\(\int \frac {2+3 x^2+5 x^4}{x^2 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx\) [25]

Optimal result

Integrand size = 43, antiderivative size = 127 \[ \int \frac {2+3 x^2+5 x^4}{x^2 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=-\frac {\left (1-9 x^2\right ) \sqrt {2+x^2}}{x \sqrt {-1+3 x} \sqrt {1+3 x}}-\frac {4 \sqrt {2} \sqrt {1-9 x^2} E\left (\arcsin (3 x)\left |-\frac {1}{18}\right .\right )}{3 \sqrt {-1+3 x} \sqrt {1+3 x}}+\frac {11 \sqrt {1-9 x^2} \operatorname {EllipticF}\left (\arcsin (3 x),-\frac {1}{18}\right )}{3 \sqrt {1+3 x} \sqrt {-2+6 x}} \] Output:

-(-9*x^2+1)*(x^2+2)^(1/2)/x/(-1+3*x)^(1/2)/(1+3*x)^(1/2)-4/3*2^(1/2)*(-9*x 
^2+1)^(1/2)*EllipticE(3*x,1/6*I*2^(1/2))/(-1+3*x)^(1/2)/(1+3*x)^(1/2)+11/3 
*(-9*x^2+1)^(1/2)*EllipticF(3*x,1/6*I*2^(1/2))/(1+3*x)^(1/2)/(-2+6*x)^(1/2 
)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 21.68 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.28 \[ \int \frac {2+3 x^2+5 x^4}{x^2 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {\sqrt {\frac {-1+3 x}{1+3 x}} \left (\frac {9 (1+3 x) \left (2+x^2\right )}{x}-12 \left (2+x^2\right ) \left (1+\frac {2}{-1+3 x}\right )+\frac {4 i (1+3 x) \sqrt {\frac {2+x^2}{(1-3 x)^2}} E\left (\arcsin \left (\frac {\sqrt {-\frac {i}{3}+\sqrt {2}+\frac {19 i}{3-9 x}}}{2^{3/4}}\right )|\frac {12 \sqrt {2}}{17 i+6 \sqrt {2}}\right )}{\sqrt {\frac {i (1+3 x)}{\left (17 i+6 \sqrt {2}\right ) (-1+3 x)}}}-122 i (-1+3 x) \sqrt {\frac {i (1+3 x)}{\left (17 i+6 \sqrt {2}\right ) (-1+3 x)}} \sqrt {\frac {2+x^2}{(1-3 x)^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-\frac {i}{3}+\sqrt {2}+\frac {19 i}{3-9 x}}}{2^{3/4}}\right ),\frac {12 \sqrt {2}}{17 i+6 \sqrt {2}}\right )\right )}{9 \sqrt {2+x^2}} \] Input:

Integrate[(2 + 3*x^2 + 5*x^4)/(x^2*Sqrt[-1 + 3*x]*Sqrt[1 + 3*x]*Sqrt[2 + x 
^2]),x]
 

Output:

(Sqrt[(-1 + 3*x)/(1 + 3*x)]*((9*(1 + 3*x)*(2 + x^2))/x - 12*(2 + x^2)*(1 + 
 2/(-1 + 3*x)) + ((4*I)*(1 + 3*x)*Sqrt[(2 + x^2)/(1 - 3*x)^2]*EllipticE[Ar 
cSin[Sqrt[-1/3*I + Sqrt[2] + (19*I)/(3 - 9*x)]/2^(3/4)], (12*Sqrt[2])/(17* 
I + 6*Sqrt[2])])/Sqrt[(I*(1 + 3*x))/((17*I + 6*Sqrt[2])*(-1 + 3*x))] - (12 
2*I)*(-1 + 3*x)*Sqrt[(I*(1 + 3*x))/((17*I + 6*Sqrt[2])*(-1 + 3*x))]*Sqrt[( 
2 + x^2)/(1 - 3*x)^2]*EllipticF[ArcSin[Sqrt[-1/3*I + Sqrt[2] + (19*I)/(3 - 
 9*x)]/2^(3/4)], (12*Sqrt[2])/(17*I + 6*Sqrt[2])]))/(9*Sqrt[2 + x^2])
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.33, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2038, 7293, 2009}

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.

