\(\int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}} \, dx\) [650]

Optimal result

Integrand size = 22, antiderivative size = 164 \[ \int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}} \, dx=-\frac {c (c x)^{3/2} \sqrt {3 a-2 a x^2}}{5 a}+\frac {9 \sqrt [4]{3} c^{5/2} \sqrt {3-2 x^2} E\left (\left .\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{5\ 2^{3/4} \sqrt {3 a-2 a x^2}}-\frac {9 \sqrt [4]{3} c^{5/2} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{5\ 2^{3/4} \sqrt {3 a-2 a x^2}} \] Output:

-1/5*c*(c*x)^(3/2)*(-2*a*x^2+3*a)^(1/2)/a+9/10*3^(1/4)*c^(5/2)*(-2*x^2+3)^ 
(1/2)*EllipticE(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2)/c^(1/2),I)*2^(1/4)/(-2*a*x 
^2+3*a)^(1/2)-9/10*3^(1/4)*c^(5/2)*(-2*x^2+3)^(1/2)*EllipticF(1/3*2^(1/4)* 
3^(3/4)*(c*x)^(1/2)/c^(1/2),I)*2^(1/4)/(-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 = 10.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.37 \[ \int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}} \, dx=\frac {c (c x)^{3/2} \left (-3+2 x^2+\sqrt {9-6 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {2 x^2}{3}\right )\right )}{5 \sqrt {a \left (3-2 x^2\right )}} \] Input:

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

Output:

(c*(c*x)^(3/2)*(-3 + 2*x^2 + Sqrt[9 - 6*x^2]*Hypergeometric2F1[1/2, 3/4, 7 
/4, (2*x^2)/3]))/(5*Sqrt[a*(3 - 2*x^2)])
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.65, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {262, 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.

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

\(\Big \downarrow \) 262

\(\displaystyle \frac {9}{10} c^2 \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}}dx-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\)

\(\Big \downarrow \) 261

\(\displaystyle \frac {9 c^2 \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}}dx}{10 \sqrt {x}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\)

\(\Big \downarrow \) 260

\(\displaystyle \frac {3 \sqrt {3} c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {3} \sqrt {x}}{\sqrt {3-2 x^2}}dx}{10 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {9 c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3-2 x^2}}dx}{10 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\)

\(\Big \downarrow \) 259

\(\displaystyle -\frac {9 \sqrt [4]{3} c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {\frac {1}{3} \left (\sqrt {6} x-3\right )+1}}{\sqrt {\frac {1}{6} \left (\sqrt {6} x-3\right )+1}}d\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}}{5\ 2^{3/4} \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {9 \sqrt [4]{3} c^2 \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\arcsin \left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{5\ 2^{3/4} \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\)

Input:

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

Output:

-1/5*(c*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2])/a - (9*3^(1/4)*c^2*Sqrt[c*x]*Sqrt 
[3 - 2*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(5*2^(3/4)* 
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 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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

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 = 0.52 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.12

method	result	size
elliptic	\(\frac {\sqrt {c x}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (-\frac {c^{2} x \sqrt {-2 a c \,x^{3}+3 a c x}}{5 a}+\frac {c^{3} \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \left (-\sqrt {6}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{60 \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{c x \sqrt {-a \left (2 x^{2}-3\right )}}\)	\(184\)
risch	\(\frac {x^{2} \left (2 x^{2}-3\right ) c^{3}}{5 \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}+\frac {\sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \left (-\sqrt {6}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right ) c^{3} \sqrt {-c x a \left (2 x^{2}-3\right )}}{60 \sqrt {-2 a c \,x^{3}+3 a c x}\, \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\)	\(185\)
default	\(\frac {c^{2} \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \left (6 \sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}-3 \sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}-16 x^{4}+24 x^{2}\right )}{40 x a \left (2 x^{2}-3\right )}\)	\(235\)

Input:

int((c*x)^(5/2)/(-2*a*x^2+3*a)^(1/2),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)*(-1/5*c^2/ 
a*x*(-2*a*c*x^3+3*a*c*x)^(1/2)+1/60*c^3*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 = 50, normalized size of antiderivative = 0.30 \[ \int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}} \, dx=-\frac {2 \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c^{2} x - 9 \, \sqrt {2} \sqrt {-a c} c^{2} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right )}{10 \, a} \] Input:

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

Output:

-1/10*(2*sqrt(-2*a*x^2 + 3*a)*sqrt(c*x)*c^2*x - 9*sqrt(2)*sqrt(-a*c)*c^2*w 
eierstrassZeta(6, 0, weierstrassPInverse(6, 0, x)))/a
 

Sympy [A] (verification not implemented)

Time = 3.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.31 \[ \int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}} \, dx=\frac {\sqrt {3} c^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} \] Input:

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

Output:

sqrt(3)*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)/(6*sqrt(a)*gamma(11/4))
 

Maxima [F]

\[ \int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}} \, dx=\int { \frac {\left (c x\right )^{\frac {5}{2}}}{\sqrt {-2 \, a x^{2} + 3 \, a}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}} \, dx=\int { \frac {\left (c x\right )^{\frac {5}{2}}}{\sqrt {-2 \, a x^{2} + 3 \, a}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}} \, dx=\int \frac {{\left (c\,x\right )}^{5/2}}{\sqrt {3\,a-2\,a\,x^2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, c^{2} \left (-2 \sqrt {x}\, \sqrt {-2 x^{2}+3}\, x -9 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{2}+3}}{2 x^{2}-3}d x \right )\right )}{10 a} \] Input:

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

Output:

(sqrt(c)*sqrt(a)*c**2*( - 2*sqrt(x)*sqrt( - 2*x**2 + 3)*x - 9*int((sqrt(x) 
*sqrt( - 2*x**2 + 3))/(2*x**2 - 3),x)))/(10*a)