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

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

[Out]

(-2*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)])

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Rubi [A]  time = 0.0459691, antiderivative size = 98, 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 = {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 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]

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 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]

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 = 0.0146599, 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)])

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Maple [A]  time = 0.018, size = 132, normalized size = 1.4 \begin{align*} -{\frac{1}{27\,ax{c}^{2} \left ( 2\,{x}^{2}-3 \right ) }\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( \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}}{\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 ) x+12\,{x}^{2}-18 \right ){\frac{1}{\sqrt{cx}}}} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

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)

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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 c^{3} x^{5} - 3 \, a c^{3} x^{3}}, x\right ) \end{align*}

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)

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Sympy [A]  time = 10.4706, size = 54, normalized size = 0.55 \begin{align*} \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 )} \end{align*}

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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

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)