3.7.43 \(\int \frac {(c x)^{5/2}}{(3 a-2 a x^2)^{3/2}} \, dx\) [643]

Optimal. Leaf size=110 \[ \frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}+\frac {3 \sqrt [4]{3} c^2 \sqrt {c x} \sqrt {3-2 x^2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{2\ 2^{3/4} a \sqrt {x} \sqrt {3 a-2 a x^2}} \]

[Out]

1/2*c*(c*x)^(3/2)/a/(-2*a*x^2+3*a)^(1/2)+3/4*3^(1/4)*c^2*EllipticE(1/6*(3-x*6^(1/2))^(1/2)*6^(1/2),2^(1/2))*(c
*x)^(1/2)*(-2*x^2+3)^(1/2)*2^(1/4)/a/x^(1/2)/(-2*a*x^2+3*a)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {294, 326, 325, 324, 435} \begin {gather*} \frac {3 \sqrt [4]{3} c^2 \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{2\ 2^{3/4} a \sqrt {x} \sqrt {3 a-2 a x^2}}+\frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 324

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 325

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 326

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 435

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 59, normalized size = 0.54 \begin {gather*} \frac {c (c x)^{3/2} \left (-3+\sqrt {9-6 x^2} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};\frac {2 x^2}{3}\right )\right )}{3 a \sqrt {a \left (3-2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(c*x)^(3/2)*(-3 + Sqrt[9 - 6*x^2]*Hypergeometric2F1[3/4, 3/2, 7/4, (2*x^2)/3]))/(3*a*Sqrt[a*(3 - 2*x^2)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs. \(2(86)=172\).
time = 0.09, size = 230, normalized size = 2.09

method	result	size
elliptic	\(\frac {\sqrt {c x}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (\frac {c^{3} x^{2}}{2 a \sqrt {-2 \left (x^{2}-\frac {3}{2}\right ) a c x}}-\frac {c^{3} \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \left (-\sqrt {6}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{72 a \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{c x \sqrt {-a \left (2 x^{2}-3\right )}}\)	\(186\)
default	\(-\frac {c^{2} \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \left (2 \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}-\sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}+8 x^{2}\right )}{16 x \,a^{2} \left (2 x^{2}-3\right )}\)	\(230\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(5/2)/(-2*a*x^2+3*a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/16*c^2/x*(c*x)^(1/2)*(-a*(2*x^2-3))^(1/2)*(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))*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))*2^(1/2)*((2*x+2^(1/2)*3^(1/2))*2^(1/2)*3^(1/2))^(1/2)+8*x^2)/a^2/(2*x^2-3)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.17, size = 74, normalized size = 0.67 \begin {gather*} -\frac {2 \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c^{2} x + 3 \, \sqrt {2} {\left (2 \, c^{2} x^{2} - 3 \, c^{2}\right )} \sqrt {-a c} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right )}{4 \, {\left (2 \, a^{2} x^{2} - 3 \, a^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(-2*a*x^2 + 3*a)*sqrt(c*x)*c^2*x + 3*sqrt(2)*(2*c^2*x^2 - 3*c^2)*sqrt(-a*c)*weierstrassZeta(6, 0,
weierstrassPInverse(6, 0, x)))/(2*a^2*x^2 - 3*a^2)

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Sympy [C] Result contains complex when optimal does not.
time = 4.03, size = 51, normalized size = 0.46 \begin {gather*} \frac {\sqrt {3} c^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*c**(5/2)*x**(7/2)*gamma(7/4)*hyper((3/2, 7/4), (11/4,), 2*x**2*exp_polar(2*I*pi)/3)/(18*a**(3/2)*gamma
(11/4))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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