Integrand size = 27, antiderivative size = 234 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {15101-8654 x}{1035276 \left (3-x+2 x^2\right )^{3/2}}-\frac {3133427+1352542 x}{523849656 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{682 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}+\frac {625 \sqrt {\frac {1}{682} \left (30463+23600 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (30463+23600 \sqrt {2}\right )}} \left (203+242 \sqrt {2}+\left (687+445 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{660176}-\frac {625 \sqrt {\frac {1}{682} \left (-30463+23600 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-30463+23600 \sqrt {2}\right )}} \left (203-242 \sqrt {2}+\left (687-445 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{660176} \]
1/1035276*(-15101+8654*x)/(2*x^2-x+3)^(3/2)+1/682*(4+65*x)/(2*x^2-x+3)^(3/ 2)/(5*x^2+3*x+2)+1/523849656*(-3133427-1352542*x)/(2*x^2-x+3)^(1/2)-625/45 0240032*arctanh(1/31*(203+x*(687-445*2^(1/2))-242*2^(1/2))*341^(1/2)/(-304 63+23600*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(-20775766+16095200*2^(1/2))^(1 /2)+625/450240032*arctan(1/31*(203+242*2^(1/2)+x*(687+445*2^(1/2)))*341^(1 /2)/(30463+23600*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(20775766+16095200*2^(1 /2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.07 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.78 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {-31010342+5712309 x-84671384 x^2-2879479 x^3-32686812 x^4-13525420 x^5}{523849656 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}+\frac {\text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-1376 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+106 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+95 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{5324}+\frac {\text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {126249 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+58712 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+10095 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{660176 \sqrt {2}} \]
(-31010342 + 5712309*x - 84671384*x^2 - 2879479*x^3 - 32686812*x^4 - 13525 420*x^5)/(523849656*(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)) + RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-1376*Log[-(Sqrt [2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 106*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 95*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) & ]/5324 + RootSum[- 56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (126249*Sqrt[2] *Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 58712*Log[-(Sqrt[2]*x) + S qrt[3 - x + 2*x^2] - #1]*#1 + 10095*Sqrt[2]*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) & ]/ (660176*Sqrt[2])
Time = 0.74 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1305, 27, 2135, 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.
| \(\displaystyle \int \frac {1}{\left (2 x^2-x+3\right )^{5/2} \left (5 x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle \frac {65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}-\frac {\int -\frac {11 \left (1040 x^2-401 x+316\right )}{2 \left (2 x^2-x+3\right )^{5/2} \left (5 x^2+3 x+2\right )}dx}{7502}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {1040 x^2-401 x+316}{\left (2 x^2-x+3\right )^{5/2} \left (5 x^2+3 x+2\right )}dx}{1364}+\frac {65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {\frac {\int \frac {11 \left (173080 x^2-284277 x+155482\right )}{2 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}dx}{8349}-\frac {15101-8654 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {173080 x^2-284277 x+155482}{\left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}dx}{1518}-\frac {15101-8654 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {10910625 (34-35 x)}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2783}-\frac {1352542 x+3133427}{253 \sqrt {2 x^2-x+3}}}{1518}-\frac {15101-8654 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {43125}{22} \int \frac {34-35 