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

Optimal. Leaf size=94 \[ \frac{c \sqrt{c x}}{2 a \sqrt{3 a-2 a x^2}}-\frac{c^{3/2} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{2 \sqrt [4]{6} a \sqrt{a \left (3-2 x^2\right )}} \]

[Out]

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

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Rubi [A]  time = 0.139715, antiderivative size = 94, 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 \[ \frac{c \sqrt{c x}}{2 a \sqrt{3 a-2 a x^2}}-\frac{c^{3/2} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{2 \sqrt [4]{6} a \sqrt{a \left (3-2 x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.7823, size = 97, normalized size = 1.03 \[ - \frac{2^{\frac{3}{4}} \sqrt [4]{3} c^{\frac{3}{2}} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{4 a \sqrt{- 2 a x^{2} + 3 a}} + \frac{c \sqrt{c x}}{2 a \sqrt{- 2 a x^{2} + 3 a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.119088, size = 91, normalized size = 0.97 \[ \frac{(c x)^{3/2} \left (6^{3/4} \left (2 x^2-3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{3}{2}}}{\sqrt{x}}\right )\right |-1\right )+6 \sqrt{2-\frac{3}{x^2}} x^{3/2}\right )}{12 a \sqrt{2-\frac{3}{x^2}} x^{5/2} \sqrt{a \left (3-2 x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

((c*x)^(3/2)*(6*Sqrt[2 - 3/x^2]*x^(3/2) + 6^(3/4)*(-3 + 2*x^2)*EllipticF[ArcSin[
(3/2)^(1/4)/Sqrt[x]], -1]))/(12*a*Sqrt[2 - 3/x^2]*x^(5/2)*Sqrt[a*(3 - 2*x^2)])

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Maple [A]  time = 0.046, size = 126, normalized size = 1.3 \[{\frac{c}{24\,{a}^{2}x \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( \sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}-12\,x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.8282, size = 51, normalized size = 0.54 \[ \frac{\sqrt{3} c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*c**(3/2)*x**(5/2)*gamma(5/4)*hyper((5/4, 3/2), (9/4,), 2*x**2*exp_polar(
2*I*pi)/3)/(18*a**(3/2)*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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