\(\int \frac {1}{x (1+2 x+\sqrt {2+3 x+5 x^2})^2} \, dx\) [25]

Optimal result

Integrand size = 25, antiderivative size = 330 \[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=-\frac {2 \left (99+70 \sqrt {2}\right ) \left (2311-1634 \sqrt {2}-\frac {\left (1755-1241 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}\right )}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )}-\frac {122 \arctan \left (\frac {3-4 \sqrt {2}-\frac {2 \left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\left (3-2 \sqrt {2}\right ) \log \left (\frac {4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}}{x}\right )-3 \log \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right ) \] Output:

-2*(99+70*2^(1/2))*(2311-1634*2^(1/2)-(1755-1241*2^(1/2))*(2^(1/2)-(5*x^2+ 
3*x+2)^(1/2))/x)/(3-15*2^(1/2)-3*(3-4*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2 
))/x+3*(1-2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))^2/x^2)-122/9*arctan(1/3*( 
3-4*2^(1/2)-2*(1-2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x)*3^(1/2))*3^(1/2 
)+(3-2*2^(1/2))*ln((4+3*x-2*2^(1/2)*(5*x^2+3*x+2)^(1/2))/x)-3*ln(1-5*2^(1/ 
2)-(3-4*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x+(1-2^(1/2))*(2^(1/2)-(5*x 
^2+3*x+2)^(1/2))^2/x^2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.15 (sec) , antiderivative size = 1106, normalized size of antiderivative = 3.35 \[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((24*(-11 + 13*x))/(1 - x + x^2) - (24*(-1 + 5*x)*Sqrt[2 + 3*x + 5*x^2])/( 
1 - x + x^2) + 244*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + (2*(149 - (43*I)*S 
qrt[3])*ArcTan[(3*(9*(21049 + (32768*I)*Sqrt[3]) + 6*(113131 + (29696*I)*S 
qrt[3])*x + (808411 + (400160*I)*Sqrt[3])*x^2 + 6*(118057 - (95288*I)*Sqrt 
[3])*x^3 + (123821 + (147920*I)*Sqrt[3])*x^4))/(-92160*I - 242151*Sqrt[3] 
+ 3*(-512560*I + 288259*Sqrt[3])*x^4 + 83244*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt 
[2 + 3*x + 5*x^2] + x^2*(1276224*I + 283307*Sqrt[3] - 194236*Sqrt[3 - (12* 
I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + 2*x*(638112*I - 354889*Sqrt[3] + 9711 
8*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + x^3*(2613552*I + 87398 
2*Sqrt[3] + 277480*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]))])/Sqrt 
[1/3 - (4*I)/Sqrt[3]] + (2*(-149*I + 43*Sqrt[3])*ArcTanh[(3*(9*(21049*I + 
32768*Sqrt[3]) + 6*(113131*I + 29696*Sqrt[3])*x + (808411*I + 400160*Sqrt[ 
3])*x^2 + (708342*I - 571728*Sqrt[3])*x^3 + (123821*I + 147920*Sqrt[3])*x^ 
4))/(92160*I - 242151*Sqrt[3] + 3*(512560*I + 288259*Sqrt[3])*x^4 + 83244* 
Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2] + x^2*(-1276224*I + 283307* 
Sqrt[3] - 194236*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + 2*x*(-6 
38112*I - 354889*Sqrt[3] + 97118*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5 
*x^2]) + x^3*(-2613552*I + 873982*Sqrt[3] + 277480*Sqrt[3 + (12*I)*Sqrt[3] 
]*Sqrt[2 + 3*x + 5*x^2]))])/Sqrt[1/3 + (4*I)/Sqrt[3]] + 108*Log[x] - 72*Sq 
rt[2]*Log[x] - 54*Log[1 - x + x^2] - ((-149*I + 43*Sqrt[3])*Log[16*(1 -...
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.71, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {3 (x-1)}{x^2-x+1}-\frac {2 \sqrt {5 x^2+3 x+2}}{x}+\frac {2 x \sqrt {5 x^2+3 x+2}}{x^2-x+1}-\frac {2 \sqrt {5 x^2+3 x+2}}{x^2-x+1}+\frac {2 x \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}-\frac {6 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}+\frac {2 (3 x+5)}{\left (x^2-x+1\right )^2}+\frac {3}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {61 \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{3 \sqrt {3}}-\sqrt {3} \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )-\frac {52 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}-3 \text {arctanh}\left (\frac {2 x+1}{\sqrt {5 x^2+3 x+2}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {3 x+4}{2 \sqrt {2} \sqrt {5 x^2+3 x+2}}\right )-\frac {2 (11-13 x)}{3 \left (x^2-x+1\right )}+\frac {2 (1-2 x) \sqrt {5 x^2+3 x+2}}{x^2-x+1}-\frac {2 (2-x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )}-\frac {3}{2} \log \left (x^2-x+1\right )+3 \log (x)\)

Input:

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

Output:

(-2*(11 - 13*x))/(3*(1 - x + x^2)) + (2*(1 - 2*x)*Sqrt[2 + 3*x + 5*x^2])/( 
1 - x + x^2) - (2*(2 - x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)) - (52*A 
rcTan[(1 - 2*x)/Sqrt[3]])/(3*Sqrt[3]) - Sqrt[3]*ArcTan[(1 - 2*x)/Sqrt[3]] 
+ (61*ArcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])])/(3*Sqrt[3]) - 3*A 
rcTanh[(1 + 2*x)/Sqrt[2 + 3*x + 5*x^2]] + 2*Sqrt[2]*ArcTanh[(4 + 3*x)/(2*S 
qrt[2]*Sqrt[2 + 3*x + 5*x^2])] + 3*Log[x] - (3*Log[1 - x + x^2])/2
 

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 = 0.52 (sec) , antiderivative size = 1515, normalized size of antiderivative = 4.59

method	result	size
trager	\(\text {Expression too large to display}\)	\(1515\)
default	\(\text {Expression too large to display}\)	\(2593\)

Input:

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

Output:

-2/3*(x-1)*(2*x-13)/(x^2-x+1)-2/3*(5*x-1)/(x^2-x+1)*(5*x^2+3*x+2)^(1/2)-2/ 
3*ln(-(23364*RootOf(3*_Z^2+27*_Z+991)^2*RootOf(4*_Z^2-36*_Z+9)^2*x-23364*R 
ootOf(3*_Z^2+27*_Z+991)^2*RootOf(4*_Z^2-36*_Z+9)^2-108972*RootOf(3*_Z^2+27 
*_Z+991)^2*RootOf(4*_Z^2-36*_Z+9)*x+852972*RootOf(3*_Z^2+27*_Z+991)*RootOf 
(4*_Z^2-36*_Z+9)^2*x+108972*RootOf(3*_Z^2+27*_Z+991)^2*RootOf(4*_Z^2-36*_Z 
+9)-125307*RootOf(3*_Z^2+27*_Z+991)^2*x-852972*RootOf(3*_Z^2+27*_Z+991)*Ro 
otOf(4*_Z^2-36*_Z+9)^2+1080432*RootOf(3*_Z^2+27*_Z+991)*RootOf(4*_Z^2-36*_ 
Z+9)*(5*x^2+3*x+2)^(1/2)-684288*RootOf(3*_Z^2+27*_Z+991)*RootOf(4*_Z^2-36* 
_Z+9)*x-7994960*RootOf(4*_Z^2-36*_Z+9)^2*x+125307*RootOf(3*_Z^2+27*_Z+991) 
^2+10919844*RootOf(3*_Z^2+27*_Z+991)*RootOf(4*_Z^2-36*_Z[phone]*(5*x^ 
2+3*x+2)^(1/2)*RootOf(3*_Z^2+27*_Z[phone]*RootOf(3*_Z^2+27*_Z+991)* 
x+7994960*RootOf(4*_Z^2-36*_Z+9)^2-39745404*RootOf(4*_Z^2-36*_Z+9)*(5*x^2+ 
3*x+2)^(1/2)-58797420*RootOf(4*_Z^2-36*_Z+9)*x-25497639*RootOf(3*_Z^2+27*_ 
Z[phone]*RootOf(4*_Z^2-36*_Z[phone]*(5*x^2+3*x+2)^(1/2)-828200 
70*x-67503816)/x)*RootOf(3*_Z^2+27*_Z+991)-6*ln(-(23364*RootOf(3*_Z^2+27*_ 
Z+991)^2*RootOf(4*_Z^2-36*_Z+9)^2*x-23364*RootOf(3*_Z^2+27*_Z+991)^2*RootO 
f(4*_Z^2-36*_Z+9)^2-108972*RootOf(3*_Z^2+27*_Z+991)^2*RootOf(4*_Z^2-36*_Z+ 
9)*x+852972*RootOf(3*_Z^2+27*_Z+991)*RootOf(4*_Z^2-36*_Z+9)^2*x+108972*Roo 
tOf(3*_Z^2+27*_Z+991)^2*RootOf(4*_Z^2-36*_Z+9)-125307*RootOf(3*_Z^2+27*_Z+ 
991)^2*x-852972*RootOf(3*_Z^2+27*_Z+991)*RootOf(4*_Z^2-36*_Z+9)^2+10804...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\frac {244 \, \sqrt {3} {\left (x^{2} - x + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 122 \, \sqrt {3} {\left (x^{2} - x + 1\right )} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} + 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + 122 \, \sqrt {3} {\left (x^{2} - x + 1\right )} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} - 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + 36 \, \sqrt {2} {\left (x^{2} - x + 1\right )} \log \left (-\frac {4 \, \sqrt {2} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (3 \, x + 4\right )} + 49 \, x^{2} + 48 \, x + 32}{x^{2}}\right ) - 54 \, {\left (x^{2} - x + 1\right )} \log \left (x^{2} - x + 1\right ) + 108 \, {\left (x^{2} - x + 1\right )} \log \left (x\right ) - 27 \, {\left (x^{2} - x + 1\right )} \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) + 27 \, {\left (x^{2} - x + 1\right )} \log \left (\frac {9 \, x^{2} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) - 24 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (5 \, x - 1\right )} + 312 \, x - 264}{36 \, {\left (x^{2} - x + 1\right )}} \] Input:

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

Output:

1/36*(244*sqrt(3)*(x^2 - x + 1)*arctan(1/3*sqrt(3)*(2*x - 1)) + 122*sqrt(3 
)*(x^2 - x + 1)*arctan(1/3*(4*sqrt(3)*sqrt(5*x^2 + 3*x + 2)*(4*x - 5) + 31 
*sqrt(3)*(x^2 - 2*x))/(11*x^2 - 12*x - 8)) + 122*sqrt(3)*(x^2 - x + 1)*arc 
tan(1/3*(4*sqrt(3)*sqrt(5*x^2 + 3*x + 2)*(4*x - 5) - 31*sqrt(3)*(x^2 - 2*x 
))/(11*x^2 - 12*x - 8)) + 36*sqrt(2)*(x^2 - x + 1)*log(-(4*sqrt(2)*sqrt(5* 
x^2 + 3*x + 2)*(3*x + 4) + 49*x^2 + 48*x + 32)/x^2) - 54*(x^2 - x + 1)*log 
(x^2 - x + 1) + 108*(x^2 - x + 1)*log(x) - 27*(x^2 - x + 1)*log((9*x^2 + 2 
*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7*x + 3)/x^2) + 27*(x^2 - x + 1)*log((9 
*x^2 - 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7*x + 3)/x^2) - 24*sqrt(5*x^2 + 
 3*x + 2)*(5*x - 1) + 312*x - 264)/(x^2 - x + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.60 \[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\frac {61}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 2 \, \sqrt {2} \log \left (-\frac {{\left | -2 \, \sqrt {5} x - 2 \, \sqrt {2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} \right |}}{2 \, {\left (\sqrt {5} x - \sqrt {2} - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}}\right ) - \frac {61 \, {\left (\sqrt {5} + 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} - 4}{\sqrt {15} + 2 \, \sqrt {3}}\right )}{3 \, {\left (\sqrt {15} + 2 \, \sqrt {3}\right )}} + \frac {61 \, {\left (\sqrt {5} - 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} + 4}{\sqrt {15} - 2 \, \sqrt {3}}\right )}{3 \, {\left (\sqrt {15} - 2 \, \sqrt {3}\right )}} + \frac {2 \, {\left (55 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 74 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 459 \, \sqrt {5} x - 121 \, \sqrt {5} + 459 \, \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}}{3 \, {\left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 2 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} + 13 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} + 16 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} + 19\right )}} + \frac {2 \, {\left (13 \, x - 11\right )}}{3 \, {\left (x^{2} - x + 1\right )}} - \frac {3}{2} \, \log \left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} {\left (\sqrt {5} + 4\right )} + 5 \, \sqrt {5} + 12\right ) + \frac {3}{2} \, \log \left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} {\left (\sqrt {5} - 4\right )} - 5 \, \sqrt {5} + 12\right ) - \frac {3}{2} \, \log \left (x^{2} - x + 1\right ) + 3 \, \log \left ({\left | x \right |}\right ) \] Input:

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

Output:

61/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 2*sqrt(2)*log(-1/2*abs(-2*sqr 
t(5)*x - 2*sqrt(2) + 2*sqrt(5*x^2 + 3*x + 2))/(sqrt(5)*x - sqrt(2) - sqrt( 
5*x^2 + 3*x + 2))) - 61/3*(sqrt(5) + 2)*arctan(-(2*sqrt(5)*x - sqrt(5) - 2 
*sqrt(5*x^2 + 3*x + 2) - 4)/(sqrt(15) + 2*sqrt(3)))/(sqrt(15) + 2*sqrt(3)) 
 + 61/3*(sqrt(5) - 2)*arctan(-(2*sqrt(5)*x - sqrt(5) - 2*sqrt(5*x^2 + 3*x 
+ 2) + 4)/(sqrt(15) - 2*sqrt(3)))/(sqrt(15) - 2*sqrt(3)) + 2/3*(55*(sqrt(5 
)*x - sqrt(5*x^2 + 3*x + 2))^3 - 74*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x 
+ 2))^2 - 459*sqrt(5)*x - 121*sqrt(5) + 459*sqrt(5*x^2 + 3*x + 2))/((sqrt( 
5)*x - sqrt(5*x^2 + 3*x + 2))^4 - 2*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x 
+ 2))^3 + 13*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 + 16*sqrt(5)*(sqrt(5)*x 
 - sqrt(5*x^2 + 3*x + 2)) + 19) + 2/3*(13*x - 11)/(x^2 - x + 1) - 3/2*log( 
(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - (sqrt(5)*x - sqrt(5*x^2 + 3*x + 2) 
)*(sqrt(5) + 4) + 5*sqrt(5) + 12) + 3/2*log((sqrt(5)*x - sqrt(5*x^2 + 3*x 
+ 2))^2 - (sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))*(sqrt(5) - 4) - 5*sqrt(5) + 
12) - 3/2*log(x^2 - x + 1) + 3*log(abs(x))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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