Integrand size = 27, antiderivative size = 176 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {13-6 x}{253 \sqrt {3-x+2 x^2}}+\frac {1}{22} \sqrt {\frac {1}{682} \left (247+500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (247+500 \sqrt {2}\right )}} \left (61+4 \sqrt {2}+\left (69+65 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\frac {1}{22} \sqrt {\frac {1}{682} \left (-247+500 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-247+500 \sqrt {2}\right )}} \left (61-4 \sqrt {2}+\left (69-65 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right ) \]
1/253*(13-6*x)/(2*x^2-x+3)^(1/2)-1/15004*arctanh(1/31*(61+x*(69-65*2^(1/2) )-4*2^(1/2))*341^(1/2)/(-247+500*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(-16845 4+341000*2^(1/2))^(1/2)+1/15004*arctan(1/31*(61+4*2^(1/2)+x*(69+65*2^(1/2) ))*341^(1/2)/(247+500*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(168454+341000*2^( 1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.34 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {13-6 x}{253 \sqrt {3-x+2 x^2}}+\frac {1}{22} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {23 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+16 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-5 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ] \]
(13 - 6*x)/(253*Sqrt[3 - x + 2*x^2]) + RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1 ^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (23*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2 ] - #1] + 16*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 5*L og[-(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) & ]/22
Time = 0.50 (sec) , antiderivative size = 182, 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 = {1305, 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.
| \(\displaystyle \int \frac {1}{\left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle \frac {13-6 x}{253 \sqrt {2 x^2-x+3}}-\frac {\int -\frac {253 (5 x+8)}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2783}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \int \frac {5 x+8}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {13-6 x}{253 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {1}{22} \left (\frac {\int -\frac {11 \left (-\left (\left (13+5 \sqrt {2}\right ) x\right )-8 \sqrt {2}+3\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (13-5 \sqrt {2}\right ) x\right )+8 \sqrt {2}+3\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )+\frac {13-6 x}{253 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {\int \frac {-\left (\left (13-5 \sqrt {2}\right ) x\right )+8 \sqrt {2}+3}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (13+5 \sqrt {2}\right ) x\right )-8 \sqrt {2}+3}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )+\frac {13-6 x}{253 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {1}{22} \left (\frac {\left (247-500 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (69-65 \sqrt {2}\right ) x-4 \sqrt {2}+61\right )^2}{2 x^2-x+3}-31 \left (247-500 \sqrt {2}\right )}d\frac {\left (69-65 \sqrt {2}\right ) x-4 \sqrt {2}+61}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (247+500 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (69+65 \sqrt {2}\right ) x+4 \sqrt {2}+61\right )^2}{2 x^2-x+3}-31 \left (247+500 \sqrt {2}\right )}d\frac {\left (69+65 \sqrt {2}\right ) x+4 \sqrt {2}+61}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}\right )+\frac {13-6 x}{253 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{22} \left (\frac {\left (247-500 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (69-65 \sqrt {2}\right ) x-4 \sqrt {2}+61\right )^2}{2 x^2-x+3}-31 \left (247-500 \sqrt {2}\right )}d\frac {\left (69-65 \sqrt {2}\right ) x-4 \sqrt {2}+61}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (247+500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (247+500 \sqrt {2}\right )}} \left (\left (69+65 \sqrt {2}\right ) x+4 \sqrt {2}+61\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {13-6 x}{253 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{22} \left (\sqrt {\frac {1}{682} \left (247+500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (247+500 \sqrt {2}\right )}} \left (\left (69+65 \sqrt {2}\right ) x+4 \sqrt {2}+61\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (247-500 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (500 \sqrt {2}-247\right )}} \left (\left (69-65 \sqrt {2}\right ) x-4 \sqrt {2}+61\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (500 \sqrt {2}-247\right )}}\right )+\frac {13-6 x}{253 \sqrt {2 x^2-x+3}}\) |
