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

Optimal. Leaf size=99 \[ \frac{2 \sqrt{3 a-2 a x^2} (c x)^{3/2}}{5 c}-\frac{6 \sqrt [4]{6} a \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{5 \sqrt{x} \sqrt{3 a-2 a x^2}} \]

[Out]

(2*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2])/(5*c) - (6*6^(1/4)*a*Sqrt[c*x]*Sqrt[3 - 2*x^
2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(5*Sqrt[x]*Sqrt[3*a - 2*a*
x^2])

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Rubi [A]  time = 0.136814, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{2 \sqrt{3 a-2 a x^2} (c x)^{3/2}}{5 c}-\frac{6 \sqrt [4]{6} a \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{5 \sqrt{x} \sqrt{3 a-2 a x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2])/(5*c) - (6*6^(1/4)*a*Sqrt[c*x]*Sqrt[3 - 2*x^
2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(5*Sqrt[x]*Sqrt[3*a - 2*a*
x^2])

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Rubi in Sympy [A]  time = 50.228, size = 173, normalized size = 1.75 \[ \frac{6 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} a \sqrt{c} \sqrt{- \frac{2 x^{2}}{3} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{5 \sqrt{- 2 a x^{2} + 3 a}} - \frac{6 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} a \sqrt{c} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{5 \sqrt{- 2 a x^{2} + 3 a}} + \frac{2 \left (c x\right )^{\frac{3}{2}} \sqrt{- 2 a x^{2} + 3 a}}{5 c} \]

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]

6*2**(1/4)*3**(3/4)*a*sqrt(c)*sqrt(-2*x**2/3 + 1)*elliptic_e(asin(2**(1/4)*3**(3
/4)*sqrt(c*x)/(3*sqrt(c))), -1)/(5*sqrt(-2*a*x**2 + 3*a)) - 6*2**(1/4)*3**(3/4)*
a*sqrt(c)*sqrt(-2*x**2/3 + 1)*elliptic_f(asin(2**(1/4)*3**(3/4)*sqrt(c*x)/(3*sqr
t(c))), -1)/(5*sqrt(-2*a*x**2 + 3*a)) + 2*(c*x)**(3/2)*sqrt(-2*a*x**2 + 3*a)/(5*
c)

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Mathematica [A]  time = 0.0959984, size = 106, normalized size = 1.07 \[ \frac{2}{5} x \sqrt{a \left (3-2 x^2\right )} \sqrt{c x}+\frac{6 \sqrt [4]{6} \sqrt{a \left (3-2 x^2\right )} \sqrt{c x} \left (E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )-F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )\right )}{5 \sqrt{3-2 x^2} \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[c*x]*Sqrt[a*(3 - 2*x^2)])/5 + (6*6^(1/4)*Sqrt[c*x]*Sqrt[a*(3 - 2*x^2)]
*(EllipticE[ArcSin[(2/3)^(1/4)*Sqrt[x]], -1] - EllipticF[ArcSin[(2/3)^(1/4)*Sqrt
[x]], -1]))/(5*Sqrt[x]*Sqrt[3 - 2*x^2])

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Maple [B]  time = 0.036, size = 229, normalized size = 2.3 \[{\frac{1}{10\,x \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 2\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}-\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}+8\,{x}^{4}-12\,{x}^{2} \right ) } \]

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]

1/10*(c*x)^(1/2)*(-a*(2*x^2-3))^(1/2)*(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)*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))*((2*x+3^(1/2)*2^(1/2))*3^(1/
2)*2^(1/2))^(1/2)*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)*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))*((2*x+3^(1/2)*2^(1/2))*3^(1/2)*2^(1/2))^(1/
2)*2^(1/2)+8*x^4-12*x^2)/x/(2*x^2-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.20509, size = 53, normalized size = 0.54 \[ \frac{\sqrt{3} \sqrt{a} \sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 \Gamma \left (\frac{7}{4}\right )} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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