Integrand size = 27, antiderivative size = 188 \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {(3+10 x) \sqrt {3-x+2 x^2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{62} \sqrt {\frac {1}{682} \left (70517+49942 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (70517+49942 \sqrt {2}\right )}} \left (419+277 \sqrt {2}+\left (973+696 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\frac {1}{62} \sqrt {\frac {1}{682} \left (-70517+49942 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-70517+49942 \sqrt {2}\right )}} \left (419-277 \sqrt {2}+\left (973-696 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right ) \]
1/31*(3+10*x)*(2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)-1/42284*arctanh(1/31*(419+x* (973-696*2^(1/2))-277*2^(1/2))*341^(1/2)/(-70517+49942*2^(1/2))^(1/2)/(2*x ^2-x+3)^(1/2))*(-48092594+34060444*2^(1/2))^(1/2)+1/42284*arctan(1/31*(419 +277*2^(1/2)+x*(973+696*2^(1/2)))*341^(1/2)/(70517+49942*2^(1/2))^(1/2)/(2 *x^2-x+3)^(1/2))*(48092594+34060444*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.49 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {\frac {50 (3+10 x) \sqrt {3-x+2 x^2}}{2+3 x+5 x^2}-6151 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+124 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {49 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+10 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-10 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {191 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+55 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{13 \sqrt {2}-17 \text {$\#$1}-9 \sqrt {2} \text {$\#$1}^2+10 \text {$\#$1}^3}\&\right ]}{1550} \]
((50*(3 + 10*x)*Sqrt[3 - x + 2*x^2])/(2 + 3*x + 5*x^2) - 6151*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3 ) & ] + 124*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^ 4 & , (49*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 10*Sqrt[2]*Log[-( Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2 ]*#1^2 - 10*#1^3) & ] - 10*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[ 2]*#1^3 - 5*#1^4 & , (191*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 55*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) & ])/1550
Time = 0.50 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1302, 27, 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.
| \(\displaystyle \int \frac {\sqrt {2 x^2-x+3}}{\left (5 x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1302 |
| \(\displaystyle \frac {(10 x+3) \sqrt {2 x^2-x+3}}{31 \left (5 x^2+3 x+2\right )}-\frac {1}{31} \int -\frac {63-22 x}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{62} \int \frac {63-22 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {1}{62} \left (\frac {\int -\frac {11 \left (-\left (\left (41-22 \sqrt {2}\right ) x\right )-63 \sqrt {2}+85\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (41+22 \sqrt {2}\right ) x\right )+63 \sqrt {2}+85\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{62} \left (\frac {\int \frac {-\left (\left (41+22 \sqrt {2}\right ) x\right )+63 \sqrt {2}+85}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (41-22 \sqrt {2}\right ) x\right )-63 \sqrt {2}+85}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {1}{62} \left (\frac {\left (70517-49942 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (973-696 \sqrt {2}\right ) x-277 \sqrt {2}+419\right )^2}{2 x^2-x+3}-31 \left (70517-49942 \sqrt {2}\right )}d\frac {\left (973-696 \sqrt {2}\right ) x-277 \sqrt {2}+419}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (70517+49942 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (973+696 \sqrt {2}\right ) x+277 \sqrt {2}+419\right )^2}{2 x^2-x+3}-31 \left (70517+49942 \sqrt {2}\right )}d\frac {\left (973+696 \sqrt {2}\right ) x+277 \sqrt {2}+419}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{62} \left (\frac {\left (70517-49942 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (973-696 \sqrt {2}\right ) x-277 \sqrt {2}+419\right )^2}{2 x^2-x+3}-31 \left (70517-49942 \sqrt {2}\right )}d\frac {\left (973-696 \sqrt {2}\right ) x-277 \sqrt {2}+419}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (70517+49942 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (70517+49942 \sqrt {2}\right )}} \left (\left (973+696 \sqrt {2}\right ) x+277 \sqrt {2}+419\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{62} \left (\sqrt {\frac {1}{682} \left (70517+49942 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (70517+49942 \sqrt {2}\right )}} \left (\left (973+696 \sqrt {2}\right ) x+277 \sqrt {2}+419\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (70517-49942 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (49942 \sqrt {2}-70517\right )}} \left (\left (973-696 \sqrt {2}\right ) x-277 \sqrt {2}+419\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (49942 \sqrt {2}-70517\right )}}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
