3.645 \(\int \frac{\sqrt{c x}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{(c x)^{3/2}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{6^{3/4} a \sqrt{x} \sqrt{3 a-2 a x^2}} \]

[Out]

(c*x)^(3/2)/(3*a*c*Sqrt[3*a - 2*a*x^2]) + (Sqrt[c*x]*Sqrt[3 - 2*x^2]*EllipticE[A
rcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(6^(3/4)*a*Sqrt[x]*Sqrt[3*a - 2*a*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.141017, antiderivative size = 101, 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 \[ \frac{(c x)^{3/2}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{6^{3/4} a \sqrt{x} \sqrt{3 a-2 a x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*x)^(3/2)/(3*a*c*Sqrt[3*a - 2*a*x^2]) + (Sqrt[c*x]*Sqrt[3 - 2*x^2]*EllipticE[A
rcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(6^(3/4)*a*Sqrt[x]*Sqrt[3*a - 2*a*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 51.4448, size = 170, normalized size = 1.68 \[ - \frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c} \sqrt{- \frac{2 x^{2}}{3} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{6 a \sqrt{- 2 a x^{2} + 3 a}} + \frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{6 a \sqrt{- 2 a x^{2} + 3 a}} + \frac{\left (c x\right )^{\frac{3}{2}}}{3 a c \sqrt{- 2 a x^{2} + 3 a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(1/2)/(-2*a*x**2+3*a)**(3/2),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)/(6*a*sqrt(-2*a*x**2 + 3*a)) + 2**(1/4)*3**(3/4)*sqr
t(c)*sqrt(-2*x**2/3 + 1)*elliptic_f(asin(2**(1/4)*3**(3/4)*sqrt(c*x)/(3*sqrt(c))
), -1)/(6*a*sqrt(-2*a*x**2 + 3*a)) + (c*x)**(3/2)/(3*a*c*sqrt(-2*a*x**2 + 3*a))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0797538, size = 107, normalized size = 1.06 \[ \frac{\sqrt{c x} \left (2 x^{3/2}+\sqrt [4]{6} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )-\sqrt [4]{6} \sqrt{3-2 x^2} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )\right )}{6 a \sqrt{x} \sqrt{a \left (3-2 x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.023, size = 227, normalized size = 2.3 \[ -{\frac{1}{72\,{a}^{2}x \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 2\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}-\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}+24\,{x}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/72*(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)+24*x^2)/a^2/x/(2*x^2-3)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 5.1292, size = 51, normalized size = 0.5 \[ \frac{\sqrt{3} \sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*sqrt(c)*x**(3/2)*gamma(3/4)*hyper((3/4, 3/2), (7/4,), 2*x**2*exp_polar(2
*I*pi)/3)/(18*a**(3/2)*gamma(7/4))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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