3.7.11 \(\int \frac {\sqrt {3 a-2 a x^2}}{(c x)^{3/2}} \, dx\) [611]

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

[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 = 0.03, 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 = {283, 326, 325, 324, 435} \begin {gather*} \frac {4 \sqrt [4]{6} a \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 )}{c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {2 \sqrt {3 a-2 a x^2}}{c \sqrt {c x}} \end {gather*}

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 283

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 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 {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 ) \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 )}{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] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.01, size = 51, normalized size = 0.52 \begin {gather*} -\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 )}{(c x)^{3/2} \sqrt {3-2 x^2}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(224\) vs. \(2(80)=160\).
time = 0.10, size = 225, normalized size = 2.30

method	result	size
risch	\(\frac {2 \left (2 x^{2}-3\right ) a}{c \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}-\frac {2 \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \left (-\sqrt {6}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right ) a \sqrt {-c x a \left (2 x^{2}-3\right )}}{27 \sqrt {-2 a c \,x^{3}+3 a c x}\, c \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\)	\(184\)
elliptic	\(-\frac {\sqrt {-a \left (2 x^{2}-3\right )}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (-\frac {2 \left (-2 c \,x^{2} a +3 a c \right )}{c^{2} \sqrt {x \left (-2 c \,x^{2} a +3 a c \right )}}-\frac {2 a \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \left (-\sqrt {6}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{27 c \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{\sqrt {c x}\, a \left (2 x^{2}-3\right )}\)	\(201\)
default	\(-\frac {\sqrt {-a \left (2 x^{2}-3\right )}\, \left (2 \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}-\sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}+12 x^{2}-18\right )}{3 c \sqrt {c x}\, \left (2 x^{2}-3\right )}\)	\(225\)

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,method=_RETURNVERBOSE)

[Out]

-1/3*(-a*(2*x^2-3))^(1/2)*(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))*2^(1/2)*((2*x+2
^(1/2)*3^(1/2))*2^(1/2)*3^(1/2))^(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.18, size = 46, normalized size = 0.47 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {2} \sqrt {-a c} x {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right ) + \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}\right )}}{c^{2} x} \end {gather*}

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]

-2*(2*sqrt(2)*sqrt(-a*c)*x*weierstrassZeta(6, 0, weierstrassPInverse(6, 0, x)) + sqrt(-2*a*x^2 + 3*a)*sqrt(c*x
))/(c^2*x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.60, size = 56, normalized size = 0.57 \begin {gather*} \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 {gather*}

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

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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