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

Optimal. Leaf size=96 \[ -\frac{4\ 2^{3/4} a \sqrt{3-2 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right ),-1\right )}{3 \sqrt [4]{3} c^{5/2} \sqrt{a \left (3-2 x^2\right )}}-\frac{2 \sqrt{3 a-2 a x^2}}{3 c (c x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0482166, antiderivative size = 96, 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 = {277, 329, 224, 221} \[ -\frac{4\ 2^{3/4} a \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{3 \sqrt [4]{3} c^{5/2} \sqrt{a \left (3-2 x^2\right )}}-\frac{2 \sqrt{3 a-2 a x^2}}{3 c (c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3*a - 2*a*x^2])/(3*c*(c*x)^(3/2)) - (4*2^(3/4)*a*Sqrt[3 - 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c
*x])/Sqrt[c]], -1])/(3*3^(1/4)*c^(5/2)*Sqrt[a*(3 - 2*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 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{\sqrt{3 a-2 a x^2}}{(c x)^{5/2}} \, dx &=-\frac{2 \sqrt{3 a-2 a x^2}}{3 c (c x)^{3/2}}-\frac{(4 a) \int \frac{1}{\sqrt{c x} \sqrt{3 a-2 a x^2}} \, dx}{3 c^2}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{3 c (c x)^{3/2}}-\frac{(8 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{3 a-\frac{2 a x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{3 c^3}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{3 c (c x)^{3/2}}-\frac{\left (8 a \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 )}{3 \sqrt{3} c^3 \sqrt{a \left (3-2 x^2\right )}}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{3 c (c x)^{3/2}}-\frac{4\ 2^{3/4} a \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 )}{3 \sqrt [4]{3} c^{5/2} \sqrt{a \left (3-2 x^2\right )}}\\ \end{align*}

Mathematica [C]  time = 0.0132876, size = 51, normalized size = 0.53 \[ -\frac{2 x \sqrt{a \left (3-2 x^2\right )} \, _2F_1\left (-\frac{3}{4},-\frac{1}{2};\frac{1}{4};\frac{2 x^2}{3}\right )}{\sqrt{9-6 x^2} (c x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.029, size = 129, normalized size = 1.3 \begin{align*}{\frac{2}{9\,x{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-6\,{x}^{2}+9 \right ){\frac{1}{\sqrt{cx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(-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-6*x^2+9)/x/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{\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((-2*a*x^2+3*a)^(1/2)/(c*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(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}}{c^{3} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-2*a*x^2 + 3*a)*sqrt(c*x)/(c^3*x^3), x)

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Sympy [A]  time = 5.29622, size = 49, normalized size = 0.51 \begin{align*} \frac{\sqrt{2} i \sqrt{a} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{3}{2 x^{2}}} \right )}}{2 c^{\frac{5}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\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((-2*a*x^2+3*a)^(1/2)/(c*x)^(5/2),x, algorithm="giac")

[Out]

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