3.7.39 \(\int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx\) [639]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 = {326, 325, 324, 435} \begin {gather*} -\frac {\sqrt [4]{6} \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{\sqrt {x} \sqrt {3 a-2 a x^2}} \end {gather*}

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 324

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 325

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 326

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 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], 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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 53, normalized size = 0.79 \begin {gather*} \frac {2 x \sqrt {c x} \sqrt {3-2 x^2} \, _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 )}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(164\) vs. \(2(53)=106\).
time = 0.06, size = 165, normalized size = 2.46

method	result	size
elliptic	\(\frac {\sqrt {c x}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \left (-\sqrt {6}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{54 x \sqrt {-a \left (2 x^{2}-3\right )}\, \sqrt {-2 a c \,x^{3}+3 a c x}}\)	\(152\)
default	\(\frac {\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, \left (2 \EllipticE \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )\right )}{12 x a \left (2 x^{2}-3\right )}\)	\(165\)

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,method=_RETURNVERBOSE)

[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)))/x/a/(2*x^2-3)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.14, size = 20, normalized size = 0.30 \begin {gather*} \frac {\sqrt {2} \sqrt {-a c} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right )}{a} \end {gather*}

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]

sqrt(2)*sqrt(-a*c)*weierstrassZeta(6, 0, weierstrassPInverse(6, 0, x))/a

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Sympy [C] Result contains complex when optimal does not.
time = 0.42, size = 51, normalized size = 0.76 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x}}{\sqrt {3\,a-2\,a\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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