\(\int \frac {1}{x^2 (1+2 x+\sqrt {-2+8 x-5 x^2})^3} \, dx\) [61]

Optimal result

Integrand size = 25, antiderivative size = 822 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx =\text {Too large to display} \] Output:

-2/58029*6^(1/2)*(1070289+323446*6^(1/2)+42050*(3464517-1461538*6^(1/2))*( 
-5*x^2+8*x-2)^(1/2)/(385088-156807*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)^2+2/5*6^(1/2)*(4-6^(1/2)-5*x)^2*(68+27*6^(1/2)-50 
*(4+6^(1/2))*(-5*x^2+8*x-2)^(1/2)/(4-6^(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)^2-5/3*6^(1/2)*(2520 
325+628478*6^(1/2)-6728*(234386673413-95626095232*6^(1/2))*(-5*x^2+8*x-2)^ 
(1/2)/(13-2*6^(1/2))^5/(4-6^(1/2))^3/(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)-6673335*(13-2* 
6^(1/2))*(5*x^2-8*x+2)/(4-6^(1/2)-5*x)^2)+44*(2700719361*2^(1/2)-220565490 
8*3^(1/2))*arctan((2*2^(1/2)-3^(1/2))*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x) 
)/(72919422747-29776341258*6^(1/2))+3598160/328509*(39476391248-1611442704 
7*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/(4-6^(1/2))^4/(4+6^(1/2))-2/27 
*ln(x*(6-4*6^(1/2)+5*x*6^(1/2))/(4-6^(1/2)-5*x)^2)+2/27*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 = 14.11 (sec) , antiderivative size = 1176, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(121670/x + (3174*(40 + 1269*x))/(3 - 4*x + 9*x^2)^2 + (46*(-11467 + 68265 
*x))/(3 - 4*x + 9*x^2) - (69*Sqrt[-2 + 8*x - 5*x^2]*(-3174 + 38760*x + 212 
65*x^2 + 7704*x^3 + 126927*x^4))/(x*(3 - 4*x + 9*x^2)^2) + 179908*Sqrt[23] 
*ArcTan[(-2 + 9*x)/Sqrt[23]] - 535348*Sqrt[2]*ArcTan[(1 - 2*x)/Sqrt[-1 + 4 
*x - (5*x^2)/2]] + (522*(2147 - (371*I)*Sqrt[23])*ArcTan[(23*(-8813295 + ( 
5358132*I)*Sqrt[23] + 24*(3096994 - (1409083*I)*Sqrt[23])*x + (-298062772 
+ (70094518*I)*Sqrt[23])*x^2 + (460991572 - (59592988*I)*Sqrt[23])*x^3 + ( 
-234188473 + (17893330*I)*Sqrt[23])*x^4))/((-2381645630*I + 287409781*Sqrt 
[23])*x^4 - 9*(-20091604*I + 3816017*Sqrt[23] + 575952*Sqrt[23*(77 - (52*I 
)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + 12*x*(-9001694*I + 30089092*Sqrt[23 
] + 1115907*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + 4*x^ 
3*(1337206637*I + 41332211*Sqrt[23] + 4859595*Sqrt[23*(77 - (52*I)*Sqrt[23 
])]*Sqrt[-2 + 8*x - 5*x^2]) - 2*x^2*(1480853413*I + 357440992*Sqrt[23] + 1 
2094992*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]))])/Sqrt[77 
/23 - (52*I)/Sqrt[23]] - ((522*I)*(-2147*I + 371*Sqrt[23])*ArcTan[(23*(9*( 
979255 + (595348*I)*Sqrt[23]) + (-74327856 - (33817992*I)*Sqrt[23])*x + (2 
98062772 + (70094518*I)*Sqrt[23])*x^2 + (-460991572 - (59592988*I)*Sqrt[23 
])*x^3 + (234188473 + (17893330*I)*Sqrt[23])*x^4))/((2381645630*I + 287409 
781*Sqrt[23])*x^4 + x^2*(2961706826*I - 714881984*Sqrt[23] - 24189984*Sqrt 
[23*(77 + (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) - 9*(20091604*I + 3...
 

