Integrand size = 27, antiderivative size = 174 \[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=-\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {1}{5} \sqrt {\frac {11}{31} \left (13+10 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (13+10 \sqrt {2}\right )}} \left (6+7 \sqrt {2}+\left (20+13 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\frac {1}{5} \sqrt {\frac {11}{31} \left (-13+10 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (-13+10 \sqrt {2}\right )}} \left (6-7 \sqrt {2}+\left (20-13 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right ) \]
-1/5*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-1/155*arctanh(1/62*(6+x*(20-13 *2^(1/2))-7*2^(1/2))*682^(1/2)/(-13+10*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*( -4433+3410*2^(1/2))^(1/2)+1/155*arctan(1/62*(6+7*2^(1/2)+x*(20+13*2^(1/2)) )*682^(1/2)/(13+10*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(4433+3410*2^(1/2))^( 1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\frac {1}{5} \left (-\sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )+11 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-2 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+2 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]\right ) \]
(-(Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]]) + 11*RootSum[-56 - 26*S qrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-2*Log[-(Sqrt[2]*x) + S qrt[3 - x + 2*x^2] - #1] + 2*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2 ] - #1]*#1 + Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1^2)/(-13*Sqrt[ 2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ])/5
Time = 0.50 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1320, 27, 1090, 222, 1368, 27, 1362, 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.
| \(\displaystyle \int \frac {\sqrt {2 x^2-x+3}}{5 x^2+3 x+2} \, dx\) |
| \(\Big \downarrow \) 1320 |
| \(\displaystyle \frac {2}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{5} \int -\frac {11 (1-x)}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {11}{5} \int \frac {1-x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {11}{5} \int \frac {1-x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {1}{5} \sqrt {\frac {2}{23}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {11}{5} \int \frac {1-x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {11}{5} \left (\frac {\int -\frac {11 \left (\sqrt {2} x-\sqrt {2}+2\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\sqrt {2} x+\sqrt {2}+2\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )+\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{5} \left (\frac {\int \frac {-\sqrt {2} x+\sqrt {2}+2}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} x-\sqrt {2}+2}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )+\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {11}{5} \left (\sqrt {2} \left (13-10 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (20-13 \sqrt {2}\right ) x-7 \sqrt {2}+6\right )^2}{2 x^2-x+3}-62 \left (13-10 \sqrt {2}\right )}d\frac {\left (20-13 \sqrt {2}\right ) x-7 \sqrt {2}+6}{\sqrt {2 x^2-x+3}}-\sqrt {2} \left (13+10 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (20+13 \sqrt {2}\right ) x+7 \sqrt {2}+6\right )^2}{2 x^2-x+3}-62 \left (13+10 \sqrt {2}\right )}d\frac {\left (20+13 \sqrt {2}\right ) x+7 \sqrt {2}+6}{\sqrt {2 x^2-x+3}}\right )+\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {11}{5} \left (\sqrt {2} \left (13-10 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (20-13 \sqrt {2}\right ) x-7 \sqrt {2}+6\right )^2}{2 x^2-x+3}-62 \left (13-10 \sqrt {2}\right )}d\frac {\left (20-13 \sqrt {2}\right ) x-7 \sqrt {2}+6}{\sqrt {2 x^2-x+3}}+\sqrt {\frac {1}{341} \left (13+10 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (13+10 \sqrt {2}\right )}} \left (\left (20+13 \sqrt {2}\right ) x+7 \sqrt {2}+6\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{5} \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )+\frac {11}{5} \left (\sqrt {\frac {1}{341} \left (13+10 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (13+10 \sqrt {2}\right )}} \left (\left (20+13 \sqrt {2}\right ) x+7 \sqrt {2}+6\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (13-10 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (10 \sqrt {2}-13\right )}} \left (\left (20-13 \sqrt {2}\right ) x-7 \sqrt {2}+6\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (10 \sqrt {2}-13\right )}}\right )\) |
(Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]])/5 + (11*(Sqrt[(13 + 10*Sqrt[2])/341 ]*ArcTan[(Sqrt[11/(62*(13 + 10*Sqrt[2]))]*(6 + 7*Sqrt[2] + (20 + 13*Sqrt[2 ])*x))/Sqrt[3 - x + 2*x^2]] + ((13 - 10*Sqrt[2])*ArcTanh[(Sqrt[11/(62*(-13 + 10*Sqrt[2]))]*(6 - 7*Sqrt[2] + (20 - 13*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2 ]])/Sqrt[341*(-13 + 10*Sqrt[2])]))/5
3.1.62.3.1 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/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]/((d_) + (e_.)*(x_) + (f_.)*(x_)^ 2), x_Symbol] :> Simp[c/f Int[1/Sqrt[a + b*x + c*x^2], x], x] - Simp[1/f Int[(c*d - a*f + (c*e - b*f)*x)/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)) , x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-2*g*(g*b - 2*a*h) Subst[I nt[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, Simp[ g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], 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] && Ne Q[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(c*e - b*f ), 0]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Simp[1/(2*q) Int[Simp[h*(b*d - a*e) - g*(c*d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqr t[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[Simp[h*(b*d - a*e) - g*(c* d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x + c*x^2) *Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.40 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.78
