Integrand size = 30, antiderivative size = 151 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x+2 x^2}} \, dx=-\frac {1}{2} \sqrt {1+\frac {7 \sqrt {\frac {2}{5}}}{5}} \text {arctanh}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (17-4 \sqrt {10}\right ) x}{2 \sqrt {55-17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right )+\frac {1}{2} \sqrt {1-\frac {7 \sqrt {\frac {2}{5}}}{5}} \text {arctanh}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (17+4 \sqrt {10}\right ) x}{2 \sqrt {55+17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right ) \]
1/10*arctanh(1/2*(12+3*10^(1/2)+x*(17+4*10^(1/2)))/(2*x^2+3*x+1)^(1/2)/(55 +17*10^(1/2))^(1/2))*(25-7*10^(1/2))^(1/2)-1/10*arctanh(1/2*(x*(17-4*10^(1 /2))+12-3*10^(1/2))/(2*x^2+3*x+1)^(1/2)/(55-17*10^(1/2))^(1/2))*(25+7*10^( 1/2))^(1/2)
Time = 0.64 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.72 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x+2 x^2}} \, dx=-\frac {1}{5} \sqrt {25+7 \sqrt {10}} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right )+\frac {1}{5} \sqrt {25-7 \sqrt {10}} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right ) \]
-1/5*(Sqrt[25 + 7*Sqrt[10]]*ArcTanh[(Sqrt[1 - Sqrt[2/5]]*Sqrt[1 + 3*x + 2* x^2])/(1 + 2*x)]) + (Sqrt[25 - 7*Sqrt[10]]*ArcTanh[(Sqrt[1 + Sqrt[2/5]]*Sq rt[1 + 3*x + 2*x^2])/(1 + 2*x)])/5
Time = 0.33 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1365, 27, 1154, 219}
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.
| \(\displaystyle \int \frac {x+2}{\left (-3 x^2+4 x+2\right ) \sqrt {2 x^2+3 x+1}} \, dx\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {1}{5} \left (5-4 \sqrt {10}\right ) \int \frac {1}{2 \left (-3 x-\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx+\frac {1}{5} \left (5+4 \sqrt {10}\right ) \int \frac {1}{2 \left (-3 x+\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (5-4 \sqrt {10}\right ) \int \frac {1}{\left (-3 x-\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx+\frac {1}{10} \left (5+4 \sqrt {10}\right ) \int \frac {1}{\left (-3 x+\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {1}{5} \left (5-4 \sqrt {10}\right ) \int \frac {1}{4 \left (55-17 \sqrt {10}\right )-\frac {\left (\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )\right )^2}{2 x^2+3 x+1}}d\left (-\frac {\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{\sqrt {2 x^2+3 x+1}}\right )-\frac {1}{5} \left (5+4 \sqrt {10}\right ) \int \frac {1}{4 \left (55+17 \sqrt {10}\right )-\frac {\left (\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )\right )^2}{2 x^2+3 x+1}}d\left (-\frac {\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{\sqrt {2 x^2+3 x+1}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (5-4 \sqrt {10}\right ) \text {arctanh}\left (\frac {\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {55-17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right )}{10 \sqrt {55-17 \sqrt {10}}}+\frac {\left (5+4 \sqrt {10}\right ) \text {arctanh}\left (\frac {\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {55+17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right )}{10 \sqrt {55+17 \sqrt {10}}}\) |
((5 - 4*Sqrt[10])*ArcTanh[(3*(4 - Sqrt[10]) + (17 - 4*Sqrt[10])*x)/(2*Sqrt [55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/(10*Sqrt[55 - 17*Sqrt[10]]) + ((5 + 4*Sqrt[10])*ArcTanh[(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt [55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/(10*Sqrt[55 + 17*Sqrt[10]])
3.1.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim p[(2*c*g - h*(b - q))/q Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] , x] - Simp[(2*c*g - h*(b + q))/q Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f *x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0 ] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
Time = 2.32 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.23
| method | result | size |
