Integrand size = 30, antiderivative size = 193 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {9}{2} \sqrt {\frac {1}{5} \left (-53+17 \sqrt {10}\right )} \arctan \left (\frac {3 \left (4-\sqrt {10}\right )+\left (1+4 \sqrt {10}\right ) x}{2 \sqrt {1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )+\frac {9}{2} \sqrt {\frac {1}{5} \left (53+17 \sqrt {10}\right )} \text {arctanh}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (1-4 \sqrt {10}\right ) x}{2 \sqrt {-1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right ) \]
-2/51*(15+14*x)/(-2*x^2+3*x+1)^(3/2)-2/867*(291+4814*x)/(-2*x^2+3*x+1)^(1/ 2)+9/10*arctan(1/2*(12-3*10^(1/2)+x*(1+4*10^(1/2)))/(-2*x^2+3*x+1)^(1/2)/( 1+10^(1/2))^(1/2))*(-265+85*10^(1/2))^(1/2)+9/10*arctanh(1/2*(x*(1-4*10^(1 /2))+12+3*10^(1/2))/(-2*x^2+3*x+1)^(1/2)/(-1+10^(1/2))^(1/2))*(265+85*10^( 1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.45 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.95 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=-\frac {2 \left (546+5925 x+13860 x^2-9628 x^3\right )}{867 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {9}{2} \text {RootSum}\left [5+20 \text {$\#$1}+8 \text {$\#$1}^2-8 \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {-13 \log (x)+13 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}-6 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 \log (x) \text {$\#$1}^2+2 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{5+4 \text {$\#$1}-6 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
(-2*(546 + 5925*x + 13860*x^2 - 9628*x^3))/(867*(1 + 3*x - 2*x^2)^(3/2)) - (9*RootSum[5 + 20*#1 + 8*#1^2 - 8*#1^3 + 2*#1^4 & , (-13*Log[x] + 13*Log[ -1 + Sqrt[1 + 3*x - 2*x^2] - x*#1] + 6*Log[x]*#1 - 6*Log[-1 + Sqrt[1 + 3*x - 2*x^2] - x*#1]*#1 - 2*Log[x]*#1^2 + 2*Log[-1 + Sqrt[1 + 3*x - 2*x^2] - x*#1]*#1^2)/(5 + 4*#1 - 6*#1^2 + 2*#1^3) & ])/2
Time = 0.44 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1349, 27, 2135, 27, 1365, 27, 1154, 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.
| \(\displaystyle \int \frac {x+2}{\left (-3 x^2+4 x+2\right ) \left (-2 x^2+3 x+1\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1349 |
| \(\displaystyle \frac {2}{51} \int -\frac {-168 x^2-235 x+112}{2 \left (-3 x^2+4 x+2\right ) \left (-2 x^2+3 x+1\right )^{3/2}}dx-\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{51} \int \frac {-168 x^2-235 x+112}{\left (-3 x^2+4 x+2\right ) \left (-2 x^2+3 x+1\right )^{3/2}}dx-\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{51} \left (-\frac {2}{17} \int -\frac {7803 (3 x+2)}{2 \left (-3 x^2+4 x+2\right ) \sqrt {-2 x^2+3 x+1}}dx-\frac {2 (4814 x+291)}{17 \sqrt {-2 x^2+3 x+1}}\right )-\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{51} \left (459 \int \frac {3 x+2}{\left (-3 x^2+4 x+2\right ) \sqrt {-2 x^2+3 x+1}}dx-\frac {2 (4814 x+291)}{17 \sqrt {-2 x^2+3 x+1}}\right )-\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {1}{51} \left (459 \left (\frac {3}{5} \left (5-2 \sqrt {10}\right ) \int \frac {1}{2 \left (-3 x-\sqrt {10}+2\right ) \sqrt {-2 x^2+3 x+1}}dx+\frac {3}{5} \left (5+2 \sqrt {10}\right ) \int \frac {1}{2 \left (-3 x+\sqrt {10}+2\right ) \sqrt {-2 x^2+3 x+1}}dx\right )-\frac {2 (4814 x+291)}{17 \sqrt {-2 x^2+3 x+1}}\right )-\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{51} \left (459 \left (\frac {3}{10} \left (5-2 \sqrt {10}\right ) \int \frac {1}{\left (-3 x-\sqrt {10}+2\right ) \sqrt {-2 x^2+3 x+1}}dx+\frac {3}{10} \left (5+2 \sqrt {10}\right ) \int \frac {1}{\left (-3 x+\sqrt {10}+2\right ) \sqrt {-2 x^2+3 x+1}}dx\right )-\frac {2 (4814 x+291)}{17 \sqrt {-2 x^2+3 x+1}}\right )-\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{51} \left (459 \left (-\frac {3}{5} \left (5+2 \sqrt {10}\right ) \int \frac {1}{-\frac {\left (\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )\right )^2}{-2 x^2+3 x+1}-4 \left (1-\sqrt {10}\right )}d\left (-\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{\sqrt {-2 x^2+3 x+1}}\right )-\frac {3}{5} \left (5-2 \sqrt {10}\right ) \int \frac {1}{-\frac {\left (\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )\right )^2}{-2 x^2+3 x+1}-4 \left (1+\sqrt {10}\right )}d\left (-\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{\sqrt {-2 x^2+3 x+1}}\right )\right )-\frac {2 (4814 x+291)}{17 \sqrt {-2 x^2+3 x+1}}\right )-\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{51} \left (459 \left (-\frac {3}{5} \left (5+2 \sqrt {10}\right ) \int \frac {1}{-\frac {\left (\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )\right )^2}{-2 x^2+3 x+1}-4 \left (1-\sqrt {10}\right )}d\left (-\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{\sqrt {-2 x^2+3 x+1}}\right )-\frac {3 \left (5-2 \sqrt {10}\right ) \arctan \left (\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )}{10 \sqrt {1+\sqrt {10}}}\right )-\frac {2 (4814 x+291)}{17 \sqrt {-2 x^2+3 x+1}}\right )-\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{51} \left (459 \left (\frac {3 \left (5+2 \sqrt {10}\right ) \text {arctanh}\left (\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {\sqrt {10}-1} \sqrt {-2 x^2+3 x+1}}\right )}{10 \sqrt {\sqrt {10}-1}}-\frac {3 \left (5-2 \sqrt {10}\right ) \arctan \left (\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )}{10 \sqrt {1+\sqrt {10}}}\right )-\frac {2 (4814 x+291)}{17 \sqrt {-2 x^2+3 x+1}}\right )-\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}\) |
(-2*(15 + 14*x))/(51*(1 + 3*x - 2*x^2)^(3/2)) + ((-2*(291 + 4814*x))/(17*S qrt[1 + 3*x - 2*x^2]) + 459*((-3*(5 - 2*Sqrt[10])*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/(10* Sqrt[1 + Sqrt[10]]) + (3*(5 + 2*Sqrt[10])*ArcTanh[(3*(4 + Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/(10*Sqrt[- 1 + Sqrt[10]])))/51
3.1.27.3.1 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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e _.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> 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)))*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b* c*d - 2*a*c*e + a*b*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*h - 2*g*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1 ) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*((-h)*c *e)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g *b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2 *a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))* (p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a *((-h)*c*e)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h* c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, 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])