\(\displaystyle \int \frac {5 x^4+3 x^2+2}{x^2 \sqrt {3 x-1} \sqrt {3 x+1} \sqrt {x^2+2}} \, dx\)

\(\Big \downarrow \) 2038

\(\displaystyle \frac {\sqrt {9 x^2-1} \int \frac {5 x^4+3 x^2+2}{x^2 \sqrt {x^2+2} \sqrt {9 x^2-1}}dx}{\sqrt {3 x-1} \sqrt {3 x+1}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {\sqrt {9 x^2-1} \int \left (\frac {5 x^2}{\sqrt {x^2+2} \sqrt {9 x^2-1}}+\frac {3}{\sqrt {x^2+2} \sqrt {9 x^2-1}}+\frac {2}{\sqrt {x^2+2} \sqrt {9 x^2-1} x^2}\right )dx}{\sqrt {3 x-1} \sqrt {3 x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {9 x^2-1} \left (\frac {4 \sqrt {2} \sqrt {1-9 x^2} \operatorname {EllipticF}\left (\arcsin (3 x),-\frac {1}{18}\right )}{3 \sqrt {9 x^2-1}}+\frac {\sqrt {1-9 x^2} \operatorname {EllipticF}\left (\arcsin (3 x),-\frac {1}{18}\right )}{\sqrt {2} \sqrt {9 x^2-1}}-\frac {4 \sqrt {2} \sqrt {1-9 x^2} E\left (\arcsin (3 x)\left |-\frac {1}{18}\right .\right )}{3 \sqrt {9 x^2-1}}+\frac {\sqrt {x^2+2} \sqrt {9 x^2-1}}{x}\right )}{\sqrt {3 x-1} \sqrt {3 x+1}}\)

Input:

Int[(2 + 3*x^2 + 5*x^4)/(x^2*Sqrt[-1 + 3*x]*Sqrt[1 + 3*x]*Sqrt[2 + x^2]),x 
]
 

Output:

(Sqrt[-1 + 9*x^2]*((Sqrt[2 + x^2]*Sqrt[-1 + 9*x^2])/x - (4*Sqrt[2]*Sqrt[1 
- 9*x^2]*EllipticE[ArcSin[3*x], -1/18])/(3*Sqrt[-1 + 9*x^2]) + (Sqrt[1 - 9 
*x^2]*EllipticF[ArcSin[3*x], -1/18])/(Sqrt[2]*Sqrt[-1 + 9*x^2]) + (4*Sqrt[ 
2]*Sqrt[1 - 9*x^2]*EllipticF[ArcSin[3*x], -1/18])/(3*Sqrt[-1 + 9*x^2])))/( 
Sqrt[-1 + 3*x]*Sqrt[1 + 3*x])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2038

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p 
_)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_), x_Symbol] :> Simp[(a1 + b1*x^(n/2)) 
^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPart[p] 
)   Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, 
 b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(Eq 
Q[n, 2] && IGtQ[q, 0])
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 4.88 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.95

method	result	size
default	\(-\frac {\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}\, \left (23 i \sqrt {x^{2}+2}\, \sqrt {-9 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right ) x +4 i \sqrt {x^{2}+2}\, \sqrt {-9 x^{2}+1}\, x \operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )-81 x^{4}-153 x^{2}+18\right )}{9 x \left (9 x^{4}+17 x^{2}-2\right )}\)	\(121\)
elliptic	\(\frac {\sqrt {\left (x^{2}+2\right ) \left (9 x^{2}-1\right )}\, \left (-\frac {3 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )}{2 \sqrt {9 x^{4}+17 x^{2}-2}}+\frac {2 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )\right )}{9 \sqrt {9 x^{4}+17 x^{2}-2}}+\frac {\sqrt {9 x^{4}+17 x^{2}-2}}{x}\right )}{\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}}\)	\(178\)
risch	\(\frac {\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}}{x}+\frac {\left (-\frac {3 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )}{2 \sqrt {9 x^{4}+17 x^{2}-2}}+\frac {2 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )\right )}{9 \sqrt {9 x^{4}+17 x^{2}-2}}\right ) \sqrt {\left (-1+3 x \right ) \left (1+3 x \right ) \left (x^{2}+2\right )}}{\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}}\)	\(189\)

Input:

int((5*x^4+3*x^2+2)/x^2/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x,metho 
d=_RETURNVERBOSE)
 

Output:

-1/9*(-1+3*x)^(1/2)*(1+3*x)^(1/2)*(x^2+2)^(1/2)*(23*I*(x^2+2)^(1/2)*(-9*x^ 
2+1)^(1/2)*EllipticF(1/2*I*x*2^(1/2),3*I*2^(1/2))*x+4*I*(x^2+2)^(1/2)*(-9* 
x^2+1)^(1/2)*x*EllipticE(1/2*I*x*2^(1/2),3*I*2^(1/2))-81*x^4-153*x^2+18)/x 
/(9*x^4+17*x^2-2)
 

