3.642 \(\int \frac {1}{(c x)^{5/2} \sqrt {3 a-2 a x^2}} \, dx\)

Optimal. Leaf size=98 \[ \frac {2\ 2^{3/4} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{9 \sqrt [4]{3} c^{5/2} \sqrt {a \left (3-2 x^2\right )}}-\frac {2 \sqrt {3 a-2 a x^2}}{9 a c (c x)^{3/2}} \]

[Out]

2/27*2^(3/4)*EllipticF(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2)/c^(1/2),I)*(-2*x^2+3)^(1/2)*3^(3/4)/c^(5/2)/(a*(-2*x^2+
3))^(1/2)-2/9*(-2*a*x^2+3*a)^(1/2)/a/c/(c*x)^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 98, 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 = {325, 329, 224, 221} \[ \frac {2\ 2^{3/4} \sqrt {3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{9 \sqrt [4]{3} c^{5/2} \sqrt {a \left (3-2 x^2\right )}}-\frac {2 \sqrt {3 a-2 a x^2}}{9 a c (c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(5/2)*Sqrt[3*a - 2*a*x^2]),x]

[Out]

(-2*Sqrt[3*a - 2*a*x^2])/(9*a*c*(c*x)^(3/2)) + (2*2^(3/4)*Sqrt[3 - 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c
*x])/Sqrt[c]], -1])/(9*3^(1/4)*c^(5/2)*Sqrt[a*(3 - 2*x^2)])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 224

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

Rule 325

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {1}{(c x)^{5/2} \sqrt {3 a-2 a x^2}} \, dx &=-\frac {2 \sqrt {3 a-2 a x^2}}{9 a c (c x)^{3/2}}+\frac {2 \int \frac {1}{\sqrt {c x} \sqrt {3 a-2 a x^2}} \, dx}{9 c^2}\\ &=-\frac {2 \sqrt {3 a-2 a x^2}}{9 a c (c x)^{3/2}}+\frac {4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {3 a-\frac {2 a x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{9 c^3}\\ &=-\frac {2 \sqrt {3 a-2 a x^2}}{9 a c (c x)^{3/2}}+\frac {\left (4 \sqrt {3-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {2 x^4}{3 c^2}}} \, dx,x,\sqrt {c x}\right )}{9 \sqrt {3} c^3 \sqrt {a \left (3-2 x^2\right )}}\\ &=-\frac {2 \sqrt {3 a-2 a x^2}}{9 a c (c x)^{3/2}}+\frac {2\ 2^{3/4} \sqrt {3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{9 \sqrt [4]{3} c^{5/2} \sqrt {a \left (3-2 x^2\right )}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.02, size = 53, normalized size = 0.54 \[ -\frac {2 x \sqrt {3-2 x^2} \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};\frac {2 x^2}{3}\right )}{3 \sqrt {a \left (9-6 x^2\right )} (c x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(5/2)*Sqrt[3*a - 2*a*x^2]),x]

[Out]

(-2*x*Sqrt[3 - 2*x^2]*Hypergeometric2F1[-3/4, 1/2, 1/4, (2*x^2)/3])/(3*(c*x)^(5/2)*Sqrt[a*(9 - 6*x^2)])

________________________________________________________________________________________

fricas [F]  time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{2 \, a c^{3} x^{5} - 3 \, a c^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(-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*c^3*x^5 - 3*a*c^3*x^3), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(-2*a*x^2+3*a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-2*a*x^2 + 3*a)*(c*x)^(5/2)), x)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 132, normalized size = 1.35 \[ -\frac {\sqrt {-\left (2 x^{2}-3\right ) a}\, \left (12 x^{2}+\sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {-\sqrt {2}\, \sqrt {3}\, x}\, x \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )-18\right )}{27 \sqrt {c x}\, \left (2 x^{2}-3\right ) a \,c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(5/2)/(-2*a*x^2+3*a)^(1/2),x)

[Out]

-1/27*(-(2*x^2-3)*a)^(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)*(-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))*x+12*x^2-18)/x/a/c^2/(c*x)^(1/2)/(2*x^2-3)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(-2*a*x^2+3*a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-2*a*x^2 + 3*a)*(c*x)^(5/2)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (c\,x\right )}^{5/2}\,\sqrt {3\,a-2\,a\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x)^(5/2)*(3*a - 2*a*x^2)^(1/2)),x)

[Out]

int(1/((c*x)^(5/2)*(3*a - 2*a*x^2)^(1/2)), x)

________________________________________________________________________________________

sympy [A]  time = 3.77, size = 54, normalized size = 0.55 \[ \frac {\sqrt {3} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt {a} c^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(5/2)/(-2*a*x**2+3*a)**(1/2),x)

[Out]

sqrt(3)*gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), 2*x**2*exp_polar(2*I*pi)/3)/(6*sqrt(a)*c**(5/2)*x**(3/2)*gamma(
1/4))

________________________________________________________________________________________