\(\int \frac {2+3 x^2+5 x^4}{x^7 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx\) [22]

Optimal result

Integrand size = 43, antiderivative size = 168 \[ \int \frac {2+3 x^2+5 x^4}{x^7 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}}{6 x^6}+\frac {103 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}}{48 x^4}+\frac {1927 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}}{64 x^2}-\frac {35231 \sqrt {-1+9 x^2} \arctan \left (\frac {\sqrt {2+x^2}}{\sqrt {2} \sqrt {-1+9 x^2}}\right )}{64 \sqrt {2} \sqrt {-1+3 x} \sqrt {1+3 x}} \] Output:

1/6*(-1+3*x)^(1/2)*(1+3*x)^(1/2)*(x^2+2)^(1/2)/x^6+103/48*(-1+3*x)^(1/2)*( 
1+3*x)^(1/2)*(x^2+2)^(1/2)/x^4+1927/64*(-1+3*x)^(1/2)*(1+3*x)^(1/2)*(x^2+2 
)^(1/2)/x^2-35231/128*2^(1/2)*(9*x^2-1)^(1/2)*arctan(1/2*(x^2+2)^(1/2)*2^( 
1/2)/(9*x^2-1)^(1/2))/(-1+3*x)^(1/2)/(1+3*x)^(1/2)
 

Mathematica [A] (verified)

Time = 10.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.55 \[ \int \frac {2+3 x^2+5 x^4}{x^7 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {2 \sqrt {2+x^2} \left (-32-124 x^2-2073 x^4+52029 x^6\right )-105693 x^6 \sqrt {-2+18 x^2} \arctan \left (\frac {\sqrt {2+x^2}}{\sqrt {-2+18 x^2}}\right )}{384 x^6 \sqrt {-1+3 x} \sqrt {1+3 x}} \] Input:

Integrate[(2 + 3*x^2 + 5*x^4)/(x^7*Sqrt[-1 + 3*x]*Sqrt[1 + 3*x]*Sqrt[2 + x 
^2]),x]
 

Output:

(2*Sqrt[2 + x^2]*(-32 - 124*x^2 - 2073*x^4 + 52029*x^6) - 105693*x^6*Sqrt[ 
-2 + 18*x^2]*ArcTan[Sqrt[2 + x^2]/Sqrt[-2 + 18*x^2]])/(384*x^6*Sqrt[-1 + 3 
*x]*Sqrt[1 + 3*x])
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.88, 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.

\(\displaystyle \int \frac {5 x^4+3 x^2+2}{x^7 \sqrt {3 x-1} \sqrt {3 x+1} \sqrt {x^2+2}} \, dx\)

\(\Big \downarrow \) 2038

\(\displaystyle \frac {\sqrt {9 x^2-1} \int \frac {5 x^4+3 x^2+2}{x^7 \sqrt {x^2+2} \sqrt {9 x^2-1}}dx}{\sqrt {3 x-1} \sqrt {3 x+1}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {\sqrt {9 x^2-1} \int \left (\frac {5}{x^3 \sqrt {x^2+2} \sqrt {9 x^2-1}}+\frac {3}{x^5 \sqrt {x^2+2} \sqrt {9 x^2-1}}+\frac {2}{x^7 \sqrt {x^2+2} \sqrt {9 x^2-1}}\right )dx}{\sqrt {3 x-1} \sqrt {3 x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {9 x^2-1} \left (-\frac {35231 \arctan \left (\frac {\sqrt {x^2+2}}{\sqrt {2} \sqrt {9 x^2-1}}\right )}{64 \sqrt {2}}+\frac {1927 \sqrt {x^2+2} \sqrt {9 x^2-1}}{64 x^2}+\frac {\sqrt {x^2+2} \sqrt {9 x^2-1}}{6 x^6}+\frac {103 \sqrt {x^2+2} \sqrt {9 x^2-1}}{48 x^4}\right )}{\sqrt {3 x-1} \sqrt {3 x+1}}\)

Input:

Int[(2 + 3*x^2 + 5*x^4)/(x^7*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])/(6*x^6) + (103*Sqrt[2 
+ x^2]*Sqrt[-1 + 9*x^2])/(48*x^4) + (1927*Sqrt[2 + x^2]*Sqrt[-1 + 9*x^2])/ 
(64*x^2) - (35231*ArcTan[Sqrt[2 + x^2]/(Sqrt[2]*Sqrt[-1 + 9*x^2])])/(64*Sq 
rt[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 = 0.87 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.66