(13 - 6*x)/(253*Sqrt[3 - x + 2*x^2]) + (Sqrt[(247 + 500*Sqrt[2])/682]*ArcT an[(Sqrt[11/(31*(247 + 500*Sqrt[2]))]*(61 + 4*Sqrt[2] + (69 + 65*Sqrt[2])* x))/Sqrt[3 - x + 2*x^2]] + ((247 - 500*Sqrt[2])*ArcTanh[(Sqrt[11/(31*(-247 + 500*Sqrt[2]))]*(61 - 4*Sqrt[2] + (69 - 65*Sqrt[2])*x))/Sqrt[3 - x + 2*x ^2]])/Sqrt[682*(-247 + 500*Sqrt[2])])/22
3.1.90.3.1 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[(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a *f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(( d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*( c*e - b*f))*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si mp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f *(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))* (2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b ^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !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 = 2.71 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.61
| method | result | size |
| trager | \(-\frac {-13+6 x}{253 \sqrt {2 x^{2}-x +3}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+150700176 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2}+168454\right ) \ln \left (\frac {649244614491 \operatorname {RootOf}\left (\textit {\_Z}^{2}+150700176 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2}+168454\right ) \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{4} x -2033209431 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+150700176 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2}+168454\right ) x +1608394722165 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2} \sqrt {2 x^{2}-x +3}+3040381575 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+150700176 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2}+168454\right )+1509120 \operatorname {RootOf}\left (\textit {\_Z}^{2}+150700176 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2}+168454\right ) x +1021170535 \sqrt {2 x^{2}-x +3}-5845875 \operatorname {RootOf}\left (\textit {\_Z}^{2}+150700176 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2}+168454\right )}{110484 x \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2}+151 x +119}\right )}{15004}-\frac {9 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right ) \ln \left (\frac {373964897946816 x \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{5}+2007171224784 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{3} x -1751259787200 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{3}-75467201040 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2} \sqrt {2 x^{2}-x +3}+2645619075 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right ) x -5324797800 \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )-36443750 \sqrt {2 x^{2}-x +3}}{220968 x \operatorname {RootOf}\left (6103357128 \textit {\_Z}^{4}+6822387 \textit {\_Z}^{2}+15625\right )^{2}-55 x -238}\right )}{11}\) | \(460\) |
| risch | \(-\frac {-13+6 x}{253 \sqrt {2 x^{2}-x +3}}+\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 (2197 \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 )+3070 \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 )+1712502 \,\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}-6617446 \,\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 )}{465124 \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}}}\) | \(704\) |
| default | \(\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 (2197 \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 )+3070 \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 )+1712502 \,\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}-6617446 \,\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 )}{465124 \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}}}-\frac {3 \left (-1+4 x \right )}{506 \sqrt {2 x^{2}-x +3}}+\frac {1}{22 \sqrt {2 x^{2}-x +3}}\) | \(718\) |