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim p[(2*c*g - h*(b - q))/q Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] , x] - Simp[(2*c*g - h*(b + q))/q Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f *x^2]), x], 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] && PosQ[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 = 2.27 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.51
| method | result | size |
| trager | \(\frac {2 \left (9628 x^{3}-13860 x^{2}-5925 x -546\right ) \sqrt {-2 x^{2}+3 x +1}}{867 \left (2 x^{2}-3 x -1\right )^{2}}-18 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right ) \ln \left (-\frac {-2105600 x \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{5}+5362400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{3} x +74880 \sqrt {-2 x^{2}+3 x +1}\, \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}+473760 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{3}-3406349 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right ) x -99945 \sqrt {-2 x^{2}+3 x +1}-632106 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )}{80 x \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-87 x -34}\right )+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right ) \ln \left (-\frac {2105600 x \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right )-217440 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right ) x +473760 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right )-1497600 \sqrt {-2 x^{2}+3 x +1}\, \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-2187 \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right ) x +4374 \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right )-14580 \sqrt {-2 x^{2}+3 x +1}}{80 x \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-19 x +34}\right )}{10}\) | \(484\) |
| default | \(\text {Expression too large to display}\) | \(868\) |
2/867*(9628*x^3-13860*x^2-5925*x-546)/(2*x^2-3*x-1)^2*(-2*x^2+3*x+1)^(1/2) -18*RootOf(6400*_Z^4-8480*_Z^2-81)*ln(-(-2105600*x*RootOf(6400*_Z^4-8480*_ Z^2-81)^5+5362400*RootOf(6400*_Z^4-8480*_Z^2-81)^3*x+74880*(-2*x^2+3*x+1)^ (1/2)*RootOf(6400*_Z^4-8480*_Z^2-81)^2+473760*RootOf(6400*_Z^4-8480*_Z^2-8 1)^3-3406349*RootOf(6400*_Z^4-8480*_Z^2-81)*x-99945*(-2*x^2+3*x+1)^(1/2)-6 32106*RootOf(6400*_Z^4-8480*_Z^2-81))/(80*x*RootOf(6400*_Z^4-8480*_Z^2-81) ^2-87*x-34))+9/10*RootOf(_Z^2+400*RootOf(6400*_Z^4-8480*_Z^2-81)^2-530)*ln (-(2105600*x*RootOf(6400*_Z^4-8480*_Z^2-81)^4*RootOf(_Z^2+400*RootOf(6400* _Z^4-8480*_Z^2-81)^2-530)-217440*RootOf(6400*_Z^4-8480*_Z^2-81)^2*RootOf(_ Z^2+400*RootOf(6400*_Z^4-8480*_Z^2-81)^2-530)*x+473760*RootOf(6400*_Z^4-84 80*_Z^2-81)^2*RootOf(_Z^2+400*RootOf(6400*_Z^4-8480*_Z^2-81)^2-530)-149760 0*(-2*x^2+3*x+1)^(1/2)*RootOf(6400*_Z^4-8480*_Z^2-81)^2-2187*RootOf(_Z^2+4 00*RootOf(6400*_Z^4-8480*_Z^2-81)^2-530)*x+4374*RootOf(_Z^2+400*RootOf(640 0*_Z^4-8480*_Z^2-81)^2-530)-14580*(-2*x^2+3*x+1)^(1/2))/(80*x*RootOf(6400* _Z^4-8480*_Z^2-81)^2-19*x+34))
Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (137) = 274\).
