3.637 \(\int \frac{(c x)^{5/2}}{\sqrt{3 a-2 a x^2}} \, dx\)

Optimal. Leaf size=107 \[ -\frac{9 \sqrt [4]{3} c^2 \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{5\ 2^{3/4} \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{c \sqrt{3 a-2 a x^2} (c x)^{3/2}}{5 a} \]

[Out]

-(c*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2])/(5*a) - (9*3^(1/4)*c^2*Sqrt[c*x]*Sqrt[3 - 2
*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(5*2^(3/4)*Sqrt[x]*Sqrt
[3*a - 2*a*x^2])

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Rubi [A]  time = 0.148801, antiderivative size = 107, 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 \[ -\frac{9 \sqrt [4]{3} c^2 \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{5\ 2^{3/4} \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{c \sqrt{3 a-2 a x^2} (c x)^{3/2}}{5 a} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(5/2)/Sqrt[3*a - 2*a*x^2],x]

[Out]

-(c*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2])/(5*a) - (9*3^(1/4)*c^2*Sqrt[c*x]*Sqrt[3 - 2
*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(5*2^(3/4)*Sqrt[x]*Sqrt
[3*a - 2*a*x^2])

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Rubi in Sympy [A]  time = 48.4722, size = 170, normalized size = 1.59 \[ \frac{9 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} c^{\frac{5}{2}} \sqrt{- \frac{2 x^{2}}{3} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{10 \sqrt{- 2 a x^{2} + 3 a}} - \frac{9 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} c^{\frac{5}{2}} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{10 \sqrt{- 2 a x^{2} + 3 a}} - \frac{c \left (c x\right )^{\frac{3}{2}} \sqrt{- 2 a x^{2} + 3 a}}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(5/2)/(-2*a*x**2+3*a)**(1/2),x)

[Out]

9*2**(1/4)*3**(3/4)*c**(5/2)*sqrt(-2*x**2/3 + 1)*elliptic_e(asin(2**(1/4)*3**(3/
4)*sqrt(c*x)/(3*sqrt(c))), -1)/(10*sqrt(-2*a*x**2 + 3*a)) - 9*2**(1/4)*3**(3/4)*
c**(5/2)*sqrt(-2*x**2/3 + 1)*elliptic_f(asin(2**(1/4)*3**(3/4)*sqrt(c*x)/(3*sqrt
(c))), -1)/(10*sqrt(-2*a*x**2 + 3*a)) - c*(c*x)**(3/2)*sqrt(-2*a*x**2 + 3*a)/(5*
a)

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Mathematica [A]  time = 0.135091, size = 112, normalized size = 1.05 \[ \frac{(c x)^{5/2} \left (-9 \sqrt [4]{6} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )+9 \sqrt [4]{6} \sqrt{3-2 x^2} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )+2 \left (2 x^2-3\right ) x^{3/2}\right )}{10 x^{5/2} \sqrt{a \left (3-2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(5/2)/Sqrt[3*a - 2*a*x^2],x]

[Out]

((c*x)^(5/2)*(2*x^(3/2)*(-3 + 2*x^2) + 9*6^(1/4)*Sqrt[3 - 2*x^2]*EllipticE[ArcSi
n[(2/3)^(1/4)*Sqrt[x]], -1] - 9*6^(1/4)*Sqrt[3 - 2*x^2]*EllipticF[ArcSin[(2/3)^(
1/4)*Sqrt[x]], -1]))/(10*x^(5/2)*Sqrt[a*(3 - 2*x^2)])

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Maple [B]  time = 0.041, size = 235, normalized size = 2.2 \[{\frac{{c}^{2}}{40\,ax \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 6\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}-3\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}-16\,{x}^{4}+24\,{x}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(5/2)/(-2*a*x^2+3*a)^(1/2),x)

[Out]

1/40*c^2/x*(c*x)^(1/2)*(-a*(2*x^2-3))^(1/2)/a*(6*((-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)-3*((-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)-16*x^4+24*x^2)/(2*x^2-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{5}{2}}}{\sqrt{-2 \, a x^{2} + 3 \, a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(5/2)/sqrt(-2*a*x^2 + 3*a),x, algorithm="maxima")

[Out]

integrate((c*x)^(5/2)/sqrt(-2*a*x^2 + 3*a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(5/2)/sqrt(-2*a*x^2 + 3*a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 139.309, size = 51, normalized size = 0.48 \[ \frac{\sqrt{3} c^{\frac{5}{2}} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt{a} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(5/2)/(-2*a*x**2+3*a)**(1/2),x)

[Out]

sqrt(3)*c**(5/2)*x**(7/2)*gamma(7/4)*hyper((1/2, 7/4), (11/4,), 2*x**2*exp_polar
(2*I*pi)/3)/(6*sqrt(a)*gamma(11/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{5}{2}}}{\sqrt{-2 \, a x^{2} + 3 \, a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(5/2)/sqrt(-2*a*x^2 + 3*a),x, algorithm="giac")

[Out]

integrate((c*x)^(5/2)/sqrt(-2*a*x^2 + 3*a), x)