\(\int \frac {x^2}{(1+2 x+\sqrt {2+3 x+5 x^2})^3} \, dx\) [27]

Optimal result

Integrand size = 25, antiderivative size = 485 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=-\frac {695+504 \sqrt {2}-\frac {4 \left (49+34 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )^2}+\frac {2 \left (419+94 \sqrt {2}-\frac {\left (173+27 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}\right )}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )}+\frac {1028 \arctan \left (\frac {3-4 \sqrt {2}-\frac {2 \left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3}}\right )}{3 \sqrt {3}}-34 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {2}-\sqrt {2+3 x+5 x^2}}{\sqrt {5} x}\right )+38 \log \left (\frac {4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}}{x^2}\right )-38 \log \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right ) \] Output:

-1/3*(695+504*2^(1/2)-4*(49+34*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x)/( 
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)^2+2*(419+94*2^(1/2)-(173+27*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)+1028/9 
*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)-34*arctanh(1/5*(2^(1/2)-(5*x^2+3*x+2)^(1/2))*5^(1/2)/x)*5^( 
1/2)+38*ln((4+3*x-2*2^(1/2)*(5*x^2+3*x+2)^(1/2))/x^2)-38*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 = 14.83 (sec) , antiderivative size = 1118, normalized size of antiderivative = 2.31 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(-37 + 17*x))/(3*(1 - x + x^2)^2) + (151 + 163*x)/(3 - 3*x + 3*x^2) + ( 
2*Sqrt[2 + 3*x + 5*x^2]*(-33 + 34*x - 47*x^2 + 3*x^3))/(3*(1 - x + x^2)^2) 
 + 17*Sqrt[5]*ArcSinh[(3 + 10*x)/Sqrt[31]] - (514*ArcTan[(-1 + 2*x)/Sqrt[3 
]])/(3*Sqrt[3]) + (((7*I)/3)*(49*I + 53*Sqrt[3])*ArcTan[(3*(9*(12329 + (92 
48*I)*Sqrt[3]) + 6*(58271 - (8024*I)*Sqrt[3])*x + (76651 + (38480*I)*Sqrt[ 
3])*x^2 + (-92058 - (297648*I)*Sqrt[3])*x^3 + (-24139 + (224720*I)*Sqrt[3] 
)*x^4))/(99*(2720*I + 41*Sqrt[3]) - 3*(207760*I + 53511*Sqrt[3])*x^4 + 324 
84*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2] + x^2*(363984*I + 472577 
*Sqrt[3] - 75796*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + 2*x^3*( 
-236904*I + 279341*Sqrt[3] + 54140*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 
 5*x^2]) + x*(363984*I + 70042*Sqrt[3] + 75796*Sqrt[3 - (12*I)*Sqrt[3]]*Sq 
rt[2 + 3*x + 5*x^2]))])/Sqrt[3 - (12*I)*Sqrt[3]] + (7*(49 + (53*I)*Sqrt[3] 
)*ArcTan[(3*(-110961 + (83232*I)*Sqrt[3] + (-349626 - (48144*I)*Sqrt[3])*x 
 + (-76651 + (38480*I)*Sqrt[3])*x^2 + 6*(15343 - (49608*I)*Sqrt[3])*x^3 + 
(24139 + (224720*I)*Sqrt[3])*x^4))/(99*(-2720*I + 41*Sqrt[3]) + (623280*I 
- 160533*Sqrt[3])*x^4 + 32484*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^ 
2] + x^2*(-363984*I + 472577*Sqrt[3] - 75796*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt 
[2 + 3*x + 5*x^2]) + 2*x^3*(236904*I + 279341*Sqrt[3] + 54140*Sqrt[3 + (12 
*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + x*(-363984*I + 70042*Sqrt[3] + 75796 
*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]))])/(3*Sqrt[3 + (12*I)*...
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 361, normalized size of antiderivative = 0.74, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-38 x-121}{x^2-x+1}+\frac {17 \sqrt {5 x^2+3 x+2}}{x^2-x+1}+\frac {49 x \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}+\frac {3 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}-\frac {12 x \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^3}-\frac {20 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^3}-\frac {5 (31 x-25)}{\left (x^2-x+1\right )^2}+\frac {4 (19 x-1)}{\left (x^2-x+1\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 17 \sqrt {5} \text {arcsinh}\left (\frac {10 x+3}{\sqrt {31}}\right )-\frac {514 \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{3 \sqrt {3}}+\frac {514 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}-38 \text {arctanh}\left (\frac {2 x+1}{\sqrt {5 x^2+3 x+2}}\right )+\frac {5 \sqrt {5 x^2+3 x+2} (101-338 x)}{147 \left (x^2-x+1\right )}+\frac {(44-237 x) \sqrt {5 x^2+3 x+2}}{49 \left (x^2-x+1\right )}-\frac {(1-2 x) \sqrt {5 x^2+3 x+2}}{x^2-x+1}-\frac {49 (2-x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )}+\frac {10 (1-2 x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )^2}+\frac {2 (2-x) \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}-\frac {34 (1-2 x)}{3 \left (x^2-x+1\right )}+\frac {5 (19 x+37)}{3 \left (x^2-x+1\right )}-\frac {2 (37-17 x)}{3 \left (x^2-x+1\right )^2}-19 \log \left (x^2-x+1\right )\)

