Integrand size = 27, antiderivative size = 223 \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {(3+10 x) \sqrt {3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(3464+13665 x) \sqrt {3-x+2 x^2}}{84568 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{682} \left (112285869463+79399380740 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (112285869463+79399380740 \sqrt {2}\right )}} \left (509587+362788 \sqrt {2}+\left (1235163+872375 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{169136}-\frac {\sqrt {\frac {1}{682} \left (-112285869463+79399380740 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-112285869463+79399380740 \sqrt {2}\right )}} \left (509587-362788 \sqrt {2}+\left (1235163-872375 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{169136} \]
1/62*(3+10*x)*(2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)^2+1/84568*(3464+13665*x)*(2* x^2-x+3)^(1/2)/(5*x^2+3*x+2)-1/115350752*arctanh(1/31*(509587+x*(1235163-8 72375*2^(1/2))-362788*2^(1/2))*341^(1/2)/(-112285869463+79399380740*2^(1/2 ))^(1/2)/(2*x^2-x+3)^(1/2))*(-76578962973766+54150377664680*2^(1/2))^(1/2) +1/115350752*arctan(1/31*(509587+362788*2^(1/2)+x*(1235163+872375*2^(1/2)) )*341^(1/2)/(112285869463+79399380740*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(7 6578962973766+54150377664680*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.65 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.76 \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {661250 \sqrt {3-x+2 x^2} \left (11020+51362 x+58315 x^2+68325 x^3\right )}{\left (2+3 x+5 x^2\right )^2}+\text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-537295920831 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+120146195680 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-45923442075 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-248 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-2139373897 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+277937160 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-228643025 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{55920590000} \]
((661250*Sqrt[3 - x + 2*x^2]*(11020 + 51362*x + 58315*x^2 + 68325*x^3))/(2 + 3*x + 5*x^2)^2 + RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-537295920831*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 120146195680*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 4 5923442075*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) & ] - 248*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-2139373897*Log[-(Sqrt[2]*x) + S qrt[3 - x + 2*x^2] - #1] + 277937160*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 228643025*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) & ])/55920590000
Time = 0.64 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1302, 27, 2135, 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.
| \(\displaystyle \int \frac {\sqrt {2 x^2-x+3}}{\left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1302 |
| \(\displaystyle \frac {(10 x+3) \sqrt {2 x^2-x+3}}{62 \left (5 x^2+3 x+2\right )^2}-\frac {1}{62} \int -\frac {80 x^2-62 x+183}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{124} \int \frac {80 x^2-62 x+183}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}dx+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{124} \left (\frac {\int \frac {11 (77456-32605 x)}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{7502}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{124} \left (\frac {\int \frac {77456-32605 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {1}{124} \left (\frac {\frac {\int -\frac {11 \left (-\left (\left (44851-32605 \sqrt {2}\right ) x\right )-77456 \sqrt {2}+110061\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (44851+32605 \sqrt {2}\right ) x\right )+77456 \sqrt {2}+110061\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{124} \left (\frac {\frac {\int \frac {-\left (\left (44851+32605 \sqrt {2}\right ) x\right )+77456 \sqrt {2}+110061}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (44851-32605 \sqrt {2}\right ) x\right )-77456 \sqrt {2}+110061}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {1}{124} \left (\frac {\frac {\left (112285869463-79399380740 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (1235163-872375 \sqrt {2}\right ) x-362788 \sqrt {2}+509587\right )^2}{2 x^2-x+3}-31 \left (112285869463-79399380740 \sqrt {2}\right )}d\frac {\left (1235163-872375 \sqrt {2}\right ) x-362788 \sqrt {2}+509587}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (112285869463+79399380740 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (1235163+872375 \sqrt {2}\right ) x+362788 \sqrt {2}+509587\right )^2}{2 x^2-x+3}-31 \left (112285869463+79399380740 \sqrt {2}\right )}d\frac {\left (1235163+872375 \sqrt {2}\right ) x+362788 \sqrt {2}+509587}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{124} \left (\frac {\frac {\left (112285869463-79399380740 