\(\int \frac {1}{x^2 (1+2 x+\sqrt {-2+8 x-5 x^2})^2} \, dx\) [56]

Optimal result

Integrand size = 25, antiderivative size = 596 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=-\frac {\sqrt {\frac {2}{3}} \left (379-154 \sqrt {6}+\frac {40 \left (3949-1511 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{\left (191-74 \sqrt {6}\right ) \left (4-\sqrt {6}-5 x\right )}\right )}{69 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )}-\frac {\sqrt {\frac {2}{3}} \left (4-\sqrt {6}-5 x\right )^2 \left (24+11 \sqrt {6}-\frac {10 \left (3+2 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )}{5 x \left (2 \left (3-2 \sqrt {6}\right )+5 \sqrt {6} x\right ) \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )}+\frac {32}{27} \sqrt {2} \arctan \left (\frac {\sqrt {11-4 \sqrt {6}} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )+\frac {6032 \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{621 \sqrt {23}}+\frac {28}{27} \log \left (\frac {x \left (2 \left (26241-10774 \sqrt {6}\right )-5 \left (8064-3371 \sqrt {6}\right ) x\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )-\frac {28}{27} \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+12 x-3 \sqrt {6} x+10 \sqrt {6} x^2+6 \sqrt {-2+8 x-5 x^2}-4 \sqrt {6} \sqrt {-2+8 x-5 x^2}+5 \sqrt {6} x \sqrt {-2+8 x-5 x^2}}{\left (4-\sqrt {6}-5 x\right )^2}\right ) \] Output:

-1/207*6^(1/2)*(379-154*6^(1/2)+40*(3949-1511*6^(1/2))*(-5*x^2+8*x-2)^(1/2 
)/(191-74*6^(1/2))/(4-6^(1/2)-5*x))/(13+2*6^(1/2)-10*6^(1/2)*(-5*x^2+8*x-2 
)^(1/2)/(4-6^(1/2)-5*x)-5*(13-2*6^(1/2))*(5*x^2-8*x+2)/(4-6^(1/2)-5*x)^2)- 
1/15*6^(1/2)*(4-6^(1/2)-5*x)^2*(24+11*6^(1/2)-10*(3+2*6^(1/2))*(-5*x^2+8*x 
-2)^(1/2)/(4-6^(1/2)-5*x))/x/(6-4*6^(1/2)+5*x*6^(1/2))/(13+2*6^(1/2)-10*6^ 
(1/2)*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x)-5*(13-2*6^(1/2))*(5*x^2-8*x+2)/ 
(4-6^(1/2)-5*x)^2)+32/27*2^(1/2)*arctan((2*2^(1/2)-3^(1/2))*(-5*x^2+8*x-2) 
^(1/2)/(4-6^(1/2)-5*x))+6032/14283*arctan(1/138*(6+(12-13*6^(1/2))*(-5*x^2 
+8*x-2)^(1/2)/(4-6^(1/2)-5*x))*138^(1/2))*23^(1/2)+28/27*ln(x*(52482-21548 
*6^(1/2)-5*(8064-3371*6^(1/2))*x)/(4-6^(1/2)-5*x)^2)-28/27*ln((6-4*6^(1/2) 
+12*x-3*x*6^(1/2)+10*6^(1/2)*x^2+6*(-5*x^2+8*x-2)^(1/2)-4*6^(1/2)*(-5*x^2+ 
8*x-2)^(1/2)+5*6^(1/2)*x*(-5*x^2+8*x-2)^(1/2))/(4-6^(1/2)-5*x)^2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 13.71 (sec) , antiderivative size = 1190, normalized size of antiderivative = 2.00 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[1/(x^2*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^2),x]
 

Output:

