\(\int \frac {2+3 x^2+5 x^4}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx\) [24]

Optimal result

Integrand size = 40, antiderivative size = 125 \[ \int \frac {2+3 x^2+5 x^4}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {5}{27} x \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}-\frac {89 \sqrt {2} \sqrt {1-9 x^2} E\left (\arcsin (3 x)\left |-\frac {1}{18}\right .\right )}{81 \sqrt {-1+3 x} \sqrt {1+3 x}}+\frac {121 \sqrt {2} \sqrt {1-9 x^2} \operatorname {EllipticF}\left (\arcsin (3 x),-\frac {1}{18}\right )}{81 \sqrt {-1+3 x} \sqrt {1+3 x}} \] Output:

5/27*x*(-1+3*x)^(1/2)*(1+3*x)^(1/2)*(x^2+2)^(1/2)-89/81*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)+121/81*2^ 
(1/2)*(-9*x^2+1)^(1/2)*EllipticF(3*x,1/6*I*2^(1/2))/(-1+3*x)^(1/2)/(1+3*x) 
^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 32.34 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.30 \[ \int \frac {2+3 x^2+5 x^4}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {1}{243} \sqrt {\frac {-1+3 x}{1+3 x}} \sqrt {2+x^2} \left (45 x (1+3 x)-267 \left (1+\frac {2}{-1+3 x}\right )+\frac {89 i (1+3 x) \sqrt {\frac {2+x^2}{(1-3 x)^2}} E\left (\arcsin \left (\frac {\sqrt {-\frac {i}{3}+\sqrt {2}+\frac {19 i}{3-9 x}}}{2^{3/4}}\right )|\frac {12 \sqrt {2}}{17 i+6 \sqrt {2}}\right )}{\sqrt {\frac {i (1+3 x)}{\left (17 i+6 \sqrt {2}\right ) (-1+3 x)}} \left (2+x^2\right )}-\frac {2665 i \sqrt {\frac {i (1+3 x)}{\left (17 i+6 \sqrt {2}\right ) (-1+3 x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-\frac {i}{3}+\sqrt {2}+\frac {19 i}{3-9 x}}}{2^{3/4}}\right ),\frac {12 \sqrt {2}}{17 i+6 \sqrt {2}}\right )}{(-1+3 x) \sqrt {\frac {2+x^2}{(1-3 x)^2}}}\right ) \] Input:

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

Output:

(Sqrt[(-1 + 3*x)/(1 + 3*x)]*Sqrt[2 + x^2]*(45*x*(1 + 3*x) - 267*(1 + 2/(-1 
 + 3*x)) + ((89*I)*(1 + 3*x)*Sqrt[(2 + x^2)/(1 - 3*x)^2]*EllipticE[ArcSin[ 
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))]*(2 + x^2) 
) - ((2665*I)*Sqrt[(I*(1 + 3*x))/((17*I + 6*Sqrt[2])*(-1 + 3*x))]*Elliptic 
F[ArcSin[Sqrt[-1/3*I + Sqrt[2] + (19*I)/(3 - 9*x)]/2^(3/4)], (12*Sqrt[2])/ 
(17*I + 6*Sqrt[2])])/((-1 + 3*x)*Sqrt[(2 + x^2)/(1 - 3*x)^2])))/243
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, 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.

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

\(\Big \downarrow \) 2038

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {9 x^2-1} \left (\frac {121 \sqrt {2} \sqrt {1-9 x^2} \operatorname {EllipticF}\left (\arcsin (3 x),-\frac {1}{18}\right )}{81 \sqrt {9 x^2-1}}-\frac {89 \sqrt {2} \sqrt {1-9 x^2} E\left (\arcsin (3 x)\left |-\frac {1}{18}\right .\right )}{81 \sqrt {9 x^2-1}}+\frac {5}{27} \sqrt {x^2+2} \sqrt {9 x^2-1} x\right )}{\sqrt {3 x-1} \sqrt {3 x+1}}\)

