Integrand size = 27, antiderivative size = 223 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {(26794+86265 x) \sqrt {3-x+2 x^2}}{1860496 \left (2+3 x+5 x^2\right )}+\frac {25 \sqrt {\frac {1}{682} \left (6414867847+4536374600 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (6414867847+4536374600 \sqrt {2}\right )}} \left (123161+85754 \sqrt {2}+\left (294669+208915 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{3720992}-\frac {25 \sqrt {\frac {1}{682} \left (-6414867847+4536374600 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-6414867847+4536374600 \sqrt {2}\right )}} \left (123161-85754 \sqrt {2}+\left (294669-208915 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{3720992} \]
1/1364*(4+65*x)*(2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)^2+1/1860496*(26794+86265*x )*(2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)-25/2537716544*arctanh(1/31*(123161+x*(29 4669-208915*2^(1/2))-85754*2^(1/2))*341^(1/2)/(-6414867847+4536374600*2^(1 /2))^(1/2)/(2*x^2-x+3)^(1/2))*(-4374939871654+3093807477200*2^(1/2))^(1/2) +25/2537716544*arctan(1/31*(123161+85754*2^(1/2)+x*(294669+208915*2^(1/2)) )*341^(1/2)/(6414867847+4536374600*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(4374 939871654+3093807477200*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.64 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.78 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {3-x+2 x^2} \left (59044+341572 x+392765 x^2+431325 x^3\right )}{1860496 \left (2+3 x+5 x^2\right )^2}+\frac {3 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-42330420383 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+11629301740 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-2992879225 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{49210119200}-\frac {16 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-720397 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+129160 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-65525 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{4509725} \]
(Sqrt[3 - x + 2*x^2]*(59044 + 341572*x + 392765*x^2 + 431325*x^3))/(186049 6*(2 + 3*x + 5*x^2)^2) + (3*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt [2]*#1^3 - 5*#1^4 & , (-42330420383*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 11629301740*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]* #1 - 2992879225*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1^2)/(-13*Sq rt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ])/49210119200 - (16*RootSum[- 56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-720397*Log[-( Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 129160*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 65525*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) & ])/45097 25
Time = 0.62 (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 = {1305, 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.
| \(\displaystyle \int \frac {1}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle \frac {(65 x+4) \sqrt {2 x^2-x+3}}{1364 \left (5 x^2+3 x+2\right )^2}-\frac {\int -\frac {11 \left (520 x^2-589 x+1050\right )}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}dx}{15004}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {520 x^2-589 x+1050}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}dx}{2728}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {\frac {\int \frac {275 (18658-7445 x)}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{7502}+\frac {\sqrt {2 x^2-x+3} (86265 x+26794)}{682 \left (5 x^2+3 x+2\right )}}{2728}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {25 \int \frac {18658-7445 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{1364}+\frac {\sqrt {2 x^2-x+3} (86265 x+26794)}{682 \left (5 x^2+3 x+2\right )}}{2728}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {\frac {25 \left (\frac {\int -\frac {11 \left (-\left (\left (11213-7445 \sqrt {2}\right ) x\right )-18658 \sqrt {2}+26103\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (11213+7445 \sqrt {2}\right ) x\right )+18658 \sqrt {2}+26103\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )}{1364}+\frac {\sqrt {2 x^2-x+3} (86265 x+26794)}{682 \left (5 x^2+3 x+2\right )}}{2728}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {25 \left (\frac {\int \frac {-\left (\left (11213+7445 \sqrt {2}\right ) x\right )+18658 \sqrt {2}+26103}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (11213-7445 \sqrt {2}\right ) x\right )-18658 \sqrt {2}+26103}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )}{1364}+\frac {\sqrt {2 x^2-x+3} (86265 x+26794)}{682 \left (5 x^2+3 x+2\right )}}{2728}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {\frac {25 \left (\frac {\left (6414867847-4536374600 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (294669-208915 \sqrt {2}\right ) x-85754 \sqrt {2}+123161\right )^2}{2 x^2-x+3}-31 \left (6414867847-4536374600 \sqrt {2}\right )}d\frac {\left (294669-208915 \sqrt {2}\right ) x-85754 \sqrt {2}+123161}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (6414867847+4536374600 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (294669+208915 \sqrt {2}\right ) x+85754 \sqrt {2}+123161\right )^2}{2 x^2-x+3}-31 \left (6414867847+4536374600 \sqrt {2}\right )}d\frac {\left (294669+208915 \sqrt {2}\right ) x+85754 \sqrt {2}+123161}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}\right )}{1364}+\frac {\sqrt {2 x^2-x+3} (86265 x+26794)}{682 \left (5 x^2+3 x+2\right )}}{2728}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {25 \left (\frac {\left (6414867847-4536374600 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (294669-208915 \sqrt {2}\right ) x-85754 \sqrt {2}+123161\right )^2}{2 x^2-x+3}-31 \left (6414867847-4536374600 \sqrt {2}\right )}d\frac {\left (294669-208915 \sqrt {2}\right ) x-85754 \sqrt {2}+123161}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (6414867847+4536374600 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (6414867847+4536374600 \sqrt {2}\right )}} \left (\left (294669+208915 \sqrt {2}\right ) x+85754 \sqrt {2}+123161\right )}{\sqrt {2 x^2-x+3}}\right )\right )}{1364}+\frac {\sqrt {2 x^2-x+3} (86265 x+26794)}{682 \left (5 x^2+3 x+2\right )}}{2728}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {25 \left (\sqrt {\frac {1}{682} \left (6414867847+4536374600 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (6414867847+4536374600 \sqrt {2}\right )}} \left (\left (294669+208915 \sqrt {2}\right ) x+85754 \sqrt {2}+123161\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (6414867847-4536374600 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (4536374600 \sqrt {2}-6414867847\right )}} \left (\left (294669-208915 \sqrt {2}\right ) x-85754 \sqrt {2}+123161\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (4536374600 \sqrt {2}-6414867847\right )}}\right )}{1364}+\frac {\sqrt {2 x^2-x+3} (86265 x+26794)}{682 \left (5 x^2+3 x+2\right )}}{2728}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}\) |
((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(1364*(2 + 3*x + 5*x^2)^2) + (((26794 + 8 6265*x)*Sqrt[3 - x + 2*x^2])/(682*(2 + 3*x + 5*x^2)) + (25*(Sqrt[(64148678 47 + 4536374600*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(6414867847 + 4536374600 *Sqrt[2]))]*(123161 + 85754*Sqrt[2] + (294669 + 208915*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + ((6414867847 - 4536374600*Sqrt[2])*ArcTanh[(Sqrt[11/(31*( -6414867847 + 4536374600*Sqrt[2]))]*(123161 - 85754*Sqrt[2] + (294669 - 20 8915*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/Sqrt[682*(-6414867847 + 4536374600 *Sqrt[2])]))/1364)/2728
3.1.85.3.1 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]
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 = 3.12 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.17
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(483\) |
| risch | \(\frac {\left (431325 x^{3}+392765 x^{2}+341572 x +59044\right ) \sqrt {2 x^{2}-x +3}}{1860496 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {25 \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 (11325170 \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 )+8008997 \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 )+11668925202 \,\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}-16645371446 \,\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 )}{78669212864 \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 | \(\text {Expression too large to display}\) | \(13040\) |
1/1860496*(431325*x^3+392765*x^2+341572*x+59044)/(5*x^2+3*x+2)^2*(2*x^2-x+ 3)^(1/2)-25/2537716544*RootOf(_Z^2+267911424*RootOf(4822405632*_Z^4+787489 17689772*_Z^2+321542101742580625)^2+4374939871654)*ln((-7411434655680*x*Ro otOf(4822405632*_Z^4+78748917689772*_Z^2+321542101742580625)^4*RootOf(_Z^2 +267911424*RootOf(4822405632*_Z^4+78748917689772*_Z^2+321542101742580625)^ 2+4374939871654)-133779516386184108*RootOf(4822405632*_Z^4+78748917689772* _Z^2+321542101742580625)^2*RootOf(_Z^2+267911424*RootOf(4822405632*_Z^4+78 748917689772*_Z^2+321542101742580625)^2+4374939871654)*x+35248490028539818 757496*(2*x^2-x+3)^(1/2)*RootOf(4822405632*_Z^4+78748917689772*_Z^2+321542 101742580625)^2-2440416054631500*RootOf(4822405632*_Z^4+78748917689772*_Z^ 2+321542101742580625)^2*RootOf(_Z^2+267911424*RootOf(4822405632*_Z^4+78748 917689772*_Z^2+321542101742580625)^2+4374939871654)-596509043121541261413* RootOf(_Z^2+267911424*RootOf(4822405632*_Z^4+78748917689772*_Z^2+321542101 742580625)^2+4374939871654)*x+287988742887575789016260666*(2*x^2-x+3)^(1/2 )-24428133718051268025*RootOf(_Z^2+267911424*RootOf(4822405632*_Z^4+787489 17689772*_Z^2+321542101742580625)^2+4374939871654))/(196416*x*RootOf(48224 05632*_Z^4+78748917689772*_Z^2+321542101742580625)^2+1614873451*x+14875319 ))-75/465124*RootOf(4822405632*_Z^4+78748917689772*_Z^2+321542101742580625 )*ln(-(5691981815562240*x*RootOf(4822405632*_Z^4+78748917689772*_Z^2+32154 2101742580625)^5+83155176470593979136*RootOf(4822405632*_Z^4+7874891768...
