Integrand size = 27, antiderivative size = 188 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{682} \left (2343727+1678700 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (2343727+1678700 \sqrt {2}\right )}} \left (2119+1816 \sqrt {2}+\left (5751+3935 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1364}-\frac {\sqrt {\frac {1}{682} \left (-2343727+1678700 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-2343727+1678700 \sqrt {2}\right )}} \left (2119-1816 \sqrt {2}+\left (5751-3935 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1364} \]
1/682*(4+65*x)*(2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)-1/930248*arctanh(1/31*(2119 +x*(5751-3935*2^(1/2))-1816*2^(1/2))*341^(1/2)/(-2343727+1678700*2^(1/2))^ (1/2)/(2*x^2-x+3)^(1/2))*(-1598421814+1144873400*2^(1/2))^(1/2)+1/930248*a rctan(1/31*(2119+1816*2^(1/2)+x*(5751+3935*2^(1/2)))*341^(1/2)/(2343727+16 78700*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(1598421814+1144873400*2^(1/2))^(1 /2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.36 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}+\frac {\text {RootSum}\left [-10580-2024 \sqrt {2} \text {$\#$1}+68 \text {$\#$1}^2+44 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-9430 \sqrt {2} \log \left (\sqrt {2} (-1+4 x)-4 \sqrt {3-x+2 x^2}+\text {$\#$1}\right )+4492 \log \left (\sqrt {2} (-1+4 x)-4 \sqrt {3-x+2 x^2}+\text {$\#$1}\right ) \text {$\#$1}+205 \sqrt {2} \log \left (\sqrt {2} (-1+4 x)-4 \sqrt {3-x+2 x^2}+\text {$\#$1}\right ) \text {$\#$1}^2}{-506 \sqrt {2}+34 \text {$\#$1}+33 \sqrt {2} \text {$\#$1}^2-5 \text {$\#$1}^3}\&\right ]}{682 \sqrt {2}} \]
((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(682*(2 + 3*x + 5*x^2)) + RootSum[-10580 - 2024*Sqrt[2]*#1 + 68*#1^2 + 44*Sqrt[2]*#1^3 - 5*#1^4 & , (-9430*Sqrt[2]* Log[Sqrt[2]*(-1 + 4*x) - 4*Sqrt[3 - x + 2*x^2] + #1] + 4492*Log[Sqrt[2]*(- 1 + 4*x) - 4*Sqrt[3 - x + 2*x^2] + #1]*#1 + 205*Sqrt[2]*Log[Sqrt[2]*(-1 + 4*x) - 4*Sqrt[3 - x + 2*x^2] + #1]*#1^2)/(-506*Sqrt[2] + 34*#1 + 33*Sqrt[2 ]*#1^2 - 5*#1^3) & ]/(682*Sqrt[2])
Time = 0.52 (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 = {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.
| \(\displaystyle \int \frac {1}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle \frac {(65 x+4) \sqrt {2 x^2-x+3}}{682 \left (5 x^2+3 x+2\right )}-\frac {\int -\frac {11 (332-205 x)}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{7502}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {332-205 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {\frac {\int -\frac {11 \left (-\left (\left (127-205 \sqrt {2}\right ) x\right )-332 \sqrt {2}+537\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (127+205 \sqrt {2}\right ) x\right )+332 \sqrt {2}+537\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-\left (\left (127+205 \sqrt {2}\right ) x\right )+332 \sqrt {2}+537}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (127-205 \sqrt {2}\right ) x\right )-332 \sqrt {2}+537}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {\frac {\left (2343727-1678700 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (5751-3935 \sqrt {2}\right ) x-1816 \sqrt {2}+2119\right )^2}{2 x^2-x+3}-31 \left (2343727-1678700 \sqrt {2}\right )}d\frac {\left (5751-3935 \sqrt {2}\right ) x-1816 \sqrt {2}+2119}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (2343727+1678700 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (5751+3935 \sqrt {2}\right ) x+1816 \sqrt {2}+2119\right )^2}{2 x^2-x+3}-31 \left (2343727+1678700 \sqrt {2}\right )}d\frac {\left (5751+3935 \sqrt {2}\right ) x+1816 \sqrt {2}+2119}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\left (2343727-1678700 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (5751-3935 \sqrt {2}\right ) x-1816 \sqrt {2}+2119\right )^2}{2 x^2-x+3}-31 \left (2343727-1678700 \sqrt {2}\right )}d\frac {\left (5751-3935 \sqrt {2}\right ) x-1816 \sqrt {2}+2119}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (2343727+1678700 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (2343727+1678700 \sqrt {2}\right )}} \left (\left (5751+3935 \sqrt {2}\right ) x+1816 \sqrt {2}+2119\right )}{\sqrt {2 x^2-x+3}}\right )}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {\frac {1}{682} \left (2343727+1678700 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (2343727+1678700 \sqrt {2}\right )}} \left (\left (5751+3935 \sqrt {2}\right ) x+1816 \sqrt {2}+2119\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (2343727-1678700 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (1678700 \sqrt {2}-2343727\right )}} \left (\left (5751-3935 \sqrt {2}\right ) x-1816 \sqrt {2}+2119\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (1678700 \sqrt {2}-2343727\right )}}}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\) |