method	result	size
risch	\(\frac {\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}\, \left (5781 x^{4}+412 x^{2}+32\right )}{192 x^{6}}+\frac {35231 \sqrt {2}\, \arctan \left (\frac {\left (17 x^{2}-4\right ) \sqrt {2}}{4 \sqrt {9 x^{4}+17 x^{2}-2}}\right ) \sqrt {\left (-1+3 x \right ) \left (1+3 x \right ) \left (x^{2}+2\right )}}{256 \sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}}\)	\(111\)
elliptic	\(\frac {\sqrt {\left (x^{2}+2\right ) \left (9 x^{2}-1\right )}\, \left (\frac {\sqrt {9 x^{4}+17 x^{2}-2}}{6 x^{6}}+\frac {103 \sqrt {9 x^{4}+17 x^{2}-2}}{48 x^{4}}+\frac {1927 \sqrt {9 x^{4}+17 x^{2}-2}}{64 x^{2}}+\frac {35231 \sqrt {2}\, \arctan \left (\frac {\left (17 x^{2}-4\right ) \sqrt {2}}{4 \sqrt {9 x^{4}+17 x^{2}-2}}\right )}{256}\right )}{\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}}\)	\(128\)
default	\(\frac {\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}\, \left (105693 \sqrt {2}\, \arctan \left (\frac {\left (17 x^{2}-4\right ) \sqrt {2}}{4 \sqrt {9 x^{4}+17 x^{2}-2}}\right ) x^{6}-1005696 \ln \left (\frac {17}{6}+3 x^{2}+\sqrt {9 x^{4}+17 x^{2}-2}\right ) x^{6}+1005696 \ln \left (6 x^{2}+\frac {17}{3}+2 \sqrt {9 x^{4}+17 x^{2}-2}\right ) x^{6}+23124 x^{4} \sqrt {9 x^{4}+17 x^{2}-2}+1648 \sqrt {9 x^{4}+17 x^{2}-2}\, x^{2}+128 \sqrt {9 x^{4}+17 x^{2}-2}\right )}{768 \sqrt {9 x^{4}+17 x^{2}-2}\, x^{6}}\)	\(187\)

Input:

int((5*x^4+3*x^2+2)/x^7/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x,metho 
d=_RETURNVERBOSE)
 

Output:

1/192*(-1+3*x)^(1/2)*(1+3*x)^(1/2)*(x^2+2)^(1/2)*(5781*x^4+412*x^2+32)/x^6 
+35231/256*2^(1/2)*arctan(1/4*(17*x^2-4)*2^(1/2)/(9*x^4+17*x^2-2)^(1/2))*( 
(-1+3*x)*(1+3*x)*(x^2+2))^(1/2)/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.54 \[ \int \frac {2+3 x^2+5 x^4}{x^7 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {105693 \, \sqrt {2} x^{6} \arctan \left (-\frac {3}{2} \, \sqrt {2} x^{2} + \frac {1}{2} \, \sqrt {2} \sqrt {x^{2} + 2} \sqrt {3 \, x + 1} \sqrt {3 \, x - 1}\right ) + 34686 \, x^{6} + 2 \, {\left (5781 \, x^{4} + 412 \, x^{2} + 32\right )} \sqrt {x^{2} + 2} \sqrt {3 \, x + 1} \sqrt {3 \, x - 1}}{384 \, x^{6}} \] Input:

integrate((5*x^4+3*x^2+2)/x^7/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x 
, algorithm="fricas")
 

Output:

1/384*(105693*sqrt(2)*x^6*arctan(-3/2*sqrt(2)*x^2 + 1/2*sqrt(2)*sqrt(x^2 + 
 2)*sqrt(3*x + 1)*sqrt(3*x - 1)) + 34686*x^6 + 2*(5781*x^4 + 412*x^2 + 32) 
*sqrt(x^2 + 2)*sqrt(3*x + 1)*sqrt(3*x - 1))/x^6
 

Sympy [F(-1)]

Timed out. \[ \int \frac {2+3 x^2+5 x^4}{x^7 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\text {Timed out} \] Input:

integrate((5*x**4+3*x**2+2)/x**7/(-1+3*x)**(1/2)/(1+3*x)**(1/2)/(x**2+2)** 
(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {2+3 x^2+5 x^4}{x^7 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int { \frac {5 \, x^{4} + 3 \, x^{2} + 2}{\sqrt {x^{2} + 2} \sqrt {3 \, x + 1} \sqrt {3 \, x - 1} x^{7}} \,d x } \] Input:

integrate((5*x^4+3*x^2+2)/x^7/(-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 
^7), x)
 

Giac [F]

\[ \int \frac {2+3 x^2+5 x^4}{x^7 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int { \frac {5 \, x^{4} + 3 \, x^{2} + 2}{\sqrt {x^{2} + 2} \sqrt {3 \, x + 1} \sqrt {3 \, x - 1} x^{7}} \,d x } \] Input:

integrate((5*x^4+3*x^2+2)/x^7/(-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 
^7), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2+3 x^2+5 x^4}{x^7 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int \frac {5\,x^4+3\,x^2+2}{x^7\,\sqrt {3\,x-1}\,\sqrt {3\,x+1}\,\sqrt {x^2+2}} \,d x \] Input:

int((3*x^2 + 5*x^4 + 2)/(x^7*(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^7*(3*x - 1)^(1/2)*(3*x + 1)^(1/2)*(x^2 + 2)^(1/ 
2)), x)
 

Reduce [B] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.65 \[ \int \frac {2+3 x^2+5 x^4}{x^7 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {-105693 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}\, \sqrt {2}}{18 x^{2}-2}\right ) x^{6}+11562 \sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}\, x^{4}+824 \sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}\, x^{2}+64 \sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}}{384 x^{6}} \] Input:

int((5*x^4+3*x^2+2)/x^7/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x)
 

Output:

( - 105693*sqrt(2)*atan((sqrt(3*x + 1)*sqrt(3*x - 1)*sqrt(x**2 + 2)*sqrt(2 
))/(18*x**2 - 2))*x**6 + 11562*sqrt(3*x + 1)*sqrt(3*x - 1)*sqrt(x**2 + 2)* 
x**4 + 824*sqrt(3*x + 1)*sqrt(3*x - 1)*sqrt(x**2 + 2)*x**2 + 64*sqrt(3*x + 
 1)*sqrt(3*x - 1)*sqrt(x**2 + 2))/(384*x**6)