((3 + 10*x)*Sqrt[3 - x + 2*x^2])/(31*(2 + 3*x + 5*x^2)) + (Sqrt[(70517 + 4 9942*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(70517 + 49942*Sqrt[2]))]*(419 + 27 7*Sqrt[2] + (973 + 696*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + ((70517 - 49942 *Sqrt[2])*ArcTanh[(Sqrt[11/(31*(-70517 + 49942*Sqrt[2]))]*(419 - 277*Sqrt[ 2] + (973 - 696*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/Sqrt[682*(-70517 + 4994 2*Sqrt[2])])/62
3.1.63.3.1 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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_), x_Symbol] :> Simp[(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*((d + e *x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 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 = 1.92 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.52
| method | result | size |
| trager | \(\frac {\left (10 x +3\right ) \sqrt {2 x^{2}-x +3}}{155 x^{2}+93 x +62}-\frac {2 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right ) \ln \left (-\frac {150744827904 x \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{5}+232524550016 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{3} x +2424286162144 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2} \sqrt {2 x^{2}-x +3}-85650470400 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{3}-988525310334 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right ) x +7810921383613 \sqrt {2 x^{2}-x +3}+163005849200 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )}{10912 x \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2}+34189 x -1426}\right )}{31}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+7441984 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2}+48092594\right ) \ln \left (\frac {-18843103488 \operatorname {RootOf}\left (\textit {\_Z}^{2}+7441984 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2}+48092594\right ) \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{4} x -214475327264 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+7441984 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2}+48092594\right ) x +826681581291104 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2} \sqrt {2 x^{2}-x +3}-10706308800 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+7441984 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2}+48092594\right )-475524326173 \operatorname {RootOf}\left (\textit {\_Z}^{2}+7441984 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2}+48092594\right ) x +2678769161185106 \sqrt {2 x^{2}-x +3}-89563485700 \operatorname {RootOf}\left (\textit {\_Z}^{2}+7441984 \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2}+48092594\right )}{5456 x \operatorname {RootOf}\left (59535872 \textit {\_Z}^{4}+384740752 \textit {\_Z}^{2}+623550841\right )^{2}+18164 x +713}\right )}{42284}\) | \(473\) |
| risch | \(\frac {\left (10 x +3\right ) \sqrt {2 x^{2}-x +3}}{155 x^{2}+93 x +62}+\frac {\sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (26569 \sqrt {-775687+549362 \sqrt {2}}\, \sqrt {2}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right )+37556 \sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right )+38140168 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right ) \sqrt {2}-56005158 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )\right )}{1310804 \sqrt {\frac {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\) | \(716\) |
| default | \(\text {Expression too large to display}\) | \(16357\) |
1/31*(10*x+3)*(2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)-2/31*RootOf(59535872*_Z^4+38 4740752*_Z^2+623550841)*ln(-(150744827904*x*RootOf(59535872*_Z^4+384740752 *_Z^2+623550841)^5+232524550016*RootOf(59535872*_Z^4+384740752*_Z^2+623550 841)^3*x+2424286162144*RootOf(59535872*_Z^4+384740752*_Z^2+623550841)^2*(2 *x^2-x+3)^(1/2)-85650470400*RootOf(59535872*_Z^4+384740752*_Z^2+623550841) ^3-988525310334*RootOf(59535872*_Z^4+384740752*_Z^2+623550841)*x+781092138 3613*(2*x^2-x+3)^(1/2)+163005849200*RootOf(59535872*_Z^4+384740752*_Z^2+62 3550841))/(10912*x*RootOf(59535872*_Z^4+384740752*_Z^2+623550841)^2+34189* x-1426))-1/42284*RootOf(_Z^2+7441984*RootOf(59535872*_Z^4+384740752*_Z^2+6 23550841)^2+48092594)*ln((-18843103488*RootOf(_Z^2+7441984*RootOf(59535872 *_Z^4+384740752*_Z^2+623550841)^2+48092594)*RootOf(59535872*_Z^4+384740752 *_Z^2+623550841)^4*x-214475327264*RootOf(59535872*_Z^4+384740752*_Z^2+6235 50841)^2*RootOf(_Z^2+7441984*RootOf(59535872*_Z^4+384740752*_Z^2+623550841 )^2+48092594)*x+826681581291104*RootOf(59535872*_Z^4+384740752*_Z^2+623550 841)^2*(2*x^2-x+3)^(1/2)-10706308800*RootOf(59535872*_Z^4+384740752*_Z^2+6 23550841)^2*RootOf(_Z^2+7441984*RootOf(59535872*_Z^4+384740752*_Z^2+623550 841)^2+48092594)-475524326173*RootOf(_Z^2+7441984*RootOf(59535872*_Z^4+384 740752*_Z^2+623550841)^2+48092594)*x+2678769161185106*(2*x^2-x+3)^(1/2)-89 563485700*RootOf(_Z^2+7441984*RootOf(59535872*_Z^4+384740752*_Z^2+62355084 1)^2+48092594))/(5456*x*RootOf(59535872*_Z^4+384740752*_Z^2+623550841)^...