((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(682*(2 + 3*x + 5*x^2)) + (Sqrt[(2343727 + 1678700*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(2343727 + 1678700*Sqrt[2]))]* (2119 + 1816*Sqrt[2] + (5751 + 3935*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + (( 2343727 - 1678700*Sqrt[2])*ArcTanh[(Sqrt[11/(31*(-2343727 + 1678700*Sqrt[2 ]))]*(2119 - 1816*Sqrt[2] + (5751 - 3935*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] )/Sqrt[682*(-2343727 + 1678700*Sqrt[2])])/1364
3.1.84.3.1 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 = 2.82 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.52
| method | result | size |
| trager | \(\frac {\left (4+65 x \right ) \sqrt {2 x^{2}-x +3}}{3410 x^{2}+2046 x +1364}-\frac {3 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right ) \ln \left (\frac {-394636742489088 x \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{5}-13779789867949248 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{3} x +32137043846114592 \sqrt {2 x^{2}-x +3}\, \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2}-2373323185094400 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{3}-96300551793960525 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right ) x +386610578208666875 \sqrt {2 x^{2}-x +3}-59939356016807400 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )}{98208 x \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2}+1273745 x +135842}\right )}{341}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+66977856 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2}+1598421814\right ) \ln \left (-\frac {-2055399700464 x \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+66977856 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2}+1598421814\right )-26334199149588 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+66977856 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2}+1598421814\right ) x +1369841493940634484 \sqrt {2 x^{2}-x +3}\, \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2}+12361058255700 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+66977856 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2}+1598421814\right )+40592603336832 \operatorname {RootOf}\left (\textit {\_Z}^{2}+66977856 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2}+1598421814\right ) x +16211894834030320271 \sqrt {2 x^{2}-x +3}-17188361642025 \operatorname {RootOf}\left (\textit {\_Z}^{2}+66977856 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2}+1598421814\right )}{49104 x \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}+28771592652 \textit {\_Z}^{2}+176127105625\right )^{2}+534991 x -67921}\right )}{930248}\) | \(473\) |
| risch | \(\frac {\left (4+65 x \right ) \sqrt {2 x^{2}-x +3}}{3410 x^{2}+2046 x +1364}+\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 (153463 \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 )+217330 \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 )+236769258 \,\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}-300787234 \,\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 )}{28837688 \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 | \(\text {Expression too large to display}\) | \(5225\) |
1/682*(4+65*x)*(2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)-3/341*RootOf(1205601408*_Z^ 4+28771592652*_Z^2+176127105625)*ln((-394636742489088*x*RootOf(1205601408* _Z^4+28771592652*_Z^2+176127105625)^5-13779789867949248*RootOf(1205601408* _Z^4+28771592652*_Z^2+176127105625)^3*x+32137043846114592*(2*x^2-x+3)^(1/2 )*RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625)^2-2373323185094400 *RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625)^3-96300551793960525 *RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625)*x+38661057820866687 5*(2*x^2-x+3)^(1/2)-59939356016807400*RootOf(1205601408*_Z^4+28771592652*_ Z^2+176127105625))/(98208*x*RootOf(1205601408*_Z^4+28771592652*_Z^2+176127 105625)^2+1273745*x+135842))+1/930248*RootOf(_Z^2+66977856*RootOf(12056014 08*_Z^4+28771592652*_Z^2+176127105625)^2+1598421814)*ln(-(-2055399700464*x *RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625)^4*RootOf(_Z^2+66977 856*RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625)^2+1598421814)-26 334199149588*RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625)^2*RootO f(_Z^2+66977856*RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625)^2+15 98421814)*x+1369841493940634484*(2*x^2-x+3)^(1/2)*RootOf(1205601408*_Z^4+2 8771592652*_Z^2+176127105625)^2+12361058255700*RootOf(1205601408*_Z^4+2877 1592652*_Z^2+176127105625)^2*RootOf(_Z^2+66977856*RootOf(1205601408*_Z^4+2 8771592652*_Z^2+176127105625)^2+1598421814)+40592603336832*RootOf(_Z^2+669 77856*RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625)^2+159842181...