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 439, normalized size of antiderivative = 0.53, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {191-126 x}{27 \left (9 x^2-4 x+3\right )^2}-\frac {8 \sqrt {-5 x^2+8 x-2}}{9 x}-\frac {\sqrt {-5 x^2+8 x-2}}{27 x^2}+\frac {18 x+37}{27 \left (9 x^2-4 x+3\right )}+\frac {8 x \sqrt {-5 x^2+8 x-2}}{9 x^2-4 x+3}-\frac {29 \sqrt {-5 x^2+8 x-2}}{9 \left (9 x^2-4 x+3\right )}-\frac {5}{27 x^2}+\frac {68 x \sqrt {-5 x^2+8 x-2}}{3 \left (9 x^2-4 x+3\right )^2}-\frac {245 \sqrt {-5 x^2+8 x-2}}{27 \left (9 x^2-4 x+3\right )^2}-\frac {4 (126 x-169)}{9 \left (9 x^2-4 x+3\right )^3}+\frac {64 x \sqrt {-5 x^2+8 x-2}}{\left (9 x^2-4 x+3\right )^3}-\frac {292 \sqrt {-5 x^2+8 x-2}}{9 \left (9 x^2-4 x+3\right )^3}-\frac {2}{27 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {89954 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{14283 \sqrt {23}}-\frac {22}{27} \sqrt {2} \arctan \left (\frac {\sqrt {2} (1-2 x)}{\sqrt {-5 x^2+8 x-2}}\right )-\frac {89954 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{14283 \sqrt {23}}+\frac {2}{27} \text {arctanh}\left (\frac {2 x+1}{\sqrt {-5 x^2+8 x-2}}\right )+\frac {73 \sqrt {-5 x^2+8 x-2} (23351-82206 x)}{4004001 \left (9 x^2-4 x+3\right )}+\frac {\sqrt {-5 x^2+8 x-2}}{27 x}-\frac {4-1467 x}{1242 \left (9 x^2-4 x+3\right )}-\frac {423 (2-9 x)}{1058 \left (9 x^2-4 x+3\right )}+\frac {245 (2-9 x) \sqrt {-5 x^2+8 x-2}}{1242 \left (9 x^2-4 x+3\right )}-\frac {34 (3-2 x) \sqrt {-5 x^2+8 x-2}}{69 \left (9 x^2-4 x+3\right )}+\frac {120 (1758 x+215) \sqrt {-5 x^2+8 x-2}}{444889 \left (9 x^2-4 x+3\right )}+\frac {1269 x+40}{207 \left (9 x^2-4 x+3\right )^2}+\frac {73 (2-9 x) \sqrt {-5 x^2+8 x-2}}{207 \left (9 x^2-4 x+3\right )^2}-\frac {16 (3-2 x) \sqrt {-5 x^2+8 x-2}}{23 \left (9 x^2-4 x+3\right )^2}+\frac {1}{27} \log \left (9 x^2-4 x+3\right )+\frac {5}{27 x}-\frac {2 \log (x)}{27}\)

Input:

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

Output:

5/(27*x) + Sqrt[-2 + 8*x - 5*x^2]/(27*x) + (40 + 1269*x)/(207*(3 - 4*x + 9 
*x^2)^2) + (73*(2 - 9*x)*Sqrt[-2 + 8*x - 5*x^2])/(207*(3 - 4*x + 9*x^2)^2) 
 - (16*(3 - 2*x)*Sqrt[-2 + 8*x - 5*x^2])/(23*(3 - 4*x + 9*x^2)^2) - (4 - 1 
467*x)/(1242*(3 - 4*x + 9*x^2)) - (423*(2 - 9*x))/(1058*(3 - 4*x + 9*x^2)) 
 + (73*(23351 - 82206*x)*Sqrt[-2 + 8*x - 5*x^2])/(4004001*(3 - 4*x + 9*x^2 
)) + (245*(2 - 9*x)*Sqrt[-2 + 8*x - 5*x^2])/(1242*(3 - 4*x + 9*x^2)) - (34 
*(3 - 2*x)*Sqrt[-2 + 8*x - 5*x^2])/(69*(3 - 4*x + 9*x^2)) + (120*(215 + 17 
58*x)*Sqrt[-2 + 8*x - 5*x^2])/(444889*(3 - 4*x + 9*x^2)) - (89954*ArcTan[( 
2 - 9*x)/Sqrt[23]])/(14283*Sqrt[23]) + (89954*ArcTan[(8 - 13*x)/(Sqrt[23]* 
Sqrt[-2 + 8*x - 5*x^2])])/(14283*Sqrt[23]) - (22*Sqrt[2]*ArcTan[(Sqrt[2]*( 
1 - 2*x))/Sqrt[-2 + 8*x - 5*x^2]])/27 + (2*ArcTanh[(1 + 2*x)/Sqrt[-2 + 8*x 
 - 5*x^2]])/27 - (2*Log[x])/27 + 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 = 1.23 (sec) , antiderivative size = 1551, normalized size of antiderivative = 1.89

method	result	size
trager	\(\text {Expression too large to display}\)	\(1551\)
default	\(\text {Expression too large to display}\)	\(18899\)

