2*x^2-3))^(1/2)*2^(1/2)*((2*x+3^(1/2)*2^(1/2))*3^(1/2)*2^(
1/2))^(1/2)*((-2*x+3^(1/2)*2^(1/2))*3^(1/2)*2^(1/2))^(1/2)*3^(1/2)*(-x*3^(1/2)*2
^(1/2))^(1/2)*(2*EllipticE(1/6*3^(1/2)*2^(1/2)*((2*x+3^(1/2)*2^(1/2))*3^(1/2)*2^
(1/2))^(1/2),1/2*2^(1/2))-EllipticF(1/6*3^(1/2)*2^(1/2)*((2*x+3^(1/2)*2^(1/2))*3
^(1/2)*2^(1/2))^(1/2),1/2*2^(1/2)))/x/a/(2*x^2-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{\sqrt{-2 \, a x^{2} + 3 \, a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x)/sqrt(-2*a*x^2 + 3*a),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x)/sqrt(-2*a*x^2 + 3*a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x}}{\sqrt{-2 \, a x^{2} + 3 \, a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x)/sqrt(-2*a*x^2 + 3*a),x, algorithm="fricas")

[Out]

integral(sqrt(c*x)/sqrt(-2*a*x^2 + 3*a), x)

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Sympy [A]  time = 2.54772, size = 51, normalized size = 0.76 \[ \frac{\sqrt{3} \sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(1/2)/(-2*a*x**2+3*a)**(1/2),x)

[Out]

sqrt(3)*sqrt(c)*x**(3/2)*gamma(3/4)*hyper((1/2, 3/4), (7/4,), 2*x**2*exp_polar(2
*I*pi)/3)/(6*sqrt(a)*gamma(7/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{\sqrt{-2 \, a x^{2} + 3 \, a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x)/sqrt(-2*a*x^2 + 3*a),x, algorithm="giac")

[Out]

integrate(sqrt(c*x)/sqrt(-2*a*x^2 + 3*a), x)