Integrand size = 30, antiderivative size = 139 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx=\frac {1}{2} \sqrt {-\frac {13}{5}+\sqrt {10}} \arctan \left (\frac {3 \left (4-\sqrt {10}\right )+\left (1+4 \sqrt {10}\right ) x}{2 \sqrt {1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )+\frac {1}{2} \sqrt {\frac {13}{5}+\sqrt {10}} \text {arctanh}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (1-4 \sqrt {10}\right ) x}{2 \sqrt {-1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right ) \]
1/10*arctan(1/2*(12-3*10^(1/2)+x*(1+4*10^(1/2)))/(-2*x^2+3*x+1)^(1/2)/(1+1 0^(1/2))^(1/2))*(-65+25*10^(1/2))^(1/2)+1/10*arctanh(1/2*(x*(1-4*10^(1/2)) +12+3*10^(1/2))/(-2*x^2+3*x+1)^(1/2)/(-1+10^(1/2))^(1/2))*(65+25*10^(1/2)) ^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.07 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx=-\frac {1}{2} \text {RootSum}\left [5+20 \text {$\#$1}+8 \text {$\#$1}^2-8 \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {-7 \log (x)+7 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}-2 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 \log (x) \text {$\#$1}^2+2 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{5+4 \text {$\#$1}-6 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
-1/2*RootSum[5 + 20*#1 + 8*#1^2 - 8*#1^3 + 2*#1^4 & , (-7*Log[x] + 7*Log[- 1 + Sqrt[1 + 3*x - 2*x^2] - x*#1] + 2*Log[x]*#1 - 2*Log[-1 + Sqrt[1 + 3*x - 2*x^2] - x*#1]*#1 - 2*Log[x]*#1^2 + 2*Log[-1 + Sqrt[1 + 3*x - 2*x^2] - x *#1]*#1^2)/(5 + 4*#1 - 6*#1^2 + 2*#1^3) & ]
Time = 0.36 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1365, 27, 1154, 217, 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 |
| \(\displaystyle -\frac {1}{5} \left (5+4 \sqrt {10}\right ) \int \frac {1}{-\frac {\left (\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )\right )^2}{-2 x^2+3 x+1}-4 \left (1-\sqrt {10}\right )}d\left (-\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{\sqrt {-2 x^2+3 x+1}}\right )-\frac {1}{5} \left (5-4 \sqrt {10}\right ) \int \frac {1}{-\frac {\left (\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )\right )^2}{-2 x^2+3 x+1}-4 \left (1+\sqrt {10}\right )}d\left (-\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{\sqrt {-2 x^2+3 x+1}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {1}{5} \left (5+4 \sqrt {10}\right ) \int \frac {1}{-\frac {\left (\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )\right )^2}{-2 x^2+3 x+1}-4 \left (1-\sqrt {10}\right )}d\left (-\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{\sqrt {-2 x^2+3 x+1}}\right )-\frac {\left (5-4 \sqrt {10}\right ) \arctan \left (\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )}{10 \sqrt {1+\sqrt {10}}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (5+4 \sqrt {10}\right ) \text {arctanh}\left (\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {\sqrt {10}-1} \sqrt {-2 x^2+3 x+1}}\right )}{10 \sqrt {\sqrt {10}-1}}-\frac {\left (5-4 \sqrt {10}\right ) \arctan \left (\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )}{10 \sqrt {1+\sqrt {10}}}\) |
-1/10*((5 - 4*Sqrt[10])*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2* Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/Sqrt[1 + Sqrt[10]] + ((5 + 4*S qrt[10])*ArcTanh[(3*(4 + Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt [10]]*Sqrt[1 + 3*x - 2*x^2])])/(10*Sqrt[-1 + Sqrt[10]])
3.1.25.3.1 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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
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 = 2.37 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.27
| method | result | size |
| default | \(-\frac {\left (-8+\sqrt {10}\right ) \sqrt {10}\, \arctan \left (\frac {-1-\sqrt {10}+\frac {9 \left (\frac {1}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {1+\sqrt {10}}\, \sqrt {-18 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {1}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )-1-\sqrt {10}}}\right )}{20 \sqrt {1+\sqrt {10}}}+\frac {\left (8+\sqrt {10}\right ) \sqrt {10}\, \operatorname {arctanh}\left (\frac {-1+\sqrt {10}+\frac {9 \left (\frac {1}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {-1+\sqrt {10}}\, \sqrt {-18 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {1}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )-1+\sqrt {10}}}\right )}{20 \sqrt {-1+\sqrt {10}}}\) | \(176\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2}-130\right ) \ln \left (\frac {129200 x \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2}-130\right )+17640 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2}-130\right ) x +34200 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2}-130\right )+208000 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2} \sqrt {-2 x^{2}+3 x +1}-53 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2}-130\right ) x +4770 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2}-130\right )+29300 \sqrt {-2 x^{2}+3 x +1}}{20 x \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2}-3 x +10}\right )}{10}+\operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right ) \ln \left (-\frac {129200 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{5} x -353560 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{3} x +20800 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2} \sqrt {-2 x^{2}+3 x +1}-34200 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{3}+241227 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right ) x -29970 \sqrt {-2 x^{2}+3 x +1}+49230 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )}{20 x \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-520 \textit {\_Z}^{2}-81\right )^{2}-23 x -10}\right )\) | \(441\) |
-1/20*(-8+10^(1/2))*10^(1/2)/(1+10^(1/2))^(1/2)*arctan(9/2*(-2/9-2/9*10^(1 /2)+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2)))/(1+10^(1/2))^(1/2)/(-18*(x-2/ 3+1/3*10^(1/2))^2+9*(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1-10^(1/2))^(1 /2))+1/20*(8+10^(1/2))*10^(1/2)/(-1+10^(1/2))^(1/2)*arctanh(9/2*(-2/9+2/9* 10^(1/2)+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(-1+10^(1/2))^(1/2)/(-18 *(x-2/3-1/3*10^(1/2))^2+9*(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1+10^(1/ 2))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (99) = 198\).
