\(\int \frac {1}{x (1+2 x+\sqrt {-2+8 x-5 x^2})^3} \, dx\) [60]

Optimal result

Integrand size = 25, antiderivative size = 635 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=-\frac {2 \sqrt {\frac {2}{3}} \left (145152+81703 \sqrt {6}-\frac {1429700 \left (513-157 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{\left (13-2 \sqrt {6}\right )^3 \left (4-\sqrt {6}-5 x\right )}\right )}{19343 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )^2}+\frac {2 \sqrt {\frac {2}{3}} \left (3730892+874903 \sqrt {6}+\frac {42050 \left (160392893-64943077 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{\left (13-2 \sqrt {6}\right )^5 \left (4-\sqrt {6}-5 x\right )}\right )}{1334667 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )}+\frac {2}{27} \sqrt {2} \arctan \left (\frac {\sqrt {11-4 \sqrt {6}} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )+\frac {18472 \left (295429693-119673902 \sqrt {6}\right ) \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{14283 \sqrt {23} \left (13-2 \sqrt {6}\right )^7}-\frac {5 \left (106257964-43200721 \sqrt {6}\right ) \log \left (\frac {x \left (2 \left (3-2 \sqrt {6}\right )+5 \sqrt {6} x\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )}{27 \left (13-2 \sqrt {6}\right )^6 \left (4-\sqrt {6}\right )}+\frac {5 \left (295429693-119673902 \sqrt {6}\right ) \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+12 x-3 \sqrt {6} x+10 \sqrt {6} x^2+6 \sqrt {-2+8 x-5 x^2}-4 \sqrt {6} \sqrt {-2+8 x-5 x^2}+5 \sqrt {6} x \sqrt {-2+8 x-5 x^2}}{\left (4-\sqrt {6}-5 x\right )^2}\right )}{27 \left (13-2 \sqrt {6}\right )^7} \] Output:

-2/58029*6^(1/2)*(145152+81703*6^(1/2)-1429700*(513-157*6^(1/2))*(-5*x^2+8 
*x-2)^(1/2)/(13-2*6^(1/2))^3/(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)^2+2/3*6^(1/2)*(3730892+874903*6^(1/2)+42050*(160392893-64943077* 
6^(1/2))*(-5*x^2+8*x-2)^(1/2)/(13-2*6^(1/2))^5/(4-6^(1/2)-5*x))/(17350671+ 
2669334*6^(1/2)-13346670*6^(1/2)*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x)-6673 
335*(13-2*6^(1/2))*(5*x^2-8*x+2)/(4-6^(1/2)-5*x)^2)+2/27*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))+18472/328509*(2954 
29693-119673902*6^(1/2))*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)/(13-2*6^(1/2))^7-5/27*(106257964- 
43200721*6^(1/2))*ln(x*(6-4*6^(1/2)+5*x*6^(1/2))/(4-6^(1/2)-5*x)^2)/(13-2* 
6^(1/2))^6/(4-6^(1/2))+5/27*(295429693-119673902*6^(1/2))*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)/(13-2*6^( 
1/2))^7
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 14.04 (sec) , antiderivative size = 1160, normalized size of antiderivative = 1.83 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[1/(x*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^3),x]
 

Output:

((6348*(-423 + 604*x))/(3 - 4*x + 9*x^2)^2 + (1242*(-533 + 1076*x))/(3 - 4 
*x + 9*x^2) - (276*Sqrt[-2 + 8*x - 5*x^2]*(-6120 + 29858*x - 30033*x^2 + 4 
5360*x^3))/(3 - 4*x + 9*x^2)^2 + 36944*Sqrt[23]*ArcTan[(-2 + 9*x)/Sqrt[23] 
] - 48668*Sqrt[2]*ArcTan[(1 - 2*x)/Sqrt[-1 + 4*x - (5*x^2)/2]] - ((18*I)*( 
-178*I + 5873*Sqrt[23])*ArcTan[(23*(-22459465744 + (1546348544*I)*Sqrt[23] 
 + 8*(2035528511 - (2539584736*I)*Sqrt[23])*x + (-10233103867 + (926756389 
52*I)*Sqrt[23])*x^2 + 24*(11991284939 - (6852780844*I)*Sqrt[23])*x^3 + (9* 
I)*(36325138529*I + 8967953540*Sqrt[23])*x^4))/(416397255040*I + 876886235 
60*Sqrt[23] - 8*(446641816361*I + 23928664948*Sqrt[23])*x + (56263105080*I 
 - 932718674466*Sqrt[23])*x^4 - 9520207812*Sqrt[23*(77 - (52*I)*Sqrt[23])] 
*Sqrt[-2 + 8*x - 5*x^2] + 24593870181*Sqrt[23*(77 - (52*I)*Sqrt[23])]*x*Sq 
rt[-2 + 8*x - 5*x^2] + x^2*(8693680422544*I - 794728685026*Sqrt[23] - 4442 
7636456*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + 9*x^3*(- 
539876082772*I + 209900528952*Sqrt[23] + 3966753255*Sqrt[23*(77 - (52*I)*S 
qrt[23])]*Sqrt[-2 + 8*x - 5*x^2]))])/Sqrt[77/23 - (52*I)/Sqrt[23]] + (18*( 
178 - (5873*I)*Sqrt[23])*ArcTan[(23*(-22459465744 - (1546348544*I)*Sqrt[23 
] + 8*(2035528511 + (2539584736*I)*Sqrt[23])*x + (-10233103867 - (92675638 
952*I)*Sqrt[23])*x^2 + 24*(11991284939 + (6852780844*I)*Sqrt[23])*x^3 + (- 
326926246761 - (80711581860*I)*Sqrt[23])*x^4))/(416397255040*I - 876886235 
60*Sqrt[23] + 18*(3125728060*I + 51817704137*Sqrt[23])*x^4 + 9520207812...
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.65, 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.

\(\displaystyle \int \frac {1}{x \left (\sqrt {-5 x^2+8 x-2}+2 x+1\right )^3} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\sqrt {-5 x^2+8 x-2} x}{3 \left (9 x^2-4 x+3\right )}+\frac {\sqrt {-5 x^2+8 x-2} x}{\left (9 x^2-4 x+3\right )^2}-\frac {4 \sqrt {-5 x^2+8 x-2} x}{\left (9 x^2-4 x+3\right )^3}+\frac {5 (9 x-4)}{27 \left (9 x^2-4 x+3\right )}-\frac {4 \sqrt {-5 x^2+8 x-2}}{27 \left (9 x^2-4 x+3\right )}+\frac {15 x-14}{3 \left (9 x^2-4 x+3\right )^2}-\frac {4 \sqrt {-5 x^2+8 x-2}}{9 \left (9 x^2-4 x+3\right )^2}+\frac {4 (113 x+42)}{9 \left (9 x^2-4 x+3\right )^3}-\frac {64 \sqrt {-5 x^2+8 x-2}}{3 \left (9 x^2-4 x+3\right )^3}-\frac {\sqrt {-5 x^2+8 x-2}}{27 x}-\frac {5}{27 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {9236 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{14283 \sqrt {23}}-\frac {1}{27} \sqrt {2} \arctan \left (\frac {\sqrt {2} (1-2 x)}{\sqrt {-5 x^2+8 x-2}}\right )-\frac {9236 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{14283 \sqrt {23}}+\frac {5}{27} \text {arctanh}\left (\frac {2 x+1}{\sqrt {-5 x^2+8 x-2}}\right )+\frac {16 \sqrt {-5 x^2+8 x-2} (23351-82206 x)}{1334667 \left (9 x^2-4 x+3\right )}-\frac {302 (2-9 x)}{1587 \left (9 x^2-4 x+3\right )}-\frac {96 x+17}{138 \left (9 x^2-4 x+3\right )}+\frac {2 (2-9 x) \sqrt {-5 x^2+8 x-2}}{207 \left (9 x^2-4 x+3\right )}-\frac {(3-2 x) \sqrt {-5 x^2+8 x-2}}{46 \left (9 x^2-4 x+3\right )}-\frac {15 (1758 x+215) \sqrt {-5 x^2+8 x-2}}{889778 \left (9 x^2-4 x+3\right )}-\frac {423-604 x}{207 \left (9 x^2-4 x+3\right )^2}+\frac {16 (2-9 x) \sqrt {-5 x^2+8 x-2}}{69 \left (9 x^2-4 x+3\right )^2}+\frac {(3-2 x) \sqrt {-5 x^2+8 x-2}}{23 \left (9 x^2-4 x+3\right )^2}+\frac {5}{54} \log \left (9 x^2-4 x+3\right )-\frac {5 \log (x)}{27}\)