Input:

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

Output:

-1/304704*(x-1)*(19277595*x^4-15535485*x^3+13213829*x^2-3663363*x+507840)/ 
x/(9*x^2-4*x+3)^2-1/9522*(126927*x^4+7704*x^3+21265*x^2+38760*x-3174)/x/(9 
*x^2-4*x+3)^2*(-5*x^2+8*x-2)^(1/2)+128/4761*RootOf(35328*_Z^2-194672*_Z+84 
556953)*ln(-(549589056749568*RootOf(12288*_Z^2+67712*_Z[phone])^2*RootOf 
(35328*_Z^2-194672*_Z[phone])^2*x-549589056749568*RootOf(12288*_Z^2+6771 
2*_Z[phone])^2*RootOf(35328*_Z^2-194672*_Z[phone])^2+21090315307843584 
*RootOf(12288*_Z^2+67712*_Z[phone])*RootOf(35328*_Z^2-194672*_Z[phone] 
)^2*x+7650482517835776*RootOf(12288*_Z^2+67712*_Z[phone])^2*RootOf(35328 
*_Z^2-194672*_Z[phone])*x+879302706938726400*RootOf(12288*_Z^2+67712*_Z+ 
22667121)*RootOf(35328*_Z^2-194672*_Z[phone])*(-5*x^2+8*x-2)^(1/2)-21090 
315307843584*RootOf(12288*_Z^2+67712*_Z[phone])*RootOf(35328*_Z^2-194672 
*_Z[phone])^2-62280238075545600*RootOf(35328*_Z^2-194672*_Z[phone])^2* 
x-7650482517835776*RootOf(12288*_Z^2+67712*_Z[phone])^2*RootOf(35328*_Z^ 
2-194672*_Z[phone])-4141440798301403136*RootOf(35328*_Z^2-194672*_Z+8455 
6953)*RootOf(12288*_Z^2+67712*_Z[phone])*x-671503012082380800*RootOf(122 
88*_Z^2+67712*_Z[phone])^2*x-63862596954902250912*(-5*x^2+8*x-2)^(1/2)*R 
ootOf(35328*_Z^2-194672*_Z[phone])-82018265245493818560*RootOf(12288*_Z^ 
2+67712*_Z[phone])*(-5*x^2+8*x-2)^(1/2)+62280238075545600*RootOf(35328*_ 
Z^2-194672*_Z[phone])^2+2244476964795863040*RootOf(12288*_Z^2+67712*_Z+2 
2667121)*RootOf(35328*_Z^2-194672*_Z[phone])-117304285137685971456*Ro...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 528, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\frac {38116980 \, x^{4} - 26068338 \, x^{3} + 179908 \, \sqrt {23} {\left (81 \, x^{5} - 72 \, x^{4} + 70 \, x^{3} - 24 \, x^{2} + 9 \, x\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) - 535348 \, \sqrt {2} {\left (81 \, x^{5} - 72 \, x^{4} + 70 \, x^{3} - 24 \, x^{2} + 9 \, 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 ) + 89954 \, \sqrt {23} {\left (81 \, x^{5} - 72 \, x^{4} + 70 \, x^{3} - 24 \, x^{2} + 9 \, 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 ) + 89954 \, \sqrt {23} {\left (81 \, x^{5} - 72 \, x^{4} + 70 \, x^{3} - 24 \, x^{2} + 9 \, 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 ) + 24075204 \, x^{2} + 24334 \, {\left (81 \, x^{5} - 72 \, x^{4} + 70 \, x^{3} - 24 \, x^{2} + 9 \, x\right )} \log \left (9 \, x^{2} - 4 \, x + 3\right ) - 48668 \, {\left (81 \, x^{5} - 72 \, x^{4} + 70 \, x^{3} - 24 \, x^{2} + 9 \, x\right )} \log \left (x\right ) - 12167 \, {\left (81 \, x^{5} - 72 \, x^{4} + 70 \, x^{3} - 24 \, x^{2} + 9 \, 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 ) + 12167 \, {\left (81 \, x^{5} - 72 \, x^{4} + 70 \, x^{3} - 24 \, x^{2} + 9 \, 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 (126927 \, x^{4} + 7704 \, x^{3} + 21265 \, x^{2} + 38760 \, x - 3174\right )} \sqrt {-5 \, x^{2} + 8 \, x - 2} - 4375566 \, x + 1095030}{657018 \, {\left (81 \, x^{5} - 72 \, x^{4} + 70 \, x^{3} - 24 \, x^{2} + 9 \, x\right )}} \] Input:

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

Output:

1/657018*(38116980*x^4 - 26068338*x^3 + 179908*sqrt(23)*(81*x^5 - 72*x^4 + 
 70*x^3 - 24*x^2 + 9*x)*arctan(1/23*sqrt(23)*(9*x - 2)) - 535348*sqrt(2)*( 
81*x^5 - 72*x^4 + 70*x^3 - 24*x^2 + 9*x)*arctan(sqrt(2)*sqrt(-5*x^2 + 8*x 
- 2)*(2*x - 1)/(5*x^2 - 8*x + 2)) + 89954*sqrt(23)*(81*x^5 - 72*x^4 + 70*x 
^3 - 24*x^2 + 9*x)*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)) + 89954*sqrt(23)*(81*x^5 - 
 72*x^4 + 70*x^3 - 24*x^2 + 9*x)*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)) + 24075204*x 
^2 + 24334*(81*x^5 - 72*x^4 + 70*x^3 - 24*x^2 + 9*x)*log(9*x^2 - 4*x + 3) 
- 48668*(81*x^5 - 72*x^4 + 70*x^3 - 24*x^2 + 9*x)*log(x) - 12167*(81*x^5 - 
 72*x^4 + 70*x^3 - 24*x^2 + 9*x)*log(-(x^2 + 2*sqrt(-5*x^2 + 8*x - 2)*(2*x 
 + 1) - 12*x + 1)/x^2) + 12167*(81*x^5 - 72*x^4 + 70*x^3 - 24*x^2 + 9*x)*l 
og(-(x^2 - 2*sqrt(-5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2) - 69*(12692 
7*x^4 + 7704*x^3 + 21265*x^2 + 38760*x - 3174)*sqrt(-5*x^2 + 8*x - 2) - 43 
75566*x + 1095030)/(81*x^5 - 72*x^4 + 70*x^3 - 24*x^2 + 9*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/x^2/(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^2), x)
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 968, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-44/135*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))) + 89954/328509*sqrt(23)*arctan(1/ 
23*sqrt(23)*(9*x - 2)) + 89954/14283*(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)) + 89954/ 
14283*(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(11 
5)))/(5*sqrt(138) - 13*sqrt(115)) - 1/54*(2*sqrt(30) - 3*sqrt(5)*(sqrt(5)* 
sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))/(sqrt(6)*(sqrt(5)*sqrt(-5*x^2 
 + 8*x - 2) - sqrt(6))/(5*x - 4) - 2*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqr 
t(6))^2/(5*x - 4)^2 - 2) - 2/275961843*(490328685778*sqrt(30) - 8266064111 
07*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^7/(5*x - 4)^7 + 1457 
577177946*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^6/(5*x - 4)^ 
6 - 8000608193067*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^5/(5* 
x - 4)^5 + 5233835542230*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6 
))^4/(5*x - 4)^4 - 12047619978645*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) 
- sqrt(6))^3/(5*x - 4)^3 + 3674568646158*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8 
*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 3575769127389*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...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 6787525974949390564512010316353156991520111831627568385306768630213551 
23926078937945634815979563139430149674643852443507863375221835015625000000 
*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**10 + 543002077995951245160960825308252 
55932160894653020547082454149041708409914086315035650785278365051154411973 
97150819548062907001774680125000000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x** 
9 - 1897825781242639475371209995614398759801820898304952701116391307865142 
07983676985019663423608705505948321355942394693093408515086717523875000000 
00*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**8 + 38055730819350815167033856951973 
92751551740330447741255279947077530497548151669051782498492200280523374887 
5782896015321535198108240128560000000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x 
**7 - 48783973749608423768177348703634575074344085779152646460634900138060 
73970364475606009570915408825816549485681626239027911063595875906390375000 
0000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**6 + 426456994276163055626765758068 
46456674159726594989978628984297203049869052546833220156753326601647983118 
197003575142217552960324900238242640000000*sqrt(5)*asin((5*x - 4)/sqrt(6)) 
*x**5 - 265106229406557471180157269747501822047016169135166249896121244009 
38685464405779831449217874543187507886190605361057498387501466792685983180 
000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**4 + 1164678991317748305799431127 
58250257661120687274955030386895632547491536344775771957572893503982323934 
20432379879155943281749303312786020800000000*sqrt(5)*asin((5*x - 4)/sqr...