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (1235163-872375 \sqrt {2}\right ) x-362788 \sqrt {2}+509587\right )^2}{2 x^2-x+3}-31 \left (112285869463-79399380740 \sqrt {2}\right )}d\frac {\left (1235163-872375 \sqrt {2}\right ) x-362788 \sqrt {2}+509587}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (112285869463+79399380740 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (112285869463+79399380740 \sqrt {2}\right )}} \left (\left (1235163+872375 \sqrt {2}\right ) x+362788 \sqrt {2}+509587\right )}{\sqrt {2 x^2-x+3}}\right )}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{124} \left (\frac {\sqrt {\frac {1}{682} \left (112285869463+79399380740 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (112285869463+79399380740 \sqrt {2}\right )}} \left (\left (1235163+872375 \sqrt {2}\right ) x+362788 \sqrt {2}+509587\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (112285869463-79399380740 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (79399380740 \sqrt {2}-112285869463\right )}} \left (\left (1235163-872375 \sqrt {2}\right ) x-362788 \sqrt {2}+509587\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (79399380740 \sqrt {2}-112285869463\right )}}}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
((3 + 10*x)*Sqrt[3 - x + 2*x^2])/(62*(2 + 3*x + 5*x^2)^2) + (((3464 + 1366 5*x)*Sqrt[3 - x + 2*x^2])/(682*(2 + 3*x + 5*x^2)) + (Sqrt[(112285869463 + 79399380740*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(112285869463 + 79399380740* Sqrt[2]))]*(509587 + 362788*Sqrt[2] + (1235163 + 872375*Sqrt[2])*x))/Sqrt[ 3 - x + 2*x^2]] + ((112285869463 - 79399380740*Sqrt[2])*ArcTanh[(Sqrt[11/( 31*(-112285869463 + 79399380740*Sqrt[2]))]*(509587 - 362788*Sqrt[2] + (123 5163 - 872375*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/Sqrt[682*(-112285869463 + 79399380740*Sqrt[2])])/1364)/124
3.1.64.3.1 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]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. )*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] , C = Coeff[Px, x, 2]}, Simp[(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)))*( (A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b *e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c* e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x), 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*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B )*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a *c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*( c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4))) *x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; F reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.31 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.17
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(483\) |
| risch | \(\frac {\left (68325 x^{3}+58315 x^{2}+51362 x +11020\right ) \sqrt {2 x^{2}-x +3}}{84568 \left (5 x^{2}+3 x +2\right )^{2}}+\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 (33504619 \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 )+47385010 \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 )+49124007834 \,\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}-69208569562 \,\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 )}{3575873312 \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}}}\) | \(726\) |
| default | \(\text {Expression too large to display}\) | \(44343\) |
1/84568*(68325*x^3+58315*x^2+51362*x+11020)/(5*x^2+3*x+2)^2*(2*x^2-x+3)^(1 /2)-1/115350752*RootOf(_Z^2+29767936*RootOf(238143488*_Z^4+612631703790128 *_Z^2+394016353868467684225)^2+76578962973766)*ln(-(8901118918912*RootOf(_ Z^2+29767936*RootOf(238143488*_Z^4+612631703790128*_Z^2+394016353868467684 225)^2+76578962973766)*RootOf(238143488*_Z^4+612631703790128*_Z^2+39401635 3868467684225)^4*x+21996649948054194864*RootOf(238143488*_Z^4+612631703790 128*_Z^2+394016353868467684225)^2*RootOf(_Z^2+29767936*RootOf(238143488*_Z ^4+612631703790128*_Z^2+394016353868467684225)^2+76578962973766)*x+1055850 3141216967088325712*RootOf(238143488*_Z^4+612631703790128*_Z^2+39401635386 8467684225)^2*(2*x^2-x+3)^(1/2)-203149073871924400*RootOf(238143488*_Z^4+6 12631703790128*_Z^2+394016353868467684225)^2*RootOf(_Z^2+29767936*RootOf(2 38143488*_Z^4+612631703790128*_Z^2+394016353868467684225)^2+76578962973766 )+13558387834041967792583352*RootOf(_Z^2+29767936*RootOf(238143488*_Z^4+61 2631703790128*_Z^2+394016353868467684225)^2+76578962973766)*x+135771321202 55119256152874400047*(2*x^2-x+3)^(1/2)-238972679575677054341025*RootOf(_Z^ 2+29767936*RootOf(238143488*_Z^4+612631703790128*_Z^2+39401635386846768422 5)^2+76578962973766))/(21824*x*RootOf(238143488*_Z^4+612631703790128*_Z^2+ 394016353868467684225)^2+27985547479*x-114559849))+1/21142*RootOf(23814348 8*_Z^4+612631703790128*_Z^2+394016353868467684225)*ln((1139343221620736*x* RootOf(238143488*_Z^4+612631703790128*_Z^2+394016353868467684225)^5+304...