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {1352542 x+3133427}{253 \sqrt {2 x^2-x+3}}}{1518}-\frac {15101-8654 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {\frac {\frac {43125}{22} \left (\frac {\int -\frac {11 \left (\left (1+35 \sqrt {2}\right ) x-34 \sqrt {2}+69\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (\left (1-35 \sqrt {2}\right ) x+34 \sqrt {2}+69\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )-\frac {1352542 x+3133427}{253 \sqrt {2 x^2-x+3}}}{1518}-\frac {15101-8654 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {43125}{22} \left (\frac {\int \frac {\left (1-35 \sqrt {2}\right ) x+34 \sqrt {2}+69}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {\left (1+35 \sqrt {2}\right ) x-34 \sqrt {2}+69}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )-\frac {1352542 x+3133427}{253 \sqrt {2 x^2-x+3}}}{1518}-\frac {15101-8654 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {\frac {\frac {43125}{22} \left (\frac {\left (30463-23600 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (687-445 \sqrt {2}\right ) x-242 \sqrt {2}+203\right )^2}{2 x^2-x+3}-31 \left (30463-23600 \sqrt {2}\right )}d\frac {\left (687-445 \sqrt {2}\right ) x-242 \sqrt {2}+203}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (30463+23600 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (687+445 \sqrt {2}\right ) x+242 \sqrt {2}+203\right )^2}{2 x^2-x+3}-31 \left (30463+23600 \sqrt {2}\right )}d\frac {\left (687+445 \sqrt {2}\right ) x+242 \sqrt {2}+203}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}\right )-\frac {1352542 x+3133427}{253 \sqrt {2 x^2-x+3}}}{1518}-\frac {15101-8654 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {43125}{22} \left (\frac {\left (30463-23600 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (687-445 \sqrt {2}\right ) x-242 \sqrt {2}+203\right )^2}{2 x^2-x+3}-31 \left (30463-23600 \sqrt {2}\right )}d\frac {\left (687-445 \sqrt {2}\right ) x-242 \sqrt {2}+203}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (30463+23600 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (30463+23600 \sqrt {2}\right )}} \left (\left (687+445 \sqrt {2}\right ) x+242 \sqrt {2}+203\right )}{\sqrt {2 x^2-x+3}}\right )\right )-\frac {1352542 x+3133427}{253 \sqrt {2 x^2-x+3}}}{1518}-\frac {15101-8654 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {43125}{22} \left (\sqrt {\frac {1}{682} \left (30463+23600 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (30463+23600 \sqrt {2}\right )}} \left (\left (687+445 \sqrt {2}\right ) x+242 \sqrt {2}+203\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (30463-23600 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (23600 \sqrt {2}-30463\right )}} \left (\left (687-445 \sqrt {2}\right ) x-242 \sqrt {2}+203\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (23600 \sqrt {2}-30463\right )}}\right )-\frac {1352542 x+3133427}{253 \sqrt {2 x^2-x+3}}}{1518}-\frac {15101-8654 x}{759 \left (2 x^2-x+3\right )^{3/2}}}{1364}+\frac {65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}\) |
(4 + 65*x)/(682*(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)) + (-1/759*(15101 - 8654*x)/(3 - x + 2*x^2)^(3/2) + (-1/253*(3133427 + 1352542*x)/Sqrt[3 - x + 2*x^2] + (43125*(Sqrt[(30463 + 23600*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31* (30463 + 23600*Sqrt[2]))]*(203 + 242*Sqrt[2] + (687 + 445*Sqrt[2])*x))/Sqr t[3 - x + 2*x^2]] + ((30463 - 23600*Sqrt[2])*ArcTanh[(Sqrt[11/(31*(-30463 + 23600*Sqrt[2]))]*(203 - 242*Sqrt[2] + (687 - 445*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/Sqrt[682*(-30463 + 23600*Sqrt[2])]))/22)/1518)/1364
3.1.98.3.1 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 = 2.96 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.11
| method | result | size |
