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

Optimal. Leaf size=98 \[ \frac{4 \sqrt [4]{6} a \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{2 \sqrt{3 a-2 a x^2}}{c \sqrt{c x}} \]

[Out]

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

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Rubi [A]  time = 0.0416983, antiderivative size = 98, 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, Rules used = {277, 320, 319, 318, 424} \[ \frac{4 \sqrt [4]{6} a \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{2 \sqrt{3 a-2 a x^2}}{c \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 320

Int[Sqrt[(c_)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[c*x]/Sqrt[x], Int[Sqrt[x]/Sqrt[a + b*x^2
], x], x] /; FreeQ[{a, b, c}, x] && GtQ[-(b/a), 0]

Rule 319

Int[Sqrt[x_]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (b*x^2)/a]/Sqrt[a + b*x^2], Int[Sqrt[x]/Sqr
t[1 + (b*x^2)/a], x], x] /; FreeQ[{a, b}, x] && GtQ[-(b/a), 0] &&  !GtQ[a, 0]

Rule 318

Int[Sqrt[x_]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Dist[-2/(Sqrt[a]*(-(b/a))^(3/4)), Subst[Int[Sqrt[1 - 2*x^
2]/Sqrt[1 - x^2], x], x, Sqrt[1 - Sqrt[-(b/a)]*x]/Sqrt[2]], x] /; FreeQ[{a, b}, x] && GtQ[-(b/a), 0] && GtQ[a,
0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{3 a-2 a x^2}}{(c x)^{3/2}} \, dx &=-\frac{2 \sqrt{3 a-2 a x^2}}{c \sqrt{c x}}-\frac{(4 a) \int \frac{\sqrt{c x}}{\sqrt{3 a-2 a x^2}} \, dx}{c^2}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{c \sqrt{c x}}-\frac{\left (4 a \sqrt{c x}\right ) \int \frac{\sqrt{x}}{\sqrt{3 a-2 a x^2}} \, dx}{c^2 \sqrt{x}}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{c \sqrt{c x}}-\frac{\left (4 a \sqrt{c x} \sqrt{1-\frac{2 x^2}{3}}\right ) \int \frac{\sqrt{x}}{\sqrt{1-\frac{2 x^2}{3}}} \, dx}{c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{c \sqrt{c x}}+\frac{\left (4 \sqrt [4]{2} 3^{3/4} a \sqrt{c x} \sqrt{1-\frac{2 x^2}{3}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-2 x^2}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{\frac{2}{3}} x}}{\sqrt{2}}\right )}{c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{c \sqrt{c x}}+\frac{4 \sqrt [4]{6} a \sqrt{c x} \sqrt{3-2 x^2} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ \end{align*}

Mathematica [C]  time = 0.011827, size = 51, normalized size = 0.52 \[ -\frac{2 x \sqrt{a \left (9-6 x^2\right )} \, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\frac{2 x^2}{3}\right )}{\sqrt{3-2 x^2} (c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*Sqrt[a*(9 - 6*x^2)]*Hypergeometric2F1[-1/2, -1/4, 3/4, (2*x^2)/3])/((c*x)^(3/2)*Sqrt[3 - 2*x^2])

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Maple [B]  time = 0.034, size = 225, normalized size = 2.3 \begin{align*} -{\frac{1}{3\,c \left ( 2\,{x}^{2}-3 \right ) }\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 2\,\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}}\sqrt{ \left ( -2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}}\sqrt{3}\sqrt{-x\sqrt{2}\sqrt{3}}{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}},1/2\,\sqrt{2} \right ) -\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}}\sqrt{ \left ( -2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}}\sqrt{3}\sqrt{-x\sqrt{2}\sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{2}\sqrt{3}}{6}\sqrt{ \left ( 2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}}},{\frac{\sqrt{2}}{2}} \right ) +12\,{x}^{2}-18 \right ){\frac{1}{\sqrt{cx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-a*(2*x^2-3))^(1/2)*(2*2^(1/2)*((2*x+2^(1/2)*3^(1/2))*2^(1/2)*3^(1/2))^(1/2)*((-2*x+2^(1/2)*3^(1/2))*2^(
1/2)*3^(1/2))^(1/2)*3^(1/2)*(-x*2^(1/2)*3^(1/2))^(1/2)*EllipticE(1/6*3^(1/2)*2^(1/2)*((2*x+2^(1/2)*3^(1/2))*2^
(1/2)*3^(1/2))^(1/2),1/2*2^(1/2))-2^(1/2)*((2*x+2^(1/2)*3^(1/2))*2^(1/2)*3^(1/2))^(1/2)*((-2*x+2^(1/2)*3^(1/2)
)*2^(1/2)*3^(1/2))^(1/2)*3^(1/2)*(-x*2^(1/2)*3^(1/2))^(1/2)*EllipticF(1/6*3^(1/2)*2^(1/2)*((2*x+2^(1/2)*3^(1/2
))*2^(1/2)*3^(1/2))^(1/2),1/2*2^(1/2))+12*x^2-18)/c/(c*x)^(1/2)/(2*x^2-3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*a*x^2+3*a)^(1/2)/(c*x)^(3/2),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 [C]  time = 1.34588, size = 56, normalized size = 0.57 \begin{align*} \frac{\sqrt{3} \sqrt{a} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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