-1/253*(-13+6*x)/(2*x^2-x+3)^(1/2)-1/15004*RootOf(_Z^2+150700176*RootOf(61 03357128*_Z^4+6822387*_Z^2+15625)^2+168454)*ln((649244614491*RootOf(_Z^2+1 50700176*RootOf(6103357128*_Z^4+6822387*_Z^2+15625)^2+168454)*RootOf(61033 57128*_Z^4+6822387*_Z^2+15625)^4*x-2033209431*RootOf(6103357128*_Z^4+68223 87*_Z^2+15625)^2*RootOf(_Z^2+150700176*RootOf(6103357128*_Z^4+6822387*_Z^2 +15625)^2+168454)*x+1608394722165*RootOf(6103357128*_Z^4+6822387*_Z^2+1562 5)^2*(2*x^2-x+3)^(1/2)+3040381575*RootOf(6103357128*_Z^4+6822387*_Z^2+1562 5)^2*RootOf(_Z^2+150700176*RootOf(6103357128*_Z^4+6822387*_Z^2+15625)^2+16 8454)+1509120*RootOf(_Z^2+150700176*RootOf(6103357128*_Z^4+6822387*_Z^2+15 625)^2+168454)*x+1021170535*(2*x^2-x+3)^(1/2)-5845875*RootOf(_Z^2+15070017 6*RootOf(6103357128*_Z^4+6822387*_Z^2+15625)^2+168454))/(110484*x*RootOf(6 103357128*_Z^4+6822387*_Z^2+15625)^2+151*x+119))-9/11*RootOf(6103357128*_Z ^4+6822387*_Z^2+15625)*ln((373964897946816*x*RootOf(6103357128*_Z^4+682238 7*_Z^2+15625)^5+2007171224784*RootOf(6103357128*_Z^4+6822387*_Z^2+15625)^3 *x-1751259787200*RootOf(6103357128*_Z^4+6822387*_Z^2+15625)^3-75467201040* RootOf(6103357128*_Z^4+6822387*_Z^2+15625)^2*(2*x^2-x+3)^(1/2)+2645619075* RootOf(6103357128*_Z^4+6822387*_Z^2+15625)*x-5324797800*RootOf(6103357128* _Z^4+6822387*_Z^2+15625)-36443750*(2*x^2-x+3)^(1/2))/(220968*x*RootOf(6103 357128*_Z^4+6822387*_Z^2+15625)^2-55*x-238))
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.92 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx=-\frac {23 \, \sqrt {341} {\left (2 \, x^{2} - x + 3\right )} \sqrt {119 i \, \sqrt {31} - 247} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {119 i \, \sqrt {31} - 247} {\left (61 i \, \sqrt {31} + 93\right )} - 7750 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 147250 \, x - 170500}{x}\right ) - 23 \, \sqrt {341} {\left (2 \, x^{2} - x + 3\right )} \sqrt {119 i \, \sqrt {31} - 247} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {119 i \, \sqrt {31} - 247} {\left (-61 i \, \sqrt {31} - 93\right )} - 7750 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 147250 \, x - 170500}{x}\right ) - 23 \, \sqrt {341} {\left (2 \, x^{2} - x + 3\right )} \sqrt {-119 i \, \sqrt {31} - 247} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (61 i \, \sqrt {31} - 93\right )} \sqrt {-119 i \, \sqrt {31} - 247} - 7750 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 147250 \, x - 170500}{x}\right ) + 23 \, \sqrt {341} {\left (2 \, x^{2} - x + 3\right )} \sqrt {-119 i \, \sqrt {31} - 247} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (-61 i \, \sqrt {31} + 93\right )} \sqrt {-119 i \, \sqrt {31} - 247} - 7750 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 147250 \, x - 170500}{x}\right ) + 2728 \, \sqrt {2 \, x^{2} - x + 3} {\left (6 \, x - 13\right )}}{690184 \, {\left (2 \, x^{2} - x + 3\right )}} \]
-1/690184*(23*sqrt(341)*(2*x^2 - x + 3)*sqrt(119*I*sqrt(31) - 247)*log((sq rt(341)*sqrt(2*x^2 - x + 3)*sqrt(119*I*sqrt(31) - 247)*(61*I*sqrt(31) + 93 ) - 7750*sqrt(31)*(I*x - 6*I) + 147250*x - 170500)/x) - 23*sqrt(341)*(2*x^ 2 - x + 3)*sqrt(119*I*sqrt(31) - 247)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*s qrt(119*I*sqrt(31) - 247)*(-61*I*sqrt(31) - 93) - 7750*sqrt(31)*(I*x - 6*I ) + 147250*x - 170500)/x) - 23*sqrt(341)*(2*x^2 - x + 3)*sqrt(-119*I*sqrt( 31) - 247)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*(61*I*sqrt(31) - 93)*sqrt(-1 19*I*sqrt(31) - 247) - 7750*sqrt(31)*(-I*x + 6*I) + 147250*x - 170500)/x) + 23*sqrt(341)*(2*x^2 - x + 3)*sqrt(-119*I*sqrt(31) - 247)*log((sqrt(341)* sqrt(2*x^2 - x + 3)*(-61*I*sqrt(31) + 93)*sqrt(-119*I*sqrt(31) - 247) - 77 50*sqrt(31)*(-I*x + 6*I) + 147250*x - 170500)/x) + 2728*sqrt(2*x^2 - x + 3 )*(6*x - 13))/(2*x^2 - x + 3)
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx=\int \frac {1}{\left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \cdot \left (5 x^{2} + 3 x + 2\right )}\, dx \]
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )} {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx=\int \frac {1}{{\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^2+3\,x+2\right )} \,d x \]