Time = 0.31 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.27 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=-\frac {43680 \, x^{4} - 131040 \, x^{3} - 867 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {-1377 \, \sqrt {10} + 4293} \log \left (-\frac {405 \, \sqrt {10} x + {\left (13 \, \sqrt {10} \sqrt {5} x + 40 \, \sqrt {5} x\right )} \sqrt {-1377 \, \sqrt {10} + 4293} + 810 \, x - 810 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} + 810}{x}\right ) + 867 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {-1377 \, \sqrt {10} + 4293} \log \left (-\frac {405 \, \sqrt {10} x - {\left (13 \, \sqrt {10} \sqrt {5} x + 40 \, \sqrt {5} x\right )} \sqrt {-1377 \, \sqrt {10} + 4293} + 810 \, x - 810 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} + 810}{x}\right ) - 7803 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {17 \, \sqrt {10} + 53} \log \left (\frac {9 \, {\left (45 \, \sqrt {10} x + {\left (13 \, \sqrt {10} \sqrt {5} x - 40 \, \sqrt {5} x\right )} \sqrt {17 \, \sqrt {10} + 53} - 90 \, x + 90 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 90\right )}}{x}\right ) + 7803 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {17 \, \sqrt {10} + 53} \log \left (\frac {9 \, {\left (45 \, \sqrt {10} x - {\left (13 \, \sqrt {10} \sqrt {5} x - 40 \, \sqrt {5} x\right )} \sqrt {17 \, \sqrt {10} + 53} - 90 \, x + 90 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 90\right )}}{x}\right ) + 54600 \, x^{2} - 20 \, {\left (9628 \, x^{3} - 13860 \, x^{2} - 5925 \, x - 546\right )} \sqrt {-2 \, x^{2} + 3 \, x + 1} + 65520 \, x + 10920}{8670 \, {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )}} \]
-1/8670*(43680*x^4 - 131040*x^3 - 867*sqrt(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6* x + 1)*sqrt(-1377*sqrt(10) + 4293)*log(-(405*sqrt(10)*x + (13*sqrt(10)*sqr t(5)*x + 40*sqrt(5)*x)*sqrt(-1377*sqrt(10) + 4293) + 810*x - 810*sqrt(-2*x ^2 + 3*x + 1) + 810)/x) + 867*sqrt(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)*s qrt(-1377*sqrt(10) + 4293)*log(-(405*sqrt(10)*x - (13*sqrt(10)*sqrt(5)*x + 40*sqrt(5)*x)*sqrt(-1377*sqrt(10) + 4293) + 810*x - 810*sqrt(-2*x^2 + 3*x + 1) + 810)/x) - 7803*sqrt(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)*sqrt(17* sqrt(10) + 53)*log(9*(45*sqrt(10)*x + (13*sqrt(10)*sqrt(5)*x - 40*sqrt(5)* x)*sqrt(17*sqrt(10) + 53) - 90*x + 90*sqrt(-2*x^2 + 3*x + 1) - 90)/x) + 78 03*sqrt(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)*sqrt(17*sqrt(10) + 53)*log(9 *(45*sqrt(10)*x - (13*sqrt(10)*sqrt(5)*x - 40*sqrt(5)*x)*sqrt(17*sqrt(10) + 53) - 90*x + 90*sqrt(-2*x^2 + 3*x + 1) - 90)/x) + 54600*x^2 - 20*(9628*x ^3 - 13860*x^2 - 5925*x - 546)*sqrt(-2*x^2 + 3*x + 1) + 65520*x + 10920)/( 4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)
\[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=- \int \frac {x}{12 x^{6} \sqrt {- 2 x^{2} + 3 x + 1} - 52 x^{5} \sqrt {- 2 x^{2} + 3 x + 1} + 55 x^{4} \sqrt {- 2 x^{2} + 3 x + 1} + 22 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 31 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 16 x \sqrt {- 2 x^{2} + 3 x + 1} - 2 \sqrt {- 2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{12 x^{6} \sqrt {- 2 x^{2} + 3 x + 1} - 52 x^{5} \sqrt {- 2 x^{2} + 3 x + 1} + 55 x^{4} \sqrt {- 2 x^{2} + 3 x + 1} + 22 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 31 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 16 x \sqrt {- 2 x^{2} + 3 x + 1} - 2 \sqrt {- 2 x^{2} + 3 x + 1}}\, dx \]
-Integral(x/(12*x**6*sqrt(-2*x**2 + 3*x + 1) - 52*x**5*sqrt(-2*x**2 + 3*x + 1) + 55*x**4*sqrt(-2*x**2 + 3*x + 1) + 22*x**3*sqrt(-2*x**2 + 3*x + 1) - 31*x**2*sqrt(-2*x**2 + 3*x + 1) - 16*x*sqrt(-2*x**2 + 3*x + 1) - 2*sqrt(- 2*x**2 + 3*x + 1)), x) - Integral(2/(12*x**6*sqrt(-2*x**2 + 3*x + 1) - 52* x**5*sqrt(-2*x**2 + 3*