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {\sqrt {341} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {67921 i \, \sqrt {31} - 2343727} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {67921 i \, \sqrt {31} - 2343727} {\left (2119 i \, \sqrt {31} - 16647\right )} - 26019850 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 494377150 \, x - 572436700}{x}\right ) - \sqrt {341} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {67921 i \, \sqrt {31} - 2343727} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {67921 i \, \sqrt {31} - 2343727} {\left (-2119 i \, \sqrt {31} + 16647\right )} - 26019850 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 494377150 \, x - 572436700}{x}\right ) - \sqrt {341} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-67921 i \, \sqrt {31} - 2343727} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (2119 i \, \sqrt {31} + 16647\right )} \sqrt {-67921 i \, \sqrt {31} - 2343727} - 26019850 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 494377150 \, x - 572436700}{x}\right ) + \sqrt {341} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-67921 i \, \sqrt {31} - 2343727} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (-2119 i \, \sqrt {31} - 16647\right )} \sqrt {-67921 i \, \sqrt {31} - 2343727} - 26019850 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 494377150 \, x - 572436700}{x}\right ) + 2728 \, \sqrt {2 \, x^{2} - x + 3} {\left (65 \, x + 4\right )}}{1860496 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]
1/1860496*(sqrt(341)*(5*x^2 + 3*x + 2)*sqrt(67921*I*sqrt(31) - 2343727)*lo g((sqrt(341)*sqrt(2*x^2 - x + 3)*sqrt(67921*I*sqrt(31) - 2343727)*(2119*I* sqrt(31) - 16647) - 26019850*sqrt(31)*(-I*x + 6*I) + 494377150*x - 5724367 00)/x) - sqrt(341)*(5*x^2 + 3*x + 2)*sqrt(67921*I*sqrt(31) - 2343727)*log( (sqrt(341)*sqrt(2*x^2 - x + 3)*sqrt(67921*I*sqrt(31) - 2343727)*(-2119*I*s qrt(31) + 16647) - 26019850*sqrt(31)*(-I*x + 6*I) + 494377150*x - 57243670 0)/x) - sqrt(341)*(5*x^2 + 3*x + 2)*sqrt(-67921*I*sqrt(31) - 2343727)*log( (sqrt(341)*sqrt(2*x^2 - x + 3)*(2119*I*sqrt(31) + 16647)*sqrt(-67921*I*sqr t(31) - 2343727) - 26019850*sqrt(31)*(I*x - 6*I) + 494377150*x - 572436700 )/x) + sqrt(341)*(5*x^2 + 3*x + 2)*sqrt(-67921*I*sqrt(31) - 2343727)*log(( sqrt(341)*sqrt(2*x^2 - x + 3)*(-2119*I*sqrt(31) - 16647)*sqrt(-67921*I*sqr t(31) - 2343727) - 26019850*sqrt(31)*(I*x - 6*I) + 494377150*x - 572436700 )/x) + 2728*sqrt(2*x^2 - x + 3)*(65*x + 4))/(5*x^2 + 3*x + 2)
\[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {1}{\sqrt {2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \]
\[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} \sqrt {2 \, x^{2} - x + 3}} \,d x } \]
Exception generated. \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {1}{\sqrt {2\,x^2-x+3}\,{\left (5\,x^2+3\,x+2\right )}^2} \,d x \]