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.79 \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\sqrt {341} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {713 i \, \sqrt {31} - 70517} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {713 i \, \sqrt {31} - 70517} {\left (419 i \, \sqrt {31} + 2635\right )} - 774101 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 14707919 \, x - 17030222}{x}\right ) - \sqrt {341} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {713 i \, \sqrt {31} - 70517} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {713 i \, \sqrt {31} - 70517} {\left (-419 i \, \sqrt {31} - 2635\right )} - 774101 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 14707919 \, x - 17030222}{x}\right ) - \sqrt {341} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-713 i \, \sqrt {31} - 70517} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (419 i \, \sqrt {31} - 2635\right )} \sqrt {-713 i \, \sqrt {31} - 70517} - 774101 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 14707919 \, x - 17030222}{x}\right ) + \sqrt {341} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-713 i \, \sqrt {31} - 70517} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (-419 i \, \sqrt {31} + 2635\right )} \sqrt {-713 i \, \sqrt {31} - 70517} - 774101 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 14707919 \, x - 17030222}{x}\right ) - 2728 \, \sqrt {2 \, x^{2} - x + 3} {\left (10 \, x + 3\right )}}{84568 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]
-1/84568*(sqrt(341)*(5*x^2 + 3*x + 2)*sqrt(713*I*sqrt(31) - 70517)*log((sq rt(341)*sqrt(2*x^2 - x + 3)*sqrt(713*I*sqrt(31) - 70517)*(419*I*sqrt(31) + 2635) - 774101*sqrt(31)*(I*x - 6*I) + 14707919*x - 17030222)/x) - sqrt(34 1)*(5*x^2 + 3*x + 2)*sqrt(713*I*sqrt(31) - 70517)*log((sqrt(341)*sqrt(2*x^ 2 - x + 3)*sqrt(713*I*sqrt(31) - 70517)*(-419*I*sqrt(31) - 2635) - 774101* sqrt(31)*(I*x - 6*I) + 14707919*x - 17030222)/x) - sqrt(341)*(5*x^2 + 3*x + 2)*sqrt(-713*I*sqrt(31) - 70517)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*(419 *I*sqrt(31) - 2635)*sqrt(-713*I*sqrt(31) - 70517) - 774101*sqrt(31)*(-I*x + 6*I) + 14707919*x - 17030222)/x) + sqrt(341)*(5*x^2 + 3*x + 2)*sqrt(-713 *I*sqrt(31) - 70517)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*(-419*I*sqrt(31) + 2635)*sqrt(-713*I*sqrt(31) - 70517) - 774101*sqrt(31)*(-I*x + 6*I) + 1470 7919*x - 17030222)/x) - 2728*sqrt(2*x^2 - x + 3)*(10*x + 3))/(5*x^2 + 3*x + 2)
\[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {\sqrt {2 x^{2} - x + 3}}{\left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \]
\[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {\sqrt {2 \, x^{2} - x + 3}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {\sqrt {2\,x^2-x+3}}{{\left (5\,x^2+3\,x+2\right )}^2} \,d x \]