Time = 0.29 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.22 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx=-\frac {1}{10} \, \sqrt {5} \sqrt {5 \, \sqrt {5} \sqrt {2} + 13} \log \left (\frac {9 \, \sqrt {5} \sqrt {2} x + {\left (4 \, \sqrt {5} x - 7 \, \sqrt {2} x\right )} \sqrt {5 \, \sqrt {5} \sqrt {2} + 13} - 18 \, x + 18 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 18}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {5 \, \sqrt {5} \sqrt {2} + 13} \log \left (\frac {9 \, \sqrt {5} \sqrt {2} x - {\left (4 \, \sqrt {5} x - 7 \, \sqrt {2} x\right )} \sqrt {5 \, \sqrt {5} \sqrt {2} + 13} - 18 \, x + 18 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 18}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {-5 \, \sqrt {5} \sqrt {2} + 13} \log \left (-\frac {9 \, \sqrt {5} \sqrt {2} x + {\left (4 \, \sqrt {5} x + 7 \, \sqrt {2} x\right )} \sqrt {-5 \, \sqrt {5} \sqrt {2} + 13} + 18 \, x - 18 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} + 18}{x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {-5 \, \sqrt {5} \sqrt {2} + 13} \log \left (-\frac {9 \, \sqrt {5} \sqrt {2} x - {\left (4 \, \sqrt {5} x + 7 \, \sqrt {2} x\right )} \sqrt {-5 \, \sqrt {5} \sqrt {2} + 13} + 18 \, x - 18 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} + 18}{x}\right ) \]
-1/10*sqrt(5)*sqrt(5*sqrt(5)*sqrt(2) + 13)*log((9*sqrt(5)*sqrt(2)*x + (4*s qrt(5)*x - 7*sqrt(2)*x)*sqrt(5*sqrt(5)*sqrt(2) + 13) - 18*x + 18*sqrt(-2*x ^2 + 3*x + 1) - 18)/x) + 1/10*sqrt(5)*sqrt(5*sqrt(5)*sqrt(2) + 13)*log((9* sqrt(5)*sqrt(2)*x - (4*sqrt(5)*x - 7*sqrt(2)*x)*sqrt(5*sqrt(5)*sqrt(2) + 1 3) - 18*x + 18*sqrt(-2*x^2 + 3*x + 1) - 18)/x) + 1/10*sqrt(5)*sqrt(-5*sqrt (5)*sqrt(2) + 13)*log(-(9*sqrt(5)*sqrt(2)*x + (4*sqrt(5)*x + 7*sqrt(2)*x)* sqrt(-5*sqrt(5)*sqrt(2) + 13) + 18*x - 18*sqrt(-2*x^2 + 3*x + 1) + 18)/x) - 1/10*sqrt(5)*sqrt(-5*sqrt(5)*sqrt(2) + 13)*log(-(9*sqrt(5)*sqrt(2)*x - ( 4*sqrt(5)*x + 7*sqrt(2)*x)*sqrt(-5*sqrt(5)*sqrt(2) + 13) + 18*x - 18*sqrt( -2*x^2 + 3*x + 1) + 18)/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. 361 vs. \(2 (99) = 198\).
