3.646 \(\int \frac{1}{\sqrt{c x} \left (3 a-2 a x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=96 \[ \frac{\sqrt{c x}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{3 \sqrt [4]{6} a \sqrt{c} \sqrt{a \left (3-2 x^2\right )}} \]

[Out]

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

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Rubi [A]  time = 0.133548, antiderivative size = 96, 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 \[ \frac{\sqrt{c x}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{3 \sqrt [4]{6} a \sqrt{c} \sqrt{a \left (3-2 x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.5306, size = 97, normalized size = 1.01 \[ \frac{\sqrt{c x}}{3 a c \sqrt{- 2 a x^{2} + 3 a}} + \frac{2^{\frac{3}{4}} \sqrt [4]{3} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{6 a \sqrt{c} \sqrt{- 2 a x^{2} + 3 a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x)/(3*a*c*sqrt(-2*a*x**2 + 3*a)) + 2**(3/4)*3**(1/4)*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(c)*sqrt(
-2*a*x**2 + 3*a))

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Mathematica [A]  time = 0.101973, size = 92, normalized size = 0.96 \[ \frac{6 \sqrt{2-\frac{3}{x^2}} x^{3/2}-6^{3/4} \left (2 x^2-3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{3}{2}}}{\sqrt{x}}\right )\right |-1\right )}{18 a \sqrt{2-\frac{3}{x^2}} \sqrt{x} \sqrt{a \left (3-2 x^2\right )} \sqrt{c x}} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[2 - 3/x^2]*x^(3/2) - 6^(3/4)*(-3 + 2*x^2)*EllipticF[ArcSin[(3/2)^(1/4)/S
qrt[x]], -1])/(18*a*Sqrt[2 - 3/x^2]*Sqrt[x]*Sqrt[c*x]*Sqrt[a*(3 - 2*x^2)])

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Maple [A]  time = 0.034, size = 122, normalized size = 1.3 \[ -{\frac{1}{36\,{a}^{2} \left ( 2\,{x}^{2}-3 \right ) }\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( \sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}+12\,x \right ){\frac{1}{\sqrt{cx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*(-a*(2*x^2-3))^(1/2)*(((-2*x+3^(1/2)*2^(1/2))*3^(1/2)*2^(1/2))^(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)+12*
x)/a^2/(c*x)^(1/2)/(2*x^2-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac{3}{2}} \sqrt{c x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-2*a*x^2 + 3*a)^(3/2)*sqrt(c*x)),x, algorithm="maxima")

[Out]

integrate(1/((-2*a*x^2 + 3*a)^(3/2)*sqrt(c*x)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (2 \, a x^{2} - 3 \, a\right )} \sqrt{-2 \, a x^{2} + 3 \, a} \sqrt{c x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-2*a*x^2 + 3*a)^(3/2)*sqrt(c*x)),x, algorithm="fricas")

[Out]

integral(-1/((2*a*x^2 - 3*a)*sqrt(-2*a*x^2 + 3*a)*sqrt(c*x)), x)

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Sympy [A]  time = 8.6981, size = 51, normalized size = 0.53 \[ \frac{\sqrt{3} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac{3}{2}} \sqrt{c} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*sqrt(x)*gamma(1/4)*hyper((1/4, 3/2), (5/4,), 2*x**2*exp_polar(2*I*pi)/3)
/(18*a**(3/2)*sqrt(c)*gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac{3}{2}} \sqrt{c x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-2*a*x^2 + 3*a)^(3/2)*sqrt(c*x)),x, algorithm="giac")

[Out]

integrate(1/((-2*a*x^2 + 3*a)^(3/2)*sqrt(c*x)), x)