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.78 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {\sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {9297074375 i \, \sqrt {31} - 4009292404375} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {9297074375 i \, \sqrt {31} - 4009292404375} {\left (123161 i \, \sqrt {31} + 809193\right )} - 1757845157500 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 33399057992500 \, x - 38672593465000}{x}\right ) - \sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {9297074375 i \, \sqrt {31} - 4009292404375} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {9297074375 i \, \sqrt {31} - 4009292404375} {\left (-123161 i \, \sqrt {31} - 809193\right )} - 1757845157500 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 33399057992500 \, x - 38672593465000}{x}\right ) - \sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-9297074375 i \, \sqrt {31} - 4009292404375} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (123161 i \, \sqrt {31} - 809193\right )} \sqrt {-9297074375 i \, \sqrt {31} - 4009292404375} - 1757845157500 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 33399057992500 \, x - 38672593465000}{x}\right ) + \sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-9297074375 i \, \sqrt {31} - 4009292404375} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (-123161 i \, \sqrt {31} + 809193\right )} \sqrt {-9297074375 i \, \sqrt {31} - 4009292404375} - 1757845157500 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 33399057992500 \, x - 38672593465000}{x}\right ) - 2728 \, {\left (431325 \, x^{3} + 392765 \, x^{2} + 341572 \, x + 59044\right )} \sqrt {2 \, x^{2} - x + 3}}{5075433088 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]
-1/5075433088*(sqrt(341)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(929707 4375*I*sqrt(31) - 4009292404375)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*sqrt(9 297074375*I*sqrt(31) - 4009292404375)*(123161*I*sqrt(31) + 809193) - 17578 45157500*sqrt(31)*(I*x - 6*I) + 33399057992500*x - 38672593465000)/x) - sq rt(341)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(9297074375*I*sqrt(31) - 4009292404375)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*sqrt(9297074375*I*sqrt( 31) - 4009292404375)*(-123161*I*sqrt(31) - 809193) - 1757845157500*sqrt(31 )*(I*x - 6*I) + 33399057992500*x - 38672593465000)/x) - sqrt(341)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-9297074375*I*sqrt(31) - 4009292404375) *log((sqrt(341)*sqrt(2*x^2 - x + 3)*(123161*I*sqrt(31) - 809193)*sqrt(-929 7074375*I*sqrt(31) - 4009292404375) - 1757845157500*sqrt(31)*(-I*x + 6*I) + 33399057992500*x - 38672593465000)/x) + sqrt(341)*(25*x^4 + 30*x^3 + 29* x^2 + 12*x + 4)*sqrt(-9297074375*I*sqrt(31) - 4009292404375)*log((sqrt(341 )*sqrt(2*x^2 - x + 3)*(-123161*I*sqrt(31) + 809193)*sqrt(-9297074375*I*sqr t(31) - 4009292404375) - 1757845157500*sqrt(31)*(-I*x + 6*I) + 33399057992 500*x - 38672593465000)/x) - 2728*(431325*x^3 + 392765*x^2 + 341572*x + 59 044)*sqrt(2*x^2 - x + 3))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
\[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {1}{\sqrt {2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
\[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3} \sqrt {2 \, x^{2} - x + 3}} \,d x } \]
Exception generated. \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {1}{\sqrt {2\,x^2-x+3}\,{\left (5\,x^2+3\,x+2\right )}^3} \,d x \]