| default | \(\frac {\left (8+\sqrt {10}\right ) \sqrt {10}\, \operatorname {arctanh}\left (\frac {55+17 \sqrt {10}+\frac {9 \left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {55+17 \sqrt {10}}\, \sqrt {18 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )+55+17 \sqrt {10}}}\right )}{20 \sqrt {55+17 \sqrt {10}}}+\frac {\left (-8+\sqrt {10}\right ) \sqrt {10}\, \operatorname {arctanh}\left (\frac {55-17 \sqrt {10}+\frac {9 \left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {55-17 \sqrt {10}}\, \sqrt {18 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )+55-17 \sqrt {10}}}\right )}{20 \sqrt {55-17 \sqrt {10}}}\) | \(186\) |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right ) \ln \left (-\frac {10000 x \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right )-22600 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right ) x -12600 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right )-3200 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2} \sqrt {2 x^{2}+3 x +1}+1105 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right ) x +630 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right )-236 \sqrt {2 x^{2}+3 x +1}}{100 x \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-39 x -14}\right )}{2}+\operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right ) \ln \left (-\frac {10000 x \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{5}+12600 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{3} x +12600 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{3}+1600 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2} \sqrt {2 x^{2}+3 x +1}-7695 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right ) x -5670 \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )-918 \sqrt {2 x^{2}+3 x +1}}{100 x \operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-11 x +14}\right )\) | \(442\) |
1/20*(8+10^(1/2))*10^(1/2)/(55+17*10^(1/2))^(1/2)*arctanh(9/2*(110/9+34/9* 10^(1/2)+(17/3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(55+17*10^(1/2))^(1/2)/ (18*(x-2/3-1/3*10^(1/2))^2+9*(17/3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))+55+1 7*10^(1/2))^(1/2))+1/20*(-8+10^(1/2))*10^(1/2)/(55-17*10^(1/2))^(1/2)*arct anh(9/2*(110/9-34/9*10^(1/2)+(17/3-4/3*10^(1/2))*(x-2/3+1/3*10^(1/2)))/(55 -17*10^(1/2))^(1/2)/(18*(x-2/3+1/3*10^(1/2))^2+9*(17/3-4/3*10^(1/2))*(x-2/ 3+1/3*10^(1/2))+55-17*10^(1/2))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (103) = 206\).
Time = 0.31 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.62 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x+2 x^2}} \, dx=\frac {1}{10} \, \sqrt {7 \, \sqrt {10} + 25} \log \left (-\frac {3 \, \sqrt {10} x + {\left (\sqrt {10} x - 4 \, x\right )} \sqrt {7 \, \sqrt {10} + 25} + 6 \, x - 6 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 6}{x}\right ) - \frac {1}{10} \, \sqrt {7 \, \sqrt {10} + 25} \log \left (-\frac {3 \, \sqrt {10} x - {\left (\sqrt {10} x - 4 \, x\right )} \sqrt {7 \, \sqrt {10} + 25} + 6 \, x - 6 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 6}{x}\right ) + \frac {1}{10} \, \sqrt {-7 \, \sqrt {10} + 25} \log \left (\frac {3 \, \sqrt {10} x + {\left (\sqrt {10} x + 4 \, x\right )} \sqrt {-7 \, \sqrt {10} + 25} - 6 \, x + 6 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 6}{x}\right ) - \frac {1}{10} \, \sqrt {-7 \, \sqrt {10} + 25} \log \left (\frac {3 \, \sqrt {10} x - {\left (\sqrt {10} x + 4 \, x\right )} \sqrt {-7 \, \sqrt {10} + 25} - 6 \, x + 6 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 6}{x}\right ) \]
1/10*sqrt(7*sqrt(10) + 25)*log(-(3*sqrt(10)*x + (sqrt(10)*x - 4*x)*sqrt(7* sqrt(10) + 25) + 6*x - 6*sqrt(2*x^2 + 3*x + 1) + 6)/x) - 1/10*sqrt(7*sqrt( 10) + 25)*log(-(3*sqrt(10)*x - (sqrt(10)*x - 4*x)*sqrt(7*sqrt(10) + 25) + 6*x - 6*sqrt(2*x^2 + 3*x + 1) + 6)/x) + 1/10*sqrt(-7*sqrt(10) + 25)*log((3 *sqrt(10)*x + (sqrt(10)*x + 4*x)*sqrt(-7*sqrt(10) + 25) - 6*x + 6*sqrt(2*x ^2 + 3*x + 1) - 6)/x) - 1/10*sqrt(-7*sqrt(10) + 25)*log((3*sqrt(10)*x - (s qrt(10)*x + 4*x)*sqrt(-7*sqrt(10) + 25) - 6*x + 6*sqrt(2*x^2 + 3*x + 1) - 6)/x)
\[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x+2 x^2}} \, dx=- \int \frac {x}{3 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 4 x \sqrt {2 x^{2} + 3 x + 1} - 2 \sqrt {2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{3 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 4 x \sqrt {2 x^{2} + 3 x + 1} - 2 \sqrt {2 x^{2} + 3 x + 1}}\, dx \]
-Integral(x/(3*x**2*sqrt(2*x**2 + 3*x + 1) - 4*x*sqrt(2*x**2 + 3*x + 1) - 2*sqrt(2*x**2 + 3*x + 1)), x) - Integral(2/(3*x**2*sqrt(2*x**2 + 3*x + 1) - 4*x*sqrt(2*x**2 + 3*x + 1) - 2*sqrt(2*x**2 + 3*x + 1)), x)
Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (103) = 206\).