Fricas [F]

\[ \int \frac {2+3 x^2+5 x^4}{x^2 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int { \frac {5 \, x^{4} + 3 \, x^{2} + 2}{\sqrt {x^{2} + 2} \sqrt {3 \, x + 1} \sqrt {3 \, x - 1} x^{2}} \,d x } \] Input:

integrate((5*x^4+3*x^2+2)/x^2/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x 
, algorithm="fricas")
 

Output:

integral((5*x^4 + 3*x^2 + 2)*sqrt(x^2 + 2)*sqrt(3*x + 1)*sqrt(3*x - 1)/(9* 
x^6 + 17*x^4 - 2*x^2), x)
 

Sympy [F]

\[ \int \frac {2+3 x^2+5 x^4}{x^2 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int \frac {5 x^{4} + 3 x^{2} + 2}{x^{2} \sqrt {3 x - 1} \sqrt {3 x + 1} \sqrt {x^{2} + 2}}\, dx \] Input:

integrate((5*x**4+3*x**2+2)/x**2/(-1+3*x)**(1/2)/(1+3*x)**(1/2)/(x**2+2)** 
(1/2),x)
 

Output:

Integral((5*x**4 + 3*x**2 + 2)/(x**2*sqrt(3*x - 1)*sqrt(3*x + 1)*sqrt(x**2 
 + 2)), x)
 

Maxima [F]

\[ \int \frac {2+3 x^2+5 x^4}{x^2 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int { \frac {5 \, x^{4} + 3 \, x^{2} + 2}{\sqrt {x^{2} + 2} \sqrt {3 \, x + 1} \sqrt {3 \, x - 1} x^{2}} \,d x } \] Input:

integrate((5*x^4+3*x^2+2)/x^2/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x 
, algorithm="maxima")
 

Output:

integrate((5*x^4 + 3*x^2 + 2)/(sqrt(x^2 + 2)*sqrt(3*x + 1)*sqrt(3*x - 1)*x 
^2), x)
 

Giac [F]

\[ \int \frac {2+3 x^2+5 x^4}{x^2 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int { \frac {5 \, x^{4} + 3 \, x^{2} + 2}{\sqrt {x^{2} + 2} \sqrt {3 \, x + 1} \sqrt {3 \, x - 1} x^{2}} \,d x } \] Input:

integrate((5*x^4+3*x^2+2)/x^2/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x 
, algorithm="giac")
 

Output:

integrate((5*x^4 + 3*x^2 + 2)/(sqrt(x^2 + 2)*sqrt(3*x + 1)*sqrt(3*x - 1)*x 
^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2+3 x^2+5 x^4}{x^2 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int \frac {5\,x^4+3\,x^2+2}{x^2\,\sqrt {3\,x-1}\,\sqrt {3\,x+1}\,\sqrt {x^2+2}} \,d x \] Input:

int((3*x^2 + 5*x^4 + 2)/(x^2*(3*x - 1)^(1/2)*(3*x + 1)^(1/2)*(x^2 + 2)^(1/ 
2)),x)
 

Output:

int((3*x^2 + 5*x^4 + 2)/(x^2*(3*x - 1)^(1/2)*(3*x + 1)^(1/2)*(x^2 + 2)^(1/ 
2)), x)
 

Reduce [F]

\[ \int \frac {2+3 x^2+5 x^4}{x^2 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {5 \sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}+8 \left (\int \frac {\sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}}{9 x^{6}+17 x^{4}-2 x^{2}}d x \right ) x +27 \left (\int \frac {\sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}}{9 x^{4}+17 x^{2}-2}d x \right ) x}{9 x} \] Input:

int((5*x^4+3*x^2+2)/x^2/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

(5*sqrt(3*x + 1)*sqrt(3*x - 1)*sqrt(x**2 + 2) + 8*int((sqrt(3*x + 1)*sqrt( 
3*x - 1)*sqrt(x**2 + 2))/(9*x**6 + 17*x**4 - 2*x**2),x)*x + 27*int((sqrt(3 
*x + 1)*sqrt(3*x - 1)*sqrt(x**2 + 2))/(9*x**4 + 17*x**2 - 2),x)*x)/(9*x)