Time = 0.31 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.60 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx=-\frac {1}{20} \, \sqrt {10} {\left (\frac {\sqrt {10} \arcsin \left (\frac {8 \, \sqrt {17} \sqrt {10} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {17} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {6 \, \sqrt {17} \sqrt {10}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {24 \, \sqrt {17}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}}\right )}{\sqrt {\sqrt {10} + 1}} - \frac {\sqrt {10} \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} \sqrt {\sqrt {10} - 1}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} - \frac {2}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {1}{18}\right )}{\sqrt {\sqrt {10} - 1}} - \frac {8 \, \arcsin \left (\frac {8 \, \sqrt {17} \sqrt {10} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {17} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {6 \, \sqrt {17} \sqrt {10}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {24 \, \sqrt {17}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}}\right )}{\sqrt {\sqrt {10} + 1}} - \frac {8 \, \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} \sqrt {\sqrt {10} - 1}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} - \frac {2}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {1}{18}\right )}{\sqrt {\sqrt {10} - 1}}\right )} \]
-1/20*sqrt(10)*(sqrt(10)*arcsin(8/17*sqrt(17)*sqrt(10)*x/abs(6*x + 2*sqrt( 10) - 4) + 2/17*sqrt(17)*x/abs(6*x + 2*sqrt(10) - 4) - 6/17*sqrt(17)*sqrt( 10)/abs(6*x + 2*sqrt(10) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/ sqrt(sqrt(10) + 1) - sqrt(10)*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(sqrt(10) - 1)/abs(6*x - 2*sqrt(10) - 4) + 2/9*sqrt(10)/abs(6*x - 2 *sqrt(10) - 4) - 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/18)/sqrt(sqrt(10) - 1) - 8*arcsin(8/17*sqrt(17)*sqrt(10)*x/abs(6*x + 2*sqrt(10) - 4) + 2/17*sqrt( 17)*x/abs(6*x + 2*sqrt(10) - 4) - 6/17*sqrt(17)*sqrt(10)/abs(6*x + 2*sqrt( 10) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/sqrt(sqrt(10) + 1) - 8*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(sqrt(10) - 1)/abs(6* x - 2*sqrt(10) - 4) + 2/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) - 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/18)/sqrt(sqrt(10) - 1))
Leaf count of result is larger than twice the leaf count of optimal. 16965 vs. \(2 (99) = 198\).
Time = 255.51 (sec) , antiderivative size = 16965, normalized size of antiderivative = 122.05 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx=\text {Too large to display} \]
1/5*sqrt(25*sqrt(10) - 65)*(arctan(25440019409633258254215013/362376882997 34789947759590083723891896320*(sqrt(34) + 2*sqrt(10) + sqrt(2) + sqrt(25*s qrt(10) - 65))^15 - 6111149415804811055946029/1288451139546125864809229869 6435161563136*(sqrt(34) + 2*sqrt(10) + sqrt(2) + sqrt(25*sqrt(10) - 65))^1 4 - 958578619223566161086771177/14495075319893915979103836033489556758528* (sqrt(34) + 2*sqrt(10) + sqrt(2) + sqrt(25*sqrt(10) - 65))^13 - 3257049408 428409227173436832461/579803012795756639164153441339582270341120*(sqrt(34) + 2*sqrt(10) + sqrt(2) + sqrt(25*sqrt(10) - 65))^12 + 9736962921134002094 58145664321/36237688299734789947759590083723891896320*(sqrt(34) + 2*sqrt(1 0) + sqrt(2) + sqrt(25*sqrt(10) - 65))^11 - 655458488897784251494816231968 7/15670351697182611869301444360529250549760*(sqrt(34) + 2*sqrt(10) + sqrt( 2) + sqrt(25*sqrt(10) - 65))^10 + 1404976614114289514312381195621513/72475 376599469579895519180167447783792640*(sqrt(34) + 2*sqrt(10) + sqrt(2) + sq rt(25*sqrt(10) - 65))^9 - 46373592974030119798255326246458659/579803012795 756639164153441339582270341120*(sqrt(34) + 2*sqrt(10) + sqrt(2) + sqrt(25* sqrt(10) - 65))^8 + 29889765665562355905164717800682989/120792294332449299 82586530027907963965440*(sqrt(34) + 2*sqrt(10) + sqrt(2) + sqrt(25*sqrt(10 ) - 65))^7 - 25871159491632638141490098189566285219/5798030127957566391641 53441339582270341120*(sqrt(34) + 2*sqrt(10) + sqrt(2) + sqrt(25*sqrt(10) - 65))^6 + 6544178176466447186063829124503215171/24158458866489859965173...
Timed out. \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx=\int \frac {x+2}{\sqrt {-2\,x^2+3\,x+1}\,\left (-3\,x^2+4\,x+2\right )} \,d x \]