Time = 0.31 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.40 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x+2 x^2}} \, dx=\frac {1}{60} \, \sqrt {10} {\left (\frac {3 \, \sqrt {10} \log \left (\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {17 \, \sqrt {10} + 55}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{\sqrt {17 \, \sqrt {10} + 55}} + \frac {\sqrt {10} \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}}{{\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{\sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}} + \frac {24 \, \log \left (\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {17 \, \sqrt {10} + 55}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{\sqrt {17 \, \sqrt {10} + 55}} - \frac {8 \, \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}}{{\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{\sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}}\right )} \]
1/60*sqrt(10)*(3*sqrt(10)*log(2/9*sqrt(10) + 2/3*sqrt(2*x^2 + 3*x + 1)*sqr t(17*sqrt(10) + 55)/abs(6*x - 2*sqrt(10) - 4) + 34/9*sqrt(10)/abs(6*x - 2* sqrt(10) - 4) + 110/9/abs(6*x - 2*sqrt(10) - 4) + 17/18)/sqrt(17*sqrt(10) + 55) + sqrt(10)*log(-2/9*sqrt(10) + 2*sqrt(2*x^2 + 3*x + 1)*sqrt(-17/9*sq rt(10) + 55/9)/abs(6*x + 2*sqrt(10) - 4) - 34/9*sqrt(10)/abs(6*x + 2*sqrt( 10) - 4) + 110/9/abs(6*x + 2*sqrt(10) - 4) + 17/18)/sqrt(-17/9*sqrt(10) + 55/9) + 24*log(2/9*sqrt(10) + 2/3*sqrt(2*x^2 + 3*x + 1)*sqrt(17*sqrt(10) + 55)/abs(6*x - 2*sqrt(10) - 4) + 34/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) + 110/9/abs(6*x - 2*sqrt(10) - 4) + 17/18)/sqrt(17*sqrt(10) + 55) - 8*log(- 2/9*sqrt(10) + 2*sqrt(2*x^2 + 3*x + 1)*sqrt(-17/9*sqrt(10) + 55/9)/abs(6*x + 2*sqrt(10) - 4) - 34/9*sqrt(10)/abs(6*x + 2*sqrt(10) - 4) + 110/9/abs(6 *x + 2*sqrt(10) - 4) + 17/18)/sqrt(-17/9*sqrt(10) + 55/9))
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.62 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x+2 x^2}} \, dx=0.169235232112667 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} + 5.90976932712000\right ) - 0.686556214893333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.176527156327000\right ) + 0.686556214893333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.919278730509000\right ) - 0.169235232112667 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 1.04272727395000\right ) \]
0.169235232112667*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) + 5.9097693271200 0) - 0.686556214893333*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 0.17652715 6327000) + 0.686556214893333*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 0.91 9278730509000) - 0.169235232112667*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 1.04272727395000)
Timed out. \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x+2 x^2}} \, dx=\int \frac {x+2}{\sqrt {2\,x^2+3\,x+1}\,\left (-3\,x^2+4\,x+2\right )} \,d x \]