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

Optimal. Leaf size=117 \[ \frac{6^{3/4} a c^{3/2} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{7 \sqrt{a \left (3-2 x^2\right )}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{5/2}}{7 c}-\frac{2}{7} c \sqrt{3 a-2 a x^2} \sqrt{c x} \]

[Out]

(-2*c*Sqrt[c*x]*Sqrt[3*a - 2*a*x^2])/7 + (2*(c*x)^(5/2)*Sqrt[3*a - 2*a*x^2])/(7*
c) + (6^(3/4)*a*c^(3/2)*Sqrt[3 - 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c*x])
/Sqrt[c]], -1])/(7*Sqrt[a*(3 - 2*x^2)])

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Rubi [A]  time = 0.195791, antiderivative size = 117, 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 \[ \frac{6^{3/4} a c^{3/2} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{7 \sqrt{a \left (3-2 x^2\right )}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{5/2}}{7 c}-\frac{2}{7} c \sqrt{3 a-2 a x^2} \sqrt{c x} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*Sqrt[c*x]*Sqrt[3*a - 2*a*x^2])/7 + (2*(c*x)^(5/2)*Sqrt[3*a - 2*a*x^2])/(7*
c) + (6^(3/4)*a*c^(3/2)*Sqrt[3 - 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c*x])
/Sqrt[c]], -1])/(7*Sqrt[a*(3 - 2*x^2)])

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Rubi in Sympy [A]  time = 15.3583, size = 124, normalized size = 1.06 \[ \frac{3 \cdot 2^{\frac{3}{4}} \sqrt [4]{3} a c^{\frac{3}{2}} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{7 \sqrt{- 2 a x^{2} + 3 a}} - \frac{2 c \sqrt{c x} \sqrt{- 2 a x^{2} + 3 a}}{7} + \frac{2 \left (c x\right )^{\frac{5}{2}} \sqrt{- 2 a x^{2} + 3 a}}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*2**(3/4)*3**(1/4)*a*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)/(7*sqrt(-2*a*x**2 + 3*a)) - 2*c*sqrt(c*x)*sqrt(
-2*a*x**2 + 3*a)/7 + 2*(c*x)**(5/2)*sqrt(-2*a*x**2 + 3*a)/(7*c)

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Mathematica [A]  time = 0.176619, size = 86, normalized size = 0.74 \[ \frac{\sqrt{a \left (3-2 x^2\right )} (c x)^{3/2} \left (2 \sqrt{2-\frac{3}{x^2}} \left (x^2-1\right ) x^{3/2}+6^{3/4} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{3}{2}}}{\sqrt{x}}\right )\right |-1\right )\right )}{7 \sqrt{2-\frac{3}{x^2}} x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((c*x)^(3/2)*Sqrt[a*(3 - 2*x^2)]*(2*Sqrt[2 - 3/x^2]*x^(3/2)*(-1 + x^2) + 6^(3/4)
*EllipticF[ArcSin[(3/2)^(1/4)/Sqrt[x]], -1]))/(7*Sqrt[2 - 3/x^2]*x^(5/2))

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Maple [A]  time = 0.041, size = 133, normalized size = 1.1 \[ -{\frac{c}{14\,x \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( -8\,{x}^{5}+\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}+20\,{x}^{3}-12\,x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/14*c*(c*x)^(1/2)*(-a*(2*x^2-3))^(1/2)*(-8*x^5+((-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)+20*x^3-12*x)/x/(2*x^2-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 21.2369, size = 53, normalized size = 0.45 \[ \frac{\sqrt{3} \sqrt{a} c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*sqrt(a)*c**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), 2*x**2*e
xp_polar(2*I*pi)/3)/(2*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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