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]

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

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Rubi [A]  time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, 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 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 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 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 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*}

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Mathematica [C]  time = 0.01, 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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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{c^{2} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-2 \, a x^{2} + 3 \, a}}{\left (c x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-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)

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maple [B]  time = 0.04, size = 225, normalized size = 2.30 \[ -\frac {\sqrt {-\left (2 x^{2}-3\right ) a}\, \left (12 x^{2}+2 \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-\sqrt {2}\, \sqrt {3}\, x}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-\sqrt {2}\, \sqrt {3}\, x}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )-18\right )}{3 \sqrt {c x}\, \left (2 x^{2}-3\right ) c} \]

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*(-(2*x^2-3)*a)^(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)*(-2^(1/2)*3^(1/2)*x)^(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)*(-2^(1/2)*3^(1/2)*x)^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-2 \, a x^{2} + 3 \, a}}{\left (c x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {3\,a-2\,a\,x^2}}{{\left (c\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 1.29, size = 56, normalized size = 0.57 \[ \frac {\sqrt {3} \sqrt {a} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]

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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