| trager | \(-\frac {13525420 x^{5}+32686812 x^{4}+2879479 x^{3}+84671384 x^{2}-5712309 x +31010342}{523849656 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} \left (5 x^{2}+3 x +2\right )}-\frac {16875 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right ) \ln \left (\frac {-5226452556429468672 x \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{5}-30671894532413376 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{3} x +1319515725838464 \sqrt {2 x^{2}-x +3}\, \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2}-13991557820486400 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{3}-23459476566825 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right ) x +2583860307500 \sqrt {2 x^{2}-x +3}-69460188136200 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )}{7954848 x \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2}+18905 x +4898}\right )}{165044}+\frac {625 \operatorname {RootOf}\left (\textit {\_Z}^{2}+5425206336 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2}+20775766\right ) \ln \left (-\frac {-756141862909356 \operatorname {RootOf}\left (\textit {\_Z}^{2}+5425206336 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2}+20775766\right ) \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{4} x -1353788597799 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+5425206336 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2}+20775766\right ) x +14061089453466132 \sqrt {2 x^{2}-x +3}\, \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2}+2024241582825 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+5425206336 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2}+20775766\right )+2510468172 \operatorname {RootOf}\left (\textit {\_Z}^{2}+5425206336 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2}+20775766\right ) x +26312520839792 \sqrt {2 x^{2}-x +3}-2297406900 \operatorname {RootOf}\left (\textit {\_Z}^{2}+5425206336 \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2}+20775766\right )}{3977424 x \operatorname {RootOf}\left (1977487709472 \textit {\_Z}^{4}+7572766707 \textit {\_Z}^{2}+8702500\right )^{2}+5779 x -2449}\right )}{450240032}\) | \(493\) |
| risch | \(-\frac {13525420 x^{5}+32686812 x^{4}+2879479 x^{3}+84671384 x^{2}-5712309 x +31010342}{523849656 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} \left (5 x^{2}+3 x +2\right )}+\frac {625 \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 (17831 \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 )+25310 \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 )+29873646 \,\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}-31691858 \,\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 )}{13957440992 \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}}}\) | \(736\) |
| default | \(\text {Expression too large to display}\) | \(5975\) |
-1/523849656*(13525420*x^5+32686812*x^4+2879479*x^3+84671384*x^2-5712309*x +31010342)/(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)-16875/165044*RootOf(19774877094 72*_Z^4+7572766707*_Z^2+8702500)*ln((-5226452556429468672*x*RootOf(1977487 709472*_Z^4+7572766707*_Z^2+8702500)^5-30671894532413376*RootOf(1977487709 472*_Z^4+7572766707*_Z^2+8702500)^3*x+1319515725838464*(2*x^2-x+3)^(1/2)*R ootOf(1977487709472*_Z^4+7572766707*_Z^2+8702500)^2-13991557820486400*Root Of(1977487709472*_Z^4+7572766707*_Z^2+8702500)^3-23459476566825*RootOf(197 7487709472*_Z^4+7572766707*_Z^2+8702500)*x+2583860307500*(2*x^2-x+3)^(1/2) -69460188136200*RootOf(1977487709472*_Z^4+7572766707*_Z^2+8702500))/(79548 48*x*RootOf(1977487709472*_Z^4+7572766707*_Z^2+8702500)^2+18905*x+4898))+6 25/450240032*RootOf(_Z^2+5425206336*RootOf(1977487709472*_Z^4+7572766707*_ Z^2+8702500)^2+20775766)*ln(-(-756141862909356*RootOf(_Z^2+5425206336*Root Of(1977487709472*_Z^4+7572766707*_Z^2+8702500)^2+20775766)*RootOf(19774877 09472*_Z^4+7572766707*_Z^2+8702500)^4*x-1353788597799*RootOf(1977487709472 *_Z^4+7572766707*_Z^2+8702500)^2*RootOf(_Z^2+5425206336*RootOf(19774877094 72*_Z^4+7572766707*_Z^2+8702500)^2+20775766)*x+14061089453466132*(2*x^2-x+ 3)^(1/2)*RootOf(1977487709472*_Z^4+7572766707*_Z^2+8702500)^2+202424158282 5*RootOf(1977487709472*_Z^4+7572766707*_Z^2+8702500)^2*RootOf(_Z^2+5425206 336*RootOf(1977487709472*_Z^4+7572766707*_Z^2+8702500)^2+20775766)+2510468 172*RootOf(_Z^2+5425206336*RootOf(1977487709472*_Z^4+7572766707*_Z^2+87...