| method | result | size |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{5}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) \ln \left (-\frac {5194205 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{4} x +446710 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) x +66745294 \sqrt {2 x^{2}-x +3}\, \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}-641080 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right )-38115 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right ) x +6024106 \sqrt {2 x^{2}-x +3}+33880 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+4433\right )}{775 x \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+55 x -22}\right )}{155}-\operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right ) \ln \left (-\frac {649275625 x \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{5}+183764900 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{3} x +80135000 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{3}-53826850 \sqrt {2 x^{2}-x +3}\, \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+7037844 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right ) x +19021200 \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )-5073772 \sqrt {2 x^{2}-x +3}}{775 x \operatorname {RootOf}\left (24025 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+242\right )^{2}+88 x +22}\right )\) | \(484\) |
| default | \(\text {Expression too large to display}\) | \(2065\) |
1/5*RootOf(_Z^2-2)*ln(4*RootOf(_Z^2-2)*x+4*(2*x^2-x+3)^(1/2)-RootOf(_Z^2-2 ))-1/155*RootOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242)^2+4433)*ln(-(5 194205*RootOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242)^2+4433)*RootOf(2 4025*_Z^4+4433*_Z^2+242)^4*x+446710*RootOf(24025*_Z^4+4433*_Z^2+242)^2*Roo tOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242)^2+4433)*x+66745294*(2*x^2- x+3)^(1/2)*RootOf(24025*_Z^4+4433*_Z^2+242)^2-641080*RootOf(24025*_Z^4+443 3*_Z^2+242)^2*RootOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242)^2+4433)-3 8115*RootOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242)^2+4433)*x+6024106* (2*x^2-x+3)^(1/2)+33880*RootOf(_Z^2+24025*RootOf(24025*_Z^4+4433*_Z^2+242) ^2+4433))/(775*x*RootOf(24025*_Z^4+4433*_Z^2+242)^2+55*x-22))-RootOf(24025 *_Z^4+4433*_Z^2+242)*ln(-(649275625*x*RootOf(24025*_Z^4+4433*_Z^2+242)^5+1 83764900*RootOf(24025*_Z^4+4433*_Z^2+242)^3*x+80135000*RootOf(24025*_Z^4+4 433*_Z^2+242)^3-53826850*(2*x^2-x+3)^(1/2)*RootOf(24025*_Z^4+4433*_Z^2+242 )^2+7037844*RootOf(24025*_Z^4+4433*_Z^2+242)*x+19021200*RootOf(24025*_Z^4+ 4433*_Z^2+242)-5073772*(2*x^2-x+3)^(1/2))/(775*x*RootOf(24025*_Z^4+4433*_Z ^2+242)^2+88*x+22))
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\frac {1}{620} \, \sqrt {31} \sqrt {22 i \, \sqrt {31} - 286} \log \left (-\frac {\sqrt {2 \, x^{2} - x + 3} {\left (\sqrt {31} - 3 i\right )} \sqrt {22 i \, \sqrt {31} - 286} + 5 \, \sqrt {31} {\left (-i \, x + 6 i\right )} - 95 \, x + 110}{x}\right ) - \frac {1}{620} \, \sqrt {31} \sqrt {22 i \, \sqrt {31} - 286} \log \left (\frac {\sqrt {2 \, x^{2} - x + 3} {\left (\sqrt {31} - 3 i\right )} \sqrt {22 i \, \sqrt {31} - 286} - 5 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 95 \, x - 110}{x}\right ) + \frac {1}{620} \, \sqrt {31} \sqrt {-22 i \, \sqrt {31} - 286} \log \left (-\frac {\sqrt {2 \, x^{2} - x + 3} {\left (\sqrt {31} + 3 i\right )} \sqrt {-22 i \, \sqrt {31} - 286} + 5 \, \sqrt {31} {\left (i \, x - 6 i\right )} - 95 \, x + 110}{x}\right ) - \frac {1}{620} \, \sqrt {31} \sqrt {-22 i \, \sqrt {31} - 286} \log \left (\frac {\sqrt {2 \, x^{2} - x + 3} {\left (\sqrt {31} + 3 i\right )} \sqrt {-22 i \, \sqrt {31} - 286} - 5 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 95 \, x - 110}{x}\right ) + \frac {1}{10} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/620*sqrt(31)*sqrt(22*I*sqrt(31) - 286)*log(-(sqrt(2*x^2 - x + 3)*(sqrt(3 1) - 3*I)*sqrt(22*I*sqrt(31) - 286) + 5*sqrt(31)*(-I*x + 6*I) - 95*x + 110 )/x) - 1/620*sqrt(31)*sqrt(22*I*sqrt(31) - 286)*log((sqrt(2*x^2 - x + 3)*( sqrt(31) - 3*I)*sqrt(22*I*sqrt(31) - 286) - 5*sqrt(31)*(-I*x + 6*I) + 95*x - 110)/x) + 1/620*sqrt(31)*sqrt(-22*I*sqrt(31) - 286)*log(-(sqrt(2*x^2 - x + 3)*(sqrt(31) + 3*I)*sqrt(-22*I*sqrt(31) - 286) + 5*sqrt(31)*(I*x - 6*I ) - 95*x + 110)/x) - 1/620*sqrt(31)*sqrt(-22*I*sqrt(31) - 286)*log((sqrt(2 *x^2 - x + 3)*(sqrt(31) + 3*I)*sqrt(-22*I*sqrt(31) - 286) - 5*sqrt(31)*(I* x - 6*I) + 95*x - 110)/x) + 1/10*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3 )*(4*x - 1) - 32*x^2 + 16*x - 25)
\[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\int \frac {\sqrt {2 x^{2} - x + 3}}{5 x^{2} + 3 x + 2}\, dx \]
\[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\int { \frac {\sqrt {2 \, x^{2} - x + 3}}{5 \, x^{2} + 3 \, x + 2} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Francis algorithm failure for[-1.0, infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,inf inity,inf
Timed out. \[ \int \frac {\sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx=\int \frac {\sqrt {2\,x^2-x+3}}{5\,x^2+3\,x+2} \,d x \]