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

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

[Out]

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

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Rubi [A]  time = 0.143461, 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 \[ \frac{2 \sqrt [4]{2} \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{3^{3/4} c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{2 \sqrt{3 a-2 a x^2}}{3 a c \sqrt{c x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.6144, size = 172, normalized size = 1.61 \[ - \frac{2 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{- \frac{2 x^{2}}{3} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{3 c^{\frac{3}{2}} \sqrt{- 2 a x^{2} + 3 a}} + \frac{2 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{3 c^{\frac{3}{2}} \sqrt{- 2 a x^{2} + 3 a}} - \frac{2 \sqrt{- 2 a x^{2} + 3 a}}{3 a c \sqrt{c x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.108577, size = 110, normalized size = 1.03 \[ \frac{x \left (4 x^2+2 \sqrt [4]{6} \sqrt{3-2 x^2} \sqrt{x} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )-2 \sqrt [4]{6} \sqrt{3-2 x^2} \sqrt{x} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )-6\right )}{3 \sqrt{a \left (3-2 x^2\right )} (c x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.029, size = 228, normalized size = 2.1 \[ -{\frac{1}{18\,ac \left ( 2\,{x}^{2}-3 \right ) }\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 2\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}-\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}+24\,{x}^{2}-36 \right ){\frac{1}{\sqrt{cx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*(-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^2-36)/c/(c*x)^(1/2)/a/(2*x^2-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(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. \[{\rm integral}\left (\frac{1}{\sqrt{-2 \, a x^{2} + 3 \, a} \sqrt{c x} c x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.69047, size = 54, normalized size = 0.5 \[ \frac{\sqrt{3} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt{a} c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), 2*x**2*exp_polar(2*I*pi)/3)/(6*sq
rt(a)*c**(3/2)*sqrt(x)*gamma(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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