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.91 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {1587 \, \sqrt {341} {\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \sqrt {956640625 i \, \sqrt {31} - 11899609375} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {956640625 i \, \sqrt {31} - 11899609375} {\left (203 i \, \sqrt {31} - 2139\right )} - 228625000 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 4343875000 \, x - 5029750000}{x}\right ) - 1587 \, \sqrt {341} {\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \sqrt {956640625 i \, \sqrt {31} - 11899609375} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {956640625 i \, \sqrt {31} - 11899609375} {\left (-203 i \, \sqrt {31} + 2139\right )} - 228625000 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 4343875000 \, x - 5029750000}{x}\right ) - 1587 \, \sqrt {341} {\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \sqrt {-956640625 i \, \sqrt {31} - 11899609375} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (203 i \, \sqrt {31} + 2139\right )} \sqrt {-956640625 i \, \sqrt {31} - 11899609375} - 228625000 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 4343875000 \, x - 5029750000}{x}\right ) + 1587 \, \sqrt {341} {\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \sqrt {-956640625 i \, \sqrt {31} - 11899609375} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (-203 i \, \sqrt {31} - 2139\right )} \sqrt {-956640625 i \, \sqrt {31} - 11899609375} - 228625000 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 4343875000 \, x - 5029750000}{x}\right ) - 2728 \, {\left (13525420 \, x^{5} + 32686812 \, x^{4} + 2879479 \, x^{3} + 84671384 \, x^{2} - 5712309 \, x + 31010342\right )} \sqrt {2 \, x^{2} - x + 3}}{1429061861568 \, {\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )}} \]
1/1429061861568*(1587*sqrt(341)*(20*x^6 - 8*x^5 + 61*x^4 + x^3 + 53*x^2 + 15*x + 18)*sqrt(956640625*I*sqrt(31) - 11899609375)*log((sqrt(341)*sqrt(2* x^2 - x + 3)*sqrt(956640625*I*sqrt(31) - 11899609375)*(203*I*sqrt(31) - 21 39) - 228625000*sqrt(31)*(-I*x + 6*I) + 4343875000*x - 5029750000)/x) - 15 87*sqrt(341)*(20*x^6 - 8*x^5 + 61*x^4 + x^3 + 53*x^2 + 15*x + 18)*sqrt(956 640625*I*sqrt(31) - 11899609375)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*sqrt(9 56640625*I*sqrt(31) - 11899609375)*(-203*I*sqrt(31) + 2139) - 228625000*sq rt(31)*(-I*x + 6*I) + 4343875000*x - 5029750000)/x) - 1587*sqrt(341)*(20*x ^6 - 8*x^5 + 61*x^4 + x^3 + 53*x^2 + 15*x + 18)*sqrt(-956640625*I*sqrt(31) - 11899609375)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*(203*I*sqrt(31) + 2139) *sqrt(-956640625*I*sqrt(31) - 11899609375) - 228625000*sqrt(31)*(I*x - 6*I ) + 4343875000*x - 5029750000)/x) + 1587*sqrt(341)*(20*x^6 - 8*x^5 + 61*x^ 4 + x^3 + 53*x^2 + 15*x + 18)*sqrt(-956640625*I*sqrt(31) - 11899609375)*lo g((sqrt(341)*sqrt(2*x^2 - x + 3)*(-203*I*sqrt(31) - 2139)*sqrt(-956640625* I*sqrt(31) - 11899609375) - 228625000*sqrt(31)*(I*x - 6*I) + 4343875000*x - 5029750000)/x) - 2728*(13525420*x^5 + 32686812*x^4 + 2879479*x^3 + 84671 384*x^2 - 5712309*x + 31010342)*sqrt(2*x^2 - x + 3))/(20*x^6 - 8*x^5 + 61* x^4 + x^3 + 53*x^2 + 15*x + 18)
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {1}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \]
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {1}{{\left (2\,x^2-x+3\right )}^{5/2}\,{\left (5\,x^2+3\,x+2\right )}^2} \,d x \]