3.648 \(\int \frac{1}{(c x)^{5/2} \left (3 a-2 a x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=132 \[ -\frac{5 \sqrt{3 a-2 a x^2}}{27 a^2 c (c x)^{3/2}}+\frac{5\ 2^{3/4} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{27 \sqrt [4]{3} a c^{5/2} \sqrt{a \left (3-2 x^2\right )}}+\frac{1}{3 a c \sqrt{3 a-2 a x^2} (c x)^{3/2}} \]

[Out]

1/(3*a*c*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2]) - (5*Sqrt[3*a - 2*a*x^2])/(27*a^2*c*(c
*x)^(3/2)) + (5*2^(3/4)*Sqrt[3 - 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c*x])
/Sqrt[c]], -1])/(27*3^(1/4)*a*c^(5/2)*Sqrt[a*(3 - 2*x^2)])

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Rubi [A]  time = 0.179135, antiderivative size = 132, 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 \[ -\frac{5 \sqrt{3 a-2 a x^2}}{27 a^2 c (c x)^{3/2}}+\frac{5\ 2^{3/4} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{27 \sqrt [4]{3} a c^{5/2} \sqrt{a \left (3-2 x^2\right )}}+\frac{1}{3 a c \sqrt{3 a-2 a x^2} (c x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

1/(3*a*c*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2]) - (5*Sqrt[3*a - 2*a*x^2])/(27*a^2*c*(c
*x)^(3/2)) + (5*2^(3/4)*Sqrt[3 - 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c*x])
/Sqrt[c]], -1])/(27*3^(1/4)*a*c^(5/2)*Sqrt[a*(3 - 2*x^2)])

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Rubi in Sympy [A]  time = 16.7497, size = 129, normalized size = 0.98 \[ \frac{1}{3 a c \left (c x\right )^{\frac{3}{2}} \sqrt{- 2 a x^{2} + 3 a}} + \frac{5 \cdot 2^{\frac{3}{4}} \sqrt [4]{3} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{27 a c^{\frac{5}{2}} \sqrt{- 2 a x^{2} + 3 a}} - \frac{5 \sqrt{- 2 a x^{2} + 3 a}}{27 a^{2} c \left (c x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(3*a*c*(c*x)**(3/2)*sqrt(-2*a*x**2 + 3*a)) + 5*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)/(27*a*c**(5
/2)*sqrt(-2*a*x**2 + 3*a)) - 5*sqrt(-2*a*x**2 + 3*a)/(27*a**2*c*(c*x)**(3/2))

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Mathematica [A]  time = 0.144872, size = 95, normalized size = 0.72 \[ \frac{x \left (6 \sqrt{2-\frac{3}{x^2}} \left (5 x^2-3\right )-5\ 6^{3/4} \sqrt{x} \left (2 x^2-3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{3}{2}}}{\sqrt{x}}\right )\right |-1\right )\right )}{81 a \sqrt{2-\frac{3}{x^2}} \sqrt{a \left (3-2 x^2\right )} (c x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(6*Sqrt[2 - 3/x^2]*(-3 + 5*x^2) - 5*6^(3/4)*Sqrt[x]*(-3 + 2*x^2)*EllipticF[Ar
cSin[(3/2)^(1/4)/Sqrt[x]], -1]))/(81*a*Sqrt[2 - 3/x^2]*(c*x)^(5/2)*Sqrt[a*(3 - 2
*x^2)])

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Maple [A]  time = 0.029, size = 133, normalized size = 1. \[ -{\frac{1}{162\,{a}^{2}x{c}^{2} \left ( 2\,{x}^{2}-3 \right ) }\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 5\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}x+60\,{x}^{2}-36 \right ){\frac{1}{\sqrt{cx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*(-a*(2*x^2-3))^(1/2)*(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)*
x+60*x^2-36)/x/a^2/c^2/(c*x)^(1/2)/(2*x^2-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-2*a*x^2 + 3*a)^(3/2)*(c*x)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/((-2*a*x^2 + 3*a)^(3/2)*(c*x)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (2 \, a c^{2} x^{4} - 3 \, a c^{2} x^{2}\right )} \sqrt{-2 \, a x^{2} + 3 \, a} \sqrt{c x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-2*a*x^2 + 3*a)^(3/2)*(c*x)^(5/2)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-2*a*x^2 + 3*a)^(3/2)*(c*x)^(5/2)),x, algorithm="giac")

[Out]

integrate(1/((-2*a*x^2 + 3*a)^(3/2)*(c*x)^(5/2)), x)