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

Optimal. Leaf size=128 \[ -\frac{3 \sqrt [4]{6} a c^2 \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{5 \sqrt{x} \sqrt{3 a-2 a x^2}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{7/2}}{9 c}-\frac{2}{15} c \sqrt{3 a-2 a x^2} (c x)^{3/2} \]

[Out]

(-2*c*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2])/15 + (2*(c*x)^(7/2)*Sqrt[3*a - 2*a*x^2])/
(9*c) - (3*6^(1/4)*a*c^2*Sqrt[c*x]*Sqrt[3 - 2*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqr
t[6]*x]/Sqrt[6]], 2])/(5*Sqrt[x]*Sqrt[3*a - 2*a*x^2])

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Rubi [A]  time = 0.194163, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{3 \sqrt [4]{6} a c^2 \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{5 \sqrt{x} \sqrt{3 a-2 a x^2}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{7/2}}{9 c}-\frac{2}{15} c \sqrt{3 a-2 a x^2} (c x)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2])/15 + (2*(c*x)^(7/2)*Sqrt[3*a - 2*a*x^2])/
(9*c) - (3*6^(1/4)*a*c^2*Sqrt[c*x]*Sqrt[3 - 2*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqr
t[6]*x]/Sqrt[6]], 2])/(5*Sqrt[x]*Sqrt[3*a - 2*a*x^2])

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Rubi in Sympy [A]  time = 54.6855, size = 199, normalized size = 1.55 \[ \frac{3 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} a c^{\frac{5}{2}} \sqrt{- \frac{2 x^{2}}{3} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{5 \sqrt{- 2 a x^{2} + 3 a}} - \frac{3 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} a c^{\frac{5}{2}} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{5 \sqrt{- 2 a x^{2} + 3 a}} - \frac{2 c \left (c x\right )^{\frac{3}{2}} \sqrt{- 2 a x^{2} + 3 a}}{15} + \frac{2 \left (c x\right )^{\frac{7}{2}} \sqrt{- 2 a x^{2} + 3 a}}{9 c} \]

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]

3*2**(1/4)*3**(3/4)*a*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)/(5*sqrt(-2*a*x**2 + 3*a)) - 3*2**(1/4)*3**(3/4)
*a*c**(5/2)*sqrt(-2*x**2/3 + 1)*elliptic_f(asin(2**(1/4)*3**(3/4)*sqrt(c*x)/(3*s
qrt(c))), -1)/(5*sqrt(-2*a*x**2 + 3*a)) - 2*c*(c*x)**(3/2)*sqrt(-2*a*x**2 + 3*a)
/15 + 2*(c*x)**(7/2)*sqrt(-2*a*x**2 + 3*a)/(9*c)

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Mathematica [A]  time = 0.187564, size = 112, normalized size = 0.88 \[ \frac{\sqrt{a \left (3-2 x^2\right )} (c x)^{5/2} \left (2 \sqrt{3-2 x^2} \left (5 x^2-3\right ) x^{3/2}-27 \sqrt [4]{6} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )+27 \sqrt [4]{6} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )\right )}{45 x^{5/2} \sqrt{3-2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((c*x)^(5/2)*Sqrt[a*(3 - 2*x^2)]*(2*x^(3/2)*Sqrt[3 - 2*x^2]*(-3 + 5*x^2) + 27*6^
(1/4)*EllipticE[ArcSin[(2/3)^(1/4)*Sqrt[x]], -1] - 27*6^(1/4)*EllipticF[ArcSin[(
2/3)^(1/4)*Sqrt[x]], -1]))/(45*x^(5/2)*Sqrt[3 - 2*x^2])

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Maple [B]  time = 0.059, size = 237, normalized size = 1.9 \[{\frac{{c}^{2}}{180\,x \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 80\,{x}^{6}+18\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}-9\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}-168\,{x}^{4}+72\,{x}^{2} \right ) } \]

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]

1/180*c^2/x*(c*x)^(1/2)*(-a*(2*x^2-3))^(1/2)*(80*x^6+18*((-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)-9*((-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)-168*x^4+72*x^2)/(2*x^2-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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