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

Optimal. Leaf size=88 \[ \frac{c^{3/2} \sqrt{3-2 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right ),-1\right )}{\sqrt [4]{6} \sqrt{a \left (3-2 x^2\right )}}-\frac{c \sqrt{3 a-2 a x^2} \sqrt{c x}}{3 a} \]

[Out]

-(c*Sqrt[c*x]*Sqrt[3*a - 2*a*x^2])/(3*a) + (c^(3/2)*Sqrt[3 - 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c*x])/S
qrt[c]], -1])/(6^(1/4)*Sqrt[a*(3 - 2*x^2)])

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Rubi [A]  time = 0.0466644, antiderivative size = 88, 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 = {321, 329, 224, 221} \[ \frac{c^{3/2} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{\sqrt [4]{6} \sqrt{a \left (3-2 x^2\right )}}-\frac{c \sqrt{3 a-2 a x^2} \sqrt{c x}}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*Sqrt[c*x]*Sqrt[3*a - 2*a*x^2])/(3*a) + (c^(3/2)*Sqrt[3 - 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c*x])/S
qrt[c]], -1])/(6^(1/4)*Sqrt[a*(3 - 2*x^2)])

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

Mathematica [C]  time = 0.0214878, size = 61, normalized size = 0.69 \[ \frac{c \sqrt{c x} \left (\sqrt{9-6 x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{2 x^2}{3}\right )+2 x^2-3\right )}{3 \sqrt{a \left (3-2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.029, size = 131, normalized size = 1.5 \begin{align*} -{\frac{c}{12\,ax \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\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 ) +8\,{x}^{3}-12\,x \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*c*(c*x)^(1/2)*(-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))+8*x^3-12*x)/x/a/(2*x^2-3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x)^(3/2)/sqrt(-2*a*x^2 + 3*a), 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 x}{2 \, a x^{2} - 3 \, a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.86659, size = 51, normalized size = 0.58 \begin{align*} \frac{\sqrt{3} c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt{a} \Gamma \left (\frac{9}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*c**(3/2)*x**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), 2*x**2*exp_polar(2*I*pi)/3)/(6*sqrt(a)*gamma(9/
4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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