Input:

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

Output:

(-2*(37 - 17*x))/(3*(1 - x + x^2)^2) - (34*(1 - 2*x))/(3*(1 - x + x^2)) + 
(5*(37 + 19*x))/(3*(1 - x + x^2)) + (10*(1 - 2*x)*Sqrt[2 + 3*x + 5*x^2])/( 
3*(1 - x + x^2)^2) + (2*(2 - x)*Sqrt[2 + 3*x + 5*x^2])/(1 - x + x^2)^2 + ( 
5*(101 - 338*x)*Sqrt[2 + 3*x + 5*x^2])/(147*(1 - x + x^2)) + ((44 - 237*x) 
*Sqrt[2 + 3*x + 5*x^2])/(49*(1 - x + x^2)) - ((1 - 2*x)*Sqrt[2 + 3*x + 5*x 
^2])/(1 - x + x^2) - (49*(2 - x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)) 
+ 17*Sqrt[5]*ArcSinh[(3 + 10*x)/Sqrt[31]] + (514*ArcTan[(1 - 2*x)/Sqrt[3]] 
)/(3*Sqrt[3]) - (514*ArcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])])/(3 
*Sqrt[3]) - 38*ArcTanh[(1 + 2*x)/Sqrt[2 + 3*x + 5*x^2]] - 19*Log[1 - x + x 
^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 = 0.62 (sec) , antiderivative size = 1178, normalized size of antiderivative = 2.43

method	result	size
trager	\(\text {Expression too large to display}\)	\(1178\)
default	\(\text {Expression too large to display}\)	\(10342\)

Input:

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

Output:

-1/3*(77*x^3-317*x^2+243*x-200)*x/(x^2-x+1)^2+2/3*(3*x^3-47*x^2+34*x-33)/( 
x^2-x+1)^2*(5*x^2+3*x+2)^(1/2)-2/3*ln(-162*RootOf(3*_Z^2+342*_Z+75796)^2*R 
ootOf(4*_Z^2-456*_Z-9)^2*x+57483*RootOf(3*_Z^2+342*_Z+75796)^2*RootOf(4*_Z 
^2-456*_Z-9)*x+209748*RootOf(3*_Z^2+342*_Z+75796)*RootOf(4*_Z^2-456*_Z-9)^ 
2*x-2164968*RootOf(3*_Z^2+342*_Z+75796)^2*x+4797162*RootOf(3*_Z^2+342*_Z+7 
5796)*RootOf(4*_Z^2-456*_Z-9)*(5*x^2+3*x+2)^(1/2)-1375902*RootOf(3*_Z^2+34 
2*_Z+75796)*RootOf(4*_Z^2-456*_Z-9)*x[phone]*RootOf(4*_Z^2-456*_Z-9)^2* 
x-13447782*RootOf(3*_Z^2+342*_Z+75796)*RootOf(4*_Z^2-456*_Z-9)-734673564*( 
5*x^2+3*x+2)^(1/2)*RootOf(3*_Z^2+342*_Z[phone]*RootOf(3*_Z^2+34 
2*_Z+75796)*x-7386528492*RootOf(4*_Z^2-456*_Z-9)*(5*x^2+3*x+2)^(1/2)-37559 
822580*RootOf(4*_Z^2-456*_Z-9)*x[phone]*RootOf(3*_Z^2+342*_Z[phone] 
8698332*RootOf(4*_Z^2-456*_Z-9)[phone]*(5*x^2+3*x+2)^(1/2)[phone] 
5520*x[phone])*RootOf(3*_Z^2+342*_Z+75796)-76*ln(-162*RootOf(3*_Z^2+ 
342*_Z+75796)^2*RootOf(4*_Z^2-456*_Z-9)^2*x+57483*RootOf(3*_Z^2+342*_Z+757 
96)^2*RootOf(4*_Z^2-456*_Z-9)*x+209748*RootOf(3*_Z^2+342*_Z+75796)*RootOf( 
4*_Z^2-456*_Z-9)^2*x-2164968*RootOf(3*_Z^2+342*_Z+75796)^2*x+4797162*RootO 
f(3*_Z^2+342*_Z+75796)*RootOf(4*_Z^2-456*_Z-9)*(5*x^2+3*x+2)^(1/2)-1375902 
*RootOf(3*_Z^2+342*_Z+75796)*RootOf(4*_Z^2-456*_Z-9)*x[phone]*RootOf(4* 
_Z^2-456*_Z-9)^2*x-13447782*RootOf(3*_Z^2+342*_Z+75796)*RootOf(4*_Z^2-456* 
_Z-9)-734673564*(5*x^2+3*x+2)^(1/2)*RootOf(3*_Z^2+342*_Z[phone]...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 439, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\frac {978 \, x^{3} - 1028 \, \sqrt {3} {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 514 \, \sqrt {3} {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} + 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) - 514 \, \sqrt {3} {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} - 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + 153 \, \sqrt {5} {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \log \left (-4 \, \sqrt {5} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (10 \, x + 3\right )} - 200 \, x^{2} - 120 \, x - 49\right ) - 72 \, x^{2} - 342 \, {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \log \left (x^{2} - x + 1\right ) - 171 \, {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) + 171 \, {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \log \left (\frac {9 \, x^{2} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) + 12 \, {\left (3 \, x^{3} - 47 \, x^{2} + 34 \, x - 33\right )} \sqrt {5 \, x^{2} + 3 \, x + 2} + 276 \, x + 462}{18 \, {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )}} \] Input:

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

Output:

1/18*(978*x^3 - 1028*sqrt(3)*(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)*arctan(1/3*sq 
rt(3)*(2*x - 1)) - 514*sqrt(3)*(x^4 - 2*x^3 + 3*x^2 - 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)) - 514*sqrt(3)*(x^4 - 2*x^3 + 3*x^2 - 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)) + 153*sqrt(5)*(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)*log(-4*sqrt(5 
)*sqrt(5*x^2 + 3*x + 2)*(10*x + 3) - 200*x^2 - 120*x - 49) - 72*x^2 - 342* 
(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)*log(x^2 - x + 1) - 171*(x^4 - 2*x^3 + 3*x^ 
2 - 2*x + 1)*log((9*x^2 + 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7*x + 3)/x^2 
) + 171*(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)*log((9*x^2 - 2*sqrt(5*x^2 + 3*x + 
2)*(2*x + 1) + 7*x + 3)/x^2) + 12*(3*x^3 - 47*x^2 + 34*x - 33)*sqrt(5*x^2 
+ 3*x + 2) + 276*x + 462)/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 607, normalized size of antiderivative = 1.25 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-514/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 17*sqrt(5)*log(-2*sqrt(5)*( 
sqrt(5)*x - sqrt(5*x^2 + 3*x + 2)) - 3) + 514/3*(sqrt(5) + 2)*arctan(-(2*s 
qrt(5)*x - sqrt(5) - 2*sqrt(5*x^2 + 3*x + 2) - 4)/(sqrt(15) + 2*sqrt(3)))/ 
(sqrt(15) + 2*sqrt(3)) - 514/3*(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*sqr 
t(3)) + 2/3*(401*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^7 - 107*sqrt(5)*(sqrt 
(5)*x - sqrt(5*x^2 + 3*x + 2))^6 + 1757*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2) 
)^5 + 6621*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^4 + 44512*(sqrt(5)* 
x - sqrt(5*x^2 + 3*x + 2))^3 + 29123*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x 
 + 2))^2 + 45133*sqrt(5)*x + 5265*sqrt(5) - 45133*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)*(sqr 
t(5)*x - sqrt(5*x^2 + 3*x + 2)) + 19)^2 + 1/3*(163*x^3 - 12*x^2 + 46*x + 7 
7)/(x^2 - x + 1)^2 - 19*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) + 19*log((sq 
rt(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) - 19*log(x^2 - x + 1)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int \frac {x^{2}}{17 \sqrt {5 x^{2}+3 x +2}\, x^{2}+15 \sqrt {5 x^{2}+3 x +2}\, x +5 \sqrt {5 x^{2}+3 x +2}+38 x^{3}+45 x^{2}+27 x +7}d x \] Input:

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

Output:

int(x**2/(17*sqrt(5*x**2 + 3*x + 2)*x**2 + 15*sqrt(5*x**2 + 3*x + 2)*x + 5 
*sqrt(5*x**2 + 3*x + 2) + 38*x**3 + 45*x**2 + 27*x + 7),x)