Input:

Int[1/(x*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^3),x]
 

Output:

-1/207*(423 - 604*x)/(3 - 4*x + 9*x^2)^2 + (16*(2 - 9*x)*Sqrt[-2 + 8*x - 5 
*x^2])/(69*(3 - 4*x + 9*x^2)^2) + ((3 - 2*x)*Sqrt[-2 + 8*x - 5*x^2])/(23*( 
3 - 4*x + 9*x^2)^2) - (302*(2 - 9*x))/(1587*(3 - 4*x + 9*x^2)) - (17 + 96* 
x)/(138*(3 - 4*x + 9*x^2)) + (16*(23351 - 82206*x)*Sqrt[-2 + 8*x - 5*x^2]) 
/(1334667*(3 - 4*x + 9*x^2)) + (2*(2 - 9*x)*Sqrt[-2 + 8*x - 5*x^2])/(207*( 
3 - 4*x + 9*x^2)) - ((3 - 2*x)*Sqrt[-2 + 8*x - 5*x^2])/(46*(3 - 4*x + 9*x^ 
2)) - (15*(215 + 1758*x)*Sqrt[-2 + 8*x - 5*x^2])/(889778*(3 - 4*x + 9*x^2) 
) - (9236*ArcTan[(2 - 9*x)/Sqrt[23]])/(14283*Sqrt[23]) + (9236*ArcTan[(8 - 
 13*x)/(Sqrt[23]*Sqrt[-2 + 8*x - 5*x^2])])/(14283*Sqrt[23]) - (Sqrt[2]*Arc 
Tan[(Sqrt[2]*(1 - 2*x))/Sqrt[-2 + 8*x - 5*x^2]])/27 + (5*ArcTanh[(1 + 2*x) 
/Sqrt[-2 + 8*x - 5*x^2]])/27 - (5*Log[x])/27 + (5*Log[3 - 4*x + 9*x^2])/54
 

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 = 1.20 (sec) , antiderivative size = 1483, normalized size of antiderivative = 2.34

method	result	size
trager	\(\text {Expression too large to display}\)	\(1483\)
default	\(\text {Expression too large to display}\)	\(17094\)

Input:

int(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x,method=_RETURNVERBOSE)
 

Output:

-1/304704*(x-1)*(1920591*x^3-2575593*x^2+1705265*x-1296567)/(9*x^2-4*x+3)^ 
2-1/4761*(45360*x^3-30033*x^2+29858*x-6120)/(9*x^2-4*x+3)^2*(-5*x^2+8*x-2) 
^(1/2)+10/27*ln(-(122130901499904*RootOf(12288*_Z^2+338560*_Z+2518569)^2*R 
ootOf(282624*_Z^2-7786880*_Z[phone])^2*x-122130901499904*RootOf(12288*_Z 
^2+338560*_Z+2518569)^2*RootOf(282624*_Z^2-7786880*_Z[phone])^2+30000749 
04797184*RootOf(12288*_Z^2+338560*_Z+2518569)*RootOf(282624*_Z^2-7786880*_ 
Z[phone])^2*x-3852275311706112*RootOf(12288*_Z^2+338560*_Z+2518569)^2*Ro 
otOf(282624*_Z^2-7786880*_Z[phone])*x+3647763313049600*RootOf(12288*_Z^2 
+338560*_Z+2518569)*RootOf(282624*_Z^2-7786880*_Z[phone])*(-5*x^2+8*x-2) 
^(1/2)-3000074904797184*RootOf(12288*_Z^2+338560*_Z+2518569)*RootOf(282624 
*_Z^2-7786880*_Z[phone])^2+17937800004759552*RootOf(282624*_Z^2-7786880* 
_Z[phone])^2*x+3852275311706112*RootOf(12288*_Z^2+338560*_Z+2518569)^2*R 
ootOf(282624*_Z^2-7786880*_Z[phone])-113027496828223488*RootOf(282624*_Z 
^2-7786880*_Z[phone])*RootOf(12288*_Z^2+338560*_Z+2518569)*x[phone] 
601408*RootOf(12288*_Z^2+338560*_Z+2518569)^2*x+75250156724959488*(-5*x^2+ 
8*x-2)^(1/2)*RootOf(282624*_Z^2-7786880*_Z[phone])+17554671764171520*Roo 
tOf(12288*_Z^2+338560*_Z+2518569)*(-5*x^2+8*x-2)^(1/2)-17937800004759552*R 
ootOf(282624*_Z^2-7786880*_Z[phone])^2+105157994069458944*RootOf(12288*_ 
Z^2+338560*_Z+2518569)*RootOf(282624*_Z^2-7786880*_Z[phone])-77995293140 
3654784*RootOf(282624*_Z^2-7786880*_Z[phone])*x-23835316211601408*Roo...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 482, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\frac {12027528 \, x^{3} + 36944 \, \sqrt {23} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) - 48668 \, \sqrt {2} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x - 1\right )}}{5 \, x^{2} - 8 \, x + 2}\right ) + 18472 \, \sqrt {23} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} + 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + 18472 \, \sqrt {23} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} - 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) - 11303442 \, x^{2} + 121670 \, {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \log \left (9 \, x^{2} - 4 \, x + 3\right ) - 243340 \, {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \log \left (x\right ) - 60835 \, {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \log \left (-\frac {x^{2} + 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) + 60835 \, {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \log \left (-\frac {x^{2} - 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) - 276 \, {\left (45360 \, x^{3} - 30033 \, x^{2} + 29858 \, x - 6120\right )} \sqrt {-5 \, x^{2} + 8 \, x - 2} + 10491312 \, x - 4671162}{1314036 \, {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )}} \] Input:

integrate(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="fricas")
 

Output:

1/1314036*(12027528*x^3 + 36944*sqrt(23)*(81*x^4 - 72*x^3 + 70*x^2 - 24*x 
+ 9)*arctan(1/23*sqrt(23)*(9*x - 2)) - 48668*sqrt(2)*(81*x^4 - 72*x^3 + 70 
*x^2 - 24*x + 9)*arctan(sqrt(2)*sqrt(-5*x^2 + 8*x - 2)*(2*x - 1)/(5*x^2 - 
8*x + 2)) + 18472*sqrt(23)*(81*x^4 - 72*x^3 + 70*x^2 - 24*x + 9)*arctan(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)) + 18472*sqrt(23)*(81*x^4 - 72*x^3 + 70*x^2 - 24*x + 9) 
*arctan(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)) - 11303442*x^2 + 121670*(81*x^4 - 72*x^3 + 7 
0*x^2 - 24*x + 9)*log(9*x^2 - 4*x + 3) - 243340*(81*x^4 - 72*x^3 + 70*x^2 
- 24*x + 9)*log(x) - 60835*(81*x^4 - 72*x^3 + 70*x^2 - 24*x + 9)*log(-(x^2 
 + 2*sqrt(-5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2) + 60835*(81*x^4 - 7 
2*x^3 + 70*x^2 - 24*x + 9)*log(-(x^2 - 2*sqrt(-5*x^2 + 8*x - 2)*(2*x + 1) 
- 12*x + 1)/x^2) - 276*(45360*x^3 - 30033*x^2 + 29858*x - 6120)*sqrt(-5*x^ 
2 + 8*x - 2) + 10491312*x - 4671162)/(81*x^4 - 72*x^3 + 70*x^2 - 24*x + 9)
 

