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

Optimal. Leaf size=128 \[ -\frac{3 \sqrt [4]{6} a c^2 \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{5 \sqrt{x} \sqrt{3 a-2 a x^2}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{7/2}}{9 c}-\frac{2}{15} c \sqrt{3 a-2 a x^2} (c x)^{3/2} \]

[Out]

(-2*c*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2])/15 + (2*(c*x)^(7/2)*Sqrt[3*a - 2*a*x^2])/(9*c) - (3*6^(1/4)*a*c^2*Sqrt[
c*x]*Sqrt[3 - 2*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(5*Sqrt[x]*Sqrt[3*a - 2*a*x^2])

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Rubi [A]  time = 0.0639806, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {279, 321, 320, 319, 318, 424} \[ -\frac{3 \sqrt [4]{6} a c^2 \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{5 \sqrt{x} \sqrt{3 a-2 a x^2}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{7/2}}{9 c}-\frac{2}{15} c \sqrt{3 a-2 a x^2} (c x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2])/15 + (2*(c*x)^(7/2)*Sqrt[3*a - 2*a*x^2])/(9*c) - (3*6^(1/4)*a*c^2*Sqrt[
c*x]*Sqrt[3 - 2*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(5*Sqrt[x]*Sqrt[3*a - 2*a*x^2])

Rule 279

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

Rule 321

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

Rule 320

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 319

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 318

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 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int (c x)^{5/2} \sqrt{3 a-2 a x^2} \, dx &=\frac{2 (c x)^{7/2} \sqrt{3 a-2 a x^2}}{9 c}+\frac{1}{3} (2 a) \int \frac{(c x)^{5/2}}{\sqrt{3 a-2 a x^2}} \, dx\\ &=-\frac{2}{15} c (c x)^{3/2} \sqrt{3 a-2 a x^2}+\frac{2 (c x)^{7/2} \sqrt{3 a-2 a x^2}}{9 c}+\frac{1}{5} \left (3 a c^2\right ) \int \frac{\sqrt{c x}}{\sqrt{3 a-2 a x^2}} \, dx\\ &=-\frac{2}{15} c (c x)^{3/2} \sqrt{3 a-2 a x^2}+\frac{2 (c x)^{7/2} \sqrt{3 a-2 a x^2}}{9 c}+\frac{\left (3 a c^2 \sqrt{c x}\right ) \int \frac{\sqrt{x}}{\sqrt{3 a-2 a x^2}} \, dx}{5 \sqrt{x}}\\ &=-\frac{2}{15} c (c x)^{3/2} \sqrt{3 a-2 a x^2}+\frac{2 (c x)^{7/2} \sqrt{3 a-2 a x^2}}{9 c}+\frac{\left (3 a c^2 \sqrt{c x} \sqrt{1-\frac{2 x^2}{3}}\right ) \int \frac{\sqrt{x}}{\sqrt{1-\frac{2 x^2}{3}}} \, dx}{5 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ &=-\frac{2}{15} c (c x)^{3/2} \sqrt{3 a-2 a x^2}+\frac{2 (c x)^{7/2} \sqrt{3 a-2 a x^2}}{9 c}-\frac{\left (3 \sqrt [4]{2} 3^{3/4} a c^2 \sqrt{c x} \sqrt{1-\frac{2 x^2}{3}}\right ) \operatorname{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 )}{5 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ &=-\frac{2}{15} c (c x)^{3/2} \sqrt{3 a-2 a x^2}+\frac{2 (c x)^{7/2} \sqrt{3 a-2 a x^2}}{9 c}-\frac{3 \sqrt [4]{6} a c^2 \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 )}{5 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0289085, size = 74, normalized size = 0.58 \[ \frac{c \sqrt{a \left (3-2 x^2\right )} (c x)^{3/2} \left (3 \sqrt{3} \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{2 x^2}{3}\right )-\left (3-2 x^2\right )^{3/2}\right )}{9 \sqrt{3-2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(c*x)^(3/2)*Sqrt[a*(3 - 2*x^2)]*(-(3 - 2*x^2)^(3/2) + 3*Sqrt[3]*Hypergeometric2F1[-1/2, 3/4, 7/4, (2*x^2)/3
]))/(9*Sqrt[3 - 2*x^2])

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Maple [B]  time = 0.045, size = 237, normalized size = 1.9 \begin{align*}{\frac{{c}^{2}}{180\,x \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 80\,{x}^{6}+18\,\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}}{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}},1/2\,\sqrt{2} \right ) -9\,\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}}{\it EllipticF} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}},1/2\,\sqrt{2} \right ) -168\,{x}^{4}+72\,{x}^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180/x*c^2*(c*x)^(1/2)*(-a*(2*x^2-3))^(1/2)*(80*x^6+18*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)*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))-9*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))-168*x^4+72*x^2)/(2*x^2-3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)*c^2*x^2, x)

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Sympy [C]  time = 43.4541, size = 53, normalized size = 0.41 \begin{align*} \frac{\sqrt{3} \sqrt{a} c^{\frac{5}{2}} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 \Gamma \left (\frac{11}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*sqrt(a)*c**(5/2)*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), 2*x**2*exp_polar(2*I*pi)/3)/(2*gamma(
11/4))

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

[Out]

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