\(\int \frac {\sqrt {c x}}{(3 a-2 a x^2)^{3/2}} \, dx\) [659]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int \frac {\sqrt {c x}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {(c x)^{3/2}}{3 a c \sqrt {3 a-2 a x^2}}-\frac {\sqrt {c} \sqrt {3-2 x^2} E\left (\left .\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{6^{3/4} a \sqrt {3 a-2 a x^2}}+\frac {\sqrt {c} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{6^{3/4} a \sqrt {3 a-2 a x^2}} \] Output:

1/3*(c*x)^(3/2)/a/c/(-2*a*x^2+3*a)^(1/2)-1/6*c^(1/2)*(-2*x^2+3)^(1/2)*Elli 
pticE(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2)/c^(1/2),I)*6^(1/4)/a/(-2*a*x^2+3*a)^ 
(1/2)+1/6*c^(1/2)*(-2*x^2+3)^(1/2)*EllipticF(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/ 
2)/c^(1/2),I)*6^(1/4)/a/(-2*a*x^2+3*a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 7.68 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {c x}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {2 x \sqrt {c x} \left (3-2 x^2\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {2 x^2}{3}\right )}{9 \sqrt {3} \left (a \left (3-2 x^2\right )\right )^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.64, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {253, 261, 260, 27, 259, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 253

\(\displaystyle \frac {(c x)^{3/2}}{3 a c \sqrt {3 a-2 a x^2}}-\frac {\int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}}dx}{6 a}\)

\(\Big \downarrow \) 261

\(\displaystyle \frac {(c x)^{3/2}}{3 a c \sqrt {3 a-2 a x^2}}-\frac {\sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}}dx}{6 a \sqrt {x}}\)

\(\Big \downarrow \) 260

\(\displaystyle \frac {(c x)^{3/2}}{3 a c \sqrt {3 a-2 a x^2}}-\frac {\sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {3} \sqrt {x}}{\sqrt {3-2 x^2}}dx}{6 \sqrt {3} a \sqrt {x} \sqrt {3 a-2 a x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(c x)^{3/2}}{3 a c \sqrt {3 a-2 a x^2}}-\frac {\sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3-2 x^2}}dx}{6 a \sqrt {x} \sqrt {3 a-2 a x^2}}\)

\(\Big \downarrow \) 259

\(\displaystyle \frac {\sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {\frac {1}{3} \left (\sqrt {6} x-3\right )+1}}{\sqrt {\frac {1}{6} \left (\sqrt {6} x-3\right )+1}}d\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}}{6^{3/4} a \sqrt {x} \sqrt {3 a-2 a x^2}}+\frac {(c x)^{3/2}}{3 a c \sqrt {3 a-2 a x^2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\arcsin \left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{6^{3/4} a \sqrt {x} \sqrt {3 a-2 a x^2}}+\frac {(c x)^{3/2}}{3 a c \sqrt {3 a-2 a x^2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 259

Int[Sqrt[x_]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[-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 260

Int[Sqrt[x_]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[1 + b*(x^2/a 
)]/Sqrt[a + b*x^2]   Int[Sqrt[x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b 
}, x] && GtQ[-b/a, 0] &&  !GtQ[a, 0]
 

rule 261

Int[Sqrt[(c_)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.16

method	result	size
elliptic	\(\frac {\sqrt {c x}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (\frac {c \,x^{2}}{3 a \sqrt {-2 \left (x^{2}-\frac {3}{2}\right ) a c x}}-\frac {c \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \left (-\sqrt {6}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{324 a \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{c x \sqrt {-a \left (2 x^{2}-3\right )}}\)	\(182\)
default	\(-\frac {\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \left (2 \sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}-\sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}+24 x^{2}\right )}{72 a^{2} x \left (2 x^{2}-3\right )}\)	\(227\)

Input:

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

Output:

1/c/x*(c*x)^(1/2)/(-a*(2*x^2-3))^(1/2)*(-c*x*a*(2*x^2-3))^(1/2)*(1/3/a*c*x 
^2/(-2*(x^2-3/2)*a*c*x)^(1/2)-1/324/a*c*6^(1/2)*3^(1/2)*((x+1/2*6^(1/2))*6 
^(1/2))^(1/2)*(-6*(x-1/2*6^(1/2))*6^(1/2))^(1/2)*(-3*6^(1/2)*x)^(1/2)/(-2* 
a*c*x^3+3*a*c*x)^(1/2)*(-6^(1/2)*EllipticE(1/3*3^(1/2)*((x+1/2*6^(1/2))*6^ 
(1/2))^(1/2),1/2*2^(1/2))+1/2*6^(1/2)*EllipticF(1/3*3^(1/2)*((x+1/2*6^(1/2 
))*6^(1/2))^(1/2),1/2*2^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.40 \[ \int \frac {\sqrt {c x}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=-\frac {\sqrt {2} \sqrt {-a c} {\left (2 \, x^{2} - 3\right )} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right ) + 2 \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} x}{6 \, {\left (2 \, a^{2} x^{2} - 3 \, a^{2}\right )}} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt {c x}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} \sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \] Input:

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

Output:

sqrt(3)*sqrt(c)*x**(3/2)*gamma(3/4)*hyper((3/4, 3/2), (7/4,), 2*x**2*exp_p 
olar(2*I*pi)/3)/(18*a**(3/2)*gamma(7/4))
 

Maxima [F]

\[ \int \frac {\sqrt {c x}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {c x}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {c x}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {c x}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c x}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c\,x}}{{\left (3\,a-2\,a\,x^2\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {c x}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{2}+3}}{4 x^{4}-12 x^{2}+9}d x \right )}{a^{2}} \] Input:

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

Output:

(sqrt(c)*sqrt(a)*int((sqrt(x)*sqrt( - 2*x**2 + 3))/(4*x**4 - 12*x**2 + 9), 
x))/a**2