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.78 \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {114559849 i \, \sqrt {31} - 112285869463} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {114559849 i \, \sqrt {31} - 112285869463} {\left (509587 i \, \sqrt {31} - 3411891\right )} - 1230690401470 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 23383117627930 \, x - 27075188832340}{x}\right ) - \sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {114559849 i \, \sqrt {31} - 112285869463} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {114559849 i \, \sqrt {31} - 112285869463} {\left (-509587 i \, \sqrt {31} + 3411891\right )} - 1230690401470 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 23383117627930 \, x - 27075188832340}{x}\right ) - \sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-114559849 i \, \sqrt {31} - 112285869463} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (509587 i \, \sqrt {31} + 3411891\right )} \sqrt {-114559849 i \, \sqrt {31} - 112285869463} - 1230690401470 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 23383117627930 \, x - 27075188832340}{x}\right ) + \sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-114559849 i \, \sqrt {31} - 112285869463} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (-509587 i \, \sqrt {31} - 3411891\right )} \sqrt {-114559849 i \, \sqrt {31} - 112285869463} - 1230690401470 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 23383117627930 \, x - 27075188832340}{x}\right ) + 2728 \, {\left (68325 \, x^{3} + 58315 \, x^{2} + 51362 \, x + 11020\right )} \sqrt {2 \, x^{2} - x + 3}}{230701504 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]
1/230701504*(sqrt(341)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(11455984 9*I*sqrt(31) - 112285869463)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*sqrt(11455 9849*I*sqrt(31) - 112285869463)*(509587*I*sqrt(31) - 3411891) - 1230690401 470*sqrt(31)*(-I*x + 6*I) + 23383117627930*x - 27075188832340)/x) - sqrt(3 41)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(114559849*I*sqrt(31) - 1122 85869463)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*sqrt(114559849*I*sqrt(31) - 1 12285869463)*(-509587*I*sqrt(31) + 3411891) - 1230690401470*sqrt(31)*(-I*x + 6*I) + 23383117627930*x - 27075188832340)/x) - sqrt(341)*(25*x^4 + 30*x ^3 + 29*x^2 + 12*x + 4)*sqrt(-114559849*I*sqrt(31) - 112285869463)*log((sq rt(341)*sqrt(2*x^2 - x + 3)*(509587*I*sqrt(31) + 3411891)*sqrt(-114559849* I*sqrt(31) - 112285869463) - 1230690401470*sqrt(31)*(I*x - 6*I) + 23383117 627930*x - 27075188832340)/x) + sqrt(341)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-114559849*I*sqrt(31) - 112285869463)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*(-509587*I*sqrt(31) - 3411891)*sqrt(-114559849*I*sqrt(31) - 1122 85869463) - 1230690401470*sqrt(31)*(I*x - 6*I) + 23383117627930*x - 270751 88832340)/x) + 2728*(68325*x^3 + 58315*x^2 + 51362*x + 11020)*sqrt(2*x^2 - x + 3))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
\[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {\sqrt {2 x^{2} - x + 3}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
\[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {\sqrt {2 \, x^{2} - x + 3}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {\sqrt {2\,x^2-x+3}}{{\left (5\,x^2+3\,x+2\right )}^3} \,d x \]