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

Optimal. Leaf size=67 \[ -\frac{\sqrt [4]{6} \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{\sqrt{x} \sqrt{3 a-2 a x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0298237, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {320, 319, 318, 424} \[ -\frac{\sqrt [4]{6} \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{\sqrt{x} \sqrt{3 a-2 a x^2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x]/Sqrt[3*a - 2*a*x^2],x]

[Out]

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

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{c x}}{\sqrt{3 a-2 a x^2}} \, dx &=\frac{\sqrt{c x} \int \frac{\sqrt{x}}{\sqrt{3 a-2 a x^2}} \, dx}{\sqrt{x}}\\ &=\frac{\left (\sqrt{c x} \sqrt{1-\frac{2 x^2}{3}}\right ) \int \frac{\sqrt{x}}{\sqrt{1-\frac{2 x^2}{3}}} \, dx}{\sqrt{x} \sqrt{3 a-2 a x^2}}\\ &=-\frac{\left (\sqrt [4]{2} 3^{3/4} \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 )}{\sqrt{x} \sqrt{3 a-2 a x^2}}\\ &=-\frac{\sqrt [4]{6} \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 )}{\sqrt{x} \sqrt{3 a-2 a x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0148194, size = 53, normalized size = 0.79 \[ \frac{2 x \sqrt{3-2 x^2} \sqrt{c x} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{2 x^2}{3}\right )}{3 \sqrt{a \left (9-6 x^2\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[c*x]*Sqrt[3 - 2*x^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (2*x^2)/3])/(3*Sqrt[a*(9 - 6*x^2)])

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Maple [B]  time = 0.016, size = 165, normalized size = 2.5 \begin{align*}{\frac{\sqrt{2}\sqrt{3}}{12\,ax \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) }\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{-x\sqrt{2}\sqrt{3}} \left ( 2\,{\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 ) -{\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 ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(c*x)^(1/2)*(-a*(2*x^2-3))^(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)*(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))-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)))/a/x/(2*x^2-3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*x)/sqrt(-2*a*x^2 + 3*a), 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}}{2 \, a x^{2} - 3 \, a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 0.735532, size = 51, normalized size = 0.76 \begin{align*} \frac{\sqrt{3} \sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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