Input:

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

Output:

(Sqrt[-1 + 9*x^2]*((5*x*Sqrt[2 + x^2]*Sqrt[-1 + 9*x^2])/27 - (89*Sqrt[2]*S 
qrt[1 - 9*x^2]*EllipticE[ArcSin[3*x], -1/18])/(81*Sqrt[-1 + 9*x^2]) + (121 
*Sqrt[2]*Sqrt[1 - 9*x^2]*EllipticF[ArcSin[3*x], -1/18])/(81*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 = 4.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}\, \left (-405 x^{5}+487 i \sqrt {x^{2}+2}\, \sqrt {-9 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )+89 i \sqrt {x^{2}+2}\, \sqrt {-9 x^{2}+1}\, \operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )-765 x^{3}+90 x \right )}{243 \left (9 x^{4}+17 x^{2}-2\right )}\)	\(118\)
elliptic	\(\frac {\sqrt {\left (x^{2}+2\right ) \left (9 x^{2}-1\right )}\, \left (\frac {5 x \sqrt {9 x^{4}+17 x^{2}-2}}{27}-\frac {32 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )}{27 \sqrt {9 x^{4}+17 x^{2}-2}}+\frac {89 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )\right )}{486 \sqrt {9 x^{4}+17 x^{2}-2}}\right )}{\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}}\)	\(177\)
risch	\(\frac {5 x \sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}}{27}+\frac {\left (-\frac {32 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )}{27 \sqrt {9 x^{4}+17 x^{2}-2}}+\frac {89 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )\right )}{486 \sqrt {9 x^{4}+17 x^{2}-2}}\right ) \sqrt {\left (-1+3 x \right ) \left (1+3 x \right ) \left (x^{2}+2\right )}}{\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}}\)	\(188\)

Input:

int((5*x^4+3*x^2+2)/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x,method=_R 
ETURNVERBOSE)
 

Output:

-1/243*(-1+3*x)^(1/2)*(1+3*x)^(1/2)*(x^2+2)^(1/2)*(-405*x^5+487*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))+89*I*(x^2+2) 
^(1/2)*(-9*x^2+1)^(1/2)*EllipticE(1/2*I*x*2^(1/2),3*I*2^(1/2))-765*x^3+90* 
x)/(9*x^4+17*x^2-2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.46 \[ \int \frac {2+3 x^2+5 x^4}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {9 \, {\left (45 \, x^{2} - 89\right )} \sqrt {x^{2} + 2} \sqrt {3 \, x + 1} \sqrt {3 \, x - 1} - 89 \, x E(\arcsin \left (\frac {1}{3 \, x}\right )\,|\,-18) - 5095 \, x F(\arcsin \left (\frac {1}{3 \, x}\right )\,|\,-18)}{2187 \, x} \] Input:

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

Output:

1/2187*(9*(45*x^2 - 89)*sqrt(x^2 + 2)*sqrt(3*x + 1)*sqrt(3*x - 1) - 89*x*e 
lliptic_e(arcsin(1/3/x), -18) - 5095*x*elliptic_f(arcsin(1/3/x), -18))/x
 

Sympy [F]

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

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Reduce [F]

\[ \int \frac {2+3 x^2+5 x^4}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {5 \sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}\, x}{27}-\frac {89 \left (\int \frac {\sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}\, x^{2}}{9 x^{4}+17 x^{2}-2}d x \right )}{27}+\frac {64 \left (\int \frac {\sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}}{9 x^{4}+17 x^{2}-2}d x \right )}{27} \] Input:

int((5*x^4+3*x^2+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)*x - 89*int((sqrt(3*x + 1)*sq 
rt(3*x - 1)*sqrt(x**2 + 2)*x**2)/(9*x**4 + 17*x**2 - 2),x) + 64*int((sqrt( 
3*x + 1)*sqrt(3*x - 1)*sqrt(x**2 + 2))/(9*x**4 + 17*x**2 - 2),x))/27