1/(9*x) + (286 + 369*x)/(207*(3 - 4*x + 9*x^2)) + Sqrt[-2 + 8*x - 5*x^2]*( 
2/(9*x) + (-244 + 63*x)/(207*(3 - 4*x + 9*x^2))) + (3016*ArcTan[(-2 + 9*x) 
/Sqrt[23]])/(621*Sqrt[23]) - (16*Sqrt[2]*ArcTan[(1 - 2*x)/Sqrt[-1 + 4*x - 
(5*x^2)/2]])/27 + (2*(1912 + (65*I)*Sqrt[23])*ArcTan[(23*(72*(-345941 + (5 
6628*I)*Sqrt[23]) + 24*(6307633 - (567996*I)*Sqrt[23])*x + (-296303111 - ( 
75400*I)*Sqrt[23])*x^2 + 40*(5759561 + (123370*I)*Sqrt[23])*x^3 + (25*I)*( 
2542541*I + 43940*Sqrt[23])*x^4))/(130*(5716880*I + 3093953*Sqrt[23])*x^4 
+ x^2*(-3657150328*I + 654251390*Sqrt[23] - 23351496*Sqrt[23*(77 - (52*I)* 
Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) - 36*(7775196*I + 984347*Sqrt[23] + 138 
997*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + 3*x*(6444311 
12*I - 5168800*Sqrt[23] + 4308907*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 
+ 8*x - 5*x^2]) + x^3*(845051740*I - 1057253912*Sqrt[23] + 18764595*Sqrt[2 
3*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]))])/(69*Sqrt[23*(77 - (52 
*I)*Sqrt[23])]) + (((2*I)/69)*(1912*I + 65*Sqrt[23])*ArcTan[(23*(72*(34594 
1 + (56628*I)*Sqrt[23]) - (24*I)*(-6307633*I + 567996*Sqrt[23])*x + 13*(22 
792547 - (5800*I)*Sqrt[23])*x^2 + (40*I)*(5759561*I + 123370*Sqrt[23])*x^3 
 + 25*(2542541 + (43940*I)*Sqrt[23])*x^4))/(130*(-5716880*I + 3093953*Sqrt 
[23])*x^4 + x^2*(3657150328*I + 654251390*Sqrt[23] - 23351496*Sqrt[23*(77 
+ (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) - 36*(-7775196*I + 984347*Sqrt 
[23] + 138997*Sqrt[23*(77 + (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) +...
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {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.

\(\displaystyle \int \frac {1}{x^2 \left (\sqrt {-5 x^2+8 x-2}+2 x+1\right )^2} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {139-252 x}{27 \left (9 x^2-4 x+3\right )}-\frac {28 \sqrt {-5 x^2+8 x-2}}{27 x}-\frac {2 \sqrt {-5 x^2+8 x-2}}{9 x^2}+\frac {28 x \sqrt {-5 x^2+8 x-2}}{3 \left (9 x^2-4 x+3\right )}-\frac {58 \sqrt {-5 x^2+8 x-2}}{27 \left (9 x^2-4 x+3\right )}-\frac {1}{9 x^2}-\frac {2 (144 x-73)}{9 \left (9 x^2-4 x+3\right )^2}+\frac {20 x \sqrt {-5 x^2+8 x-2}}{\left (9 x^2-4 x+3\right )^2}-\frac {26 \sqrt {-5 x^2+8 x-2}}{9 \left (9 x^2-4 x+3\right )^2}+\frac {28}{27 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3016 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{621 \sqrt {23}}-\frac {16}{27} \sqrt {2} \arctan \left (\frac {\sqrt {2} (1-2 x)}{\sqrt {-5 x^2+8 x-2}}\right )-\frac {3016 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{621 \sqrt {23}}-\frac {28}{27} \text {arctanh}\left (\frac {2 x+1}{\sqrt {-5 x^2+8 x-2}}\right )+\frac {13 \sqrt {-5 x^2+8 x-2} (2-9 x)}{207 \left (9 x^2-4 x+3\right )}+\frac {2 \sqrt {-5 x^2+8 x-2}}{9 x}+\frac {369 x+286}{207 \left (9 x^2-4 x+3\right )}-\frac {10 (3-2 x) \sqrt {-5 x^2+8 x-2}}{23 \left (9 x^2-4 x+3\right )}-\frac {14}{27} \log \left (9 x^2-4 x+3\right )+\frac {1}{9 x}+\frac {28 \log (x)}{27}\)

Input:

Int[1/(x^2*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^2),x]
 

Output:

1/(9*x) + (2*Sqrt[-2 + 8*x - 5*x^2])/(9*x) + (286 + 369*x)/(207*(3 - 4*x + 
 9*x^2)) + (13*(2 - 9*x)*Sqrt[-2 + 8*x - 5*x^2])/(207*(3 - 4*x + 9*x^2)) - 
 (10*(3 - 2*x)*Sqrt[-2 + 8*x - 5*x^2])/(23*(3 - 4*x + 9*x^2)) - (3016*ArcT 
an[(2 - 9*x)/Sqrt[23]])/(621*Sqrt[23]) + (3016*ArcTan[(8 - 13*x)/(Sqrt[23] 
*Sqrt[-2 + 8*x - 5*x^2])])/(621*Sqrt[23]) - (16*Sqrt[2]*ArcTan[(Sqrt[2]*(1 
 - 2*x))/Sqrt[-2 + 8*x - 5*x^2]])/27 - (28*ArcTanh[(1 + 2*x)/Sqrt[-2 + 8*x 
 - 5*x^2]])/27 + (28*Log[x])/27 - (14*Log[3 - 4*x + 9*x^2])/27
 

Defintions of rubi rules used

rule 2009

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

rule 7293

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.99 (sec) , antiderivative size = 1479, normalized size of antiderivative = 2.48

method	result	size
trager	\(\text {Expression too large to display}\)	\(1479\)
default	\(\text {Expression too large to display}\)	\(6067\)

Input:

int(1/x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/1656*(x-1)*(7551*x^2-413*x+552)/x/(9*x^2-4*x+3)+1/207*(477*x^2-428*x+13 
8)/x/(9*x^2-4*x+3)*(-5*x^2+8*x-2)^(1/2)-56/27*ln(-(29817114624*RootOf(192* 
_Z^2-2576*_Z+14283)^2*RootOf(4416*_Z^2+59248*_Z+388233)^2*x-29817114624*Ro 
otOf(192*_Z^2-2576*_Z+14283)^2*RootOf(4416*_Z^2+59248*_Z+388233)^2-5239884 
99456*RootOf(192*_Z^2-2576*_Z+14283)*RootOf(4416*_Z^2+59248*_Z+388233)^2*x 
[phone]*RootOf(192*_Z^2-2576*_Z+14283)^2*RootOf(4416*_Z^2+59248*_Z+3 
88233)*x[phone]*RootOf(192*_Z^2-2576*_Z+14283)*RootOf(4416*_Z^2+5924 
8*_Z+388233)*(-5*x^2+8*x-2)^(1/2)[phone]*RootOf(192*_Z^2-2576*_Z+142 
83)*RootOf(4416*_Z^2+59248*_Z+388233)^2+2072399464512*RootOf(4416*_Z^2+592 
48*_Z+388233)^2*x-322345248768*RootOf(192*_Z^2-2576*_Z+14283)^2*RootOf(441 
6*_Z^2+59248*_Z+388233)-9746245893120*RootOf(4416*_Z^2+59248*_Z+388233)*Ro 
otOf(192*_Z^2-2576*_Z+14283)*x[phone]*RootOf(192*_Z^2-2576*_Z+14283) 
^2*x[phone]*(-5*x^2+8*x-2)^(1/2)*RootOf(4416*_Z^2+59248*_Z+388233)+ 
15252499878720*RootOf(192*_Z^2-2576*_Z+14283)*(-5*x^2+8*x-2)^(1/2)-2072399 
464512*RootOf(4416*_Z^2+59248*_Z+388233)^2+8000476420608*RootOf(192*_Z^2-2 
576*_Z+14283)*RootOf(4416*_Z^2+59248*_Z+388233)[phone]*RootOf(4416 
*_Z^2+59248*_Z+388233)*x-189952217280*RootOf(192*_Z^2-2576*_Z+14283)^2-191 
17285857168*RootOf(192*_Z^2-2576*_Z+14283)*x-84760478467704*(-5*x^2+8*x-2) 
^(1/2)-41730920513568*RootOf(4416*_Z^2+59248*_Z+388233)[phone]*Roo 
tOf(192*_Z^2-2576*_Z+14283)[phone]*x-148983165585891)/x)-32/20...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\frac {3016 \, \sqrt {23} {\left (9 \, x^{3} - 4 \, x^{2} + 3 \, x\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) - 8464 \, \sqrt {2} {\left (9 \, x^{3} - 4 \, x^{2} + 3 \, x\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x - 1\right )}}{5 \, x^{2} - 8 \, x + 2}\right ) + 1508 \, \sqrt {23} {\left (9 \, x^{3} - 4 \, x^{2} + 3 \, x\right )} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} + 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + 1508 \, \sqrt {23} {\left (9 \, x^{3} - 4 \, x^{2} + 3 \, x\right )} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} - 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + 39744 \, x^{2} - 7406 \, {\left (9 \, x^{3} - 4 \, x^{2} + 3 \, x\right )} \log \left (9 \, x^{2} - 4 \, x + 3\right ) + 14812 \, {\left (9 \, x^{3} - 4 \, x^{2} + 3 \, x\right )} \log \left (x\right ) + 3703 \, {\left (9 \, x^{3} - 4 \, x^{2} + 3 \, x\right )} \log \left (-\frac {x^{2} + 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) - 3703 \, {\left (9 \, x^{3} - 4 \, x^{2} + 3 \, x\right )} \log \left (-\frac {x^{2} - 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) + 69 \, {\left (477 \, x^{2} - 428 \, x + 138\right )} \sqrt {-5 \, x^{2} + 8 \, x - 2} + 13386 \, x + 4761}{14283 \, {\left (9 \, x^{3} - 4 \, x^{2} + 3 \, x\right )}} \] Input:

integrate(1/x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="fricas")
 

Output:

1/14283*(3016*sqrt(23)*(9*x^3 - 4*x^2 + 3*x)*arctan(1/23*sqrt(23)*(9*x - 2 
)) - 8464*sqrt(2)*(9*x^3 - 4*x^2 + 3*x)*arctan(sqrt(2)*sqrt(-5*x^2 + 8*x - 
 2)*(2*x - 1)/(5*x^2 - 8*x + 2)) + 1508*sqrt(23)*(9*x^3 - 4*x^2 + 3*x)*arc 
tan(1/23*(sqrt(23)*sqrt(-5*x^2 + 8*x - 2)*(13*x - 8) + 2*sqrt(23)*(2*x^2 - 
 3*x))/(7*x^2 - 8*x + 2)) + 1508*sqrt(23)*(9*x^3 - 4*x^2 + 3*x)*arctan(1/2 
3*(sqrt(23)*sqrt(-5*x^2 + 8*x - 2)*(13*x - 8) - 2*sqrt(23)*(2*x^2 - 3*x))/ 
(7*x^2 - 8*x + 2)) + 39744*x^2 - 7406*(9*x^3 - 4*x^2 + 3*x)*log(9*x^2 - 4* 
x + 3) + 14812*(9*x^3 - 4*x^2 + 3*x)*log(x) + 3703*(9*x^3 - 4*x^2 + 3*x)*l 
og(-(x^2 + 2*sqrt(-5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2) - 3703*(9*x 
^3 - 4*x^2 + 3*x)*log(-(x^2 - 2*sqrt(-5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 
1)/x^2) + 69*(477*x^2 - 428*x + 138)*sqrt(-5*x^2 + 8*x - 2) + 13386*x + 47 
61)/(9*x^3 - 4*x^2 + 3*x)
 

Sympy [F]

\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\int \frac {1}{x^{2} \left (2 x + \sqrt {- 5 x^{2} + 8 x - 2} + 1\right )^{2}}\, dx \] Input:

integrate(1/x**2/(1+2*x+(-5*x**2+8*x-2)**(1/2))**2,x)
 

Output:

Integral(1/(x**2*(2*x + sqrt(-5*x**2 + 8*x - 2) + 1)**2), x)
 

Maxima [F]

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

integrate(1/x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="maxima")
 