Sympy [F]

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

integrate(1/x/(1+2*x+(-5*x**2+8*x-2)**(1/2))**3,x)
 

Output:

Integral(1/(x*(2*x + sqrt(-5*x**2 + 8*x - 2) + 1)**3), x)
 

Maxima [F]

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

integrate(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="maxima")
 

Output:

integrate(1/((2*x + sqrt(-5*x^2 + 8*x - 2) + 1)^3*x), x)
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 848, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="giac")
 

Output:

-2/135*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))) + 9236/328509*sqrt(23)*arctan(1/23 
*sqrt(23)*(9*x - 2)) + 9236/14283*(5*sqrt(6) + 13*sqrt(5))*arctan(-(26*sqr 
t(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)) + 9236/1428 
3*(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)) + 1/9522*(87156*x^3 - 81909*x^2 + 76024*x - 
33849)/(9*x^2 - 4*x + 3)^2 - 40/91987281*(2628815260*sqrt(30) - 1817695077 
*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^7/(5*x - 4)^7 + 655947 
4327*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^6/(5*x - 4)^6 - 3 
7448922405*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^5/(5*x - 4)^ 
5 + 28060334100*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^4/(5*x 
 - 4)^4 - 70036300635*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^3 
/(5*x - 4)^3 + 20955667353*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt 
(6))^2/(5*x - 4)^2 - 21784905243*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - 
 sqrt(6))/(5*x - 4))/(104*sqrt(6)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6 
))^3/(5*x - 4)^3 + 104*sqrt(6)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/ 
(5*x - 4) - 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^4/(5*x - 4)^4 - 
 494*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 139)^2 ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\int \frac {1}{x\,{\left (2\,x+\sqrt {-5\,x^2+8\,x-2}+1\right )}^3} \,d x \] Input:

int(1/(x*(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^3),x)
 

Output:

int(1/(x*(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^3), x)
 

Reduce [F]

\[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\text {too large to display} \] Input:

int(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x)
 

Output:

(3281081299159115718947032867935693027047558996480231571185722683442010739 
65132031613936789971705723926975811015867637941750000000*sqrt(5)*asin((5*x 
 - 4)/sqrt(6))*x**8 - 2099892031461834060126101035478843537310437757747348 
20555886251740288687337684500232919545581891663313264519050155288282720000 
0000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**7 + 581423809081608436833063345348 
16991892629435570220557876754801369201537472418754678688927265899694801953 
58025922663861679880000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**6 - 90933102 
34019171789257175298821718014160606764289627859034896501286871601304471024 
901093803494360472069217617534872705931712000000*sqrt(5)*asin((5*x - 4)/sq 
rt(6))*x**5 + 903281359601446691480298456134653773872132723688332285418922 
8485017138698930536993949598447410576309042273112895211408327280800000*sqr 
t(5)*asin((5*x - 4)/sqrt(6))*x**4 - 61623743290508161390550876918027644106 
26957334426079001465627706291010986601548649536099365935233562821066233665 
359429717721600000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**3 + 2955401999551656 
51608506574167146225107558233544432626023312127593136265421847582186904118 
1687943468337704622140738730721224000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x 
**2 - 90140001828994884360946622921776610977537287767313603867993400111462 
7788583079594616657549318313847211642543978662018278515200000*sqrt(5)*asin 
((5*x - 4)/sqrt(6))*x + 17360907779041509501003448687732341523262607855812 
36120522414727565176167080120610447615808245131141400038529394049624639...