Output:

integrate(1/((2*x + sqrt(-5*x^2 + 8*x - 2) + 1)^2*x^2), x)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.42 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="giac")
 

Output:

-32/135*sqrt(10)*sqrt(5)*arctan(-1/10*sqrt(10)*(sqrt(6) - 4*(sqrt(5)*sqrt( 
-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))) + 3016/14283*sqrt(23)*arctan(1/23 
*sqrt(23)*(9*x - 2)) + 3016/621*(5*sqrt(6) + 13*sqrt(5))*arctan(-(26*sqrt( 
6) + 12*sqrt(5) - 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4) 
)/(5*sqrt(138) + 13*sqrt(115)))/(5*sqrt(138) + 13*sqrt(115)) + 3016/621*(5 
*sqrt(6) - 13*sqrt(5))*arctan((26*sqrt(6) - 12*sqrt(5) - 139*(sqrt(5)*sqrt 
(-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))/(5*sqrt(138) - 13*sqrt(115)))/(5* 
sqrt(138) - 13*sqrt(115)) + 1/207*(576*x^2 + 194*x + 69)/(9*x^3 - 4*x^2 + 
3*x) - 1/28773*(350558*sqrt(30) + 85083*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x 
 - 2) - sqrt(6))^5/(5*x - 4)^5 + 220730*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8* 
x - 2) - sqrt(6))^4/(5*x - 4)^4 - 3442530*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8 
*x - 2) - sqrt(6))^3/(5*x - 4)^3 + 2162560*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 
 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 2710485*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 
+ 8*x - 2) - sqrt(6))/(5*x - 4))/(347*sqrt(6)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 
 2) - sqrt(6))^5/(5*x - 4)^5 + 910*sqrt(6)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) 
 - sqrt(6))^3/(5*x - 4)^3 + 347*sqrt(6)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - 
sqrt(6))/(5*x - 4) - 278*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^6/(5*x 
 - 4)^6 - 1890*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^4/(5*x - 4)^4 - 
1890*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 278) - 14/ 
27*log(9*x^2 - 4*x + 3) - 14/27*log(-4*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\int \frac {1}{x^2\,{\left (2\,x+\sqrt {-5\,x^2+8\,x-2}+1\right )}^2} \,d x \] Input:

int(1/(x^2*(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^2),x)
 

Output:

int(1/(x^2*(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^2), x)
 

Reduce [F]

\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\text {too large to display} \] Input:

int(1/x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x)
 

Output:

(8240432737453414511572801174788082190003102897077594247620834070055876085 
47439283630115000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**6 - 39554077139776 
38965554944563898279451201489390597245238858000353626820521027708561424552 
000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**5 + 7301226872858724106994528329 
780136822372131959474873876089531105429260677806496605872357200000*sqrt(5) 
*asin((5*x - 4)/sqrt(6))*x**4 - 648751974684332422268139384933044550646446 
7537104683384359031361899199945929232610046787840000*sqrt(5)*asin((5*x - 4 
)/sqrt(6))*x**3 + 30892059794125358480569759426314196117130644608827085911 
79693815280107541494829914938101240000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x** 
2 - 9590642901545781393024581633945951656987315016213567468045818141386512 
96703508594399730880000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x + 76702891293469 
38743098265080471422419734732415919288963996531821361947419569512639191800 
00000*sqrt(6)*asin((5*x - 4)/sqrt(6))*x**6 - 36817387820865305966871672386 
26282761472671559641258702718335274253734761393366066812064000000*sqrt(6)* 
asin((5*x - 4)/sqrt(6))*x**5 + 6796065558456433329810818374015716595302744 
148465376918124729577228447437131663988759470400000*sqrt(6)*asin((5*x - 4) 
/sqrt(6))*x**4 - 603865765015805610952937191204481998935204928728650233973 
4237476981763483542962878965498880000*sqrt(6)*asin((5*x - 4)/sqrt(6))*x**3 
 + 28754682911863876925607984954020126497369998845535649501868282305524487 
72391111024649907680000*sqrt(6)*asin((5*x - 4)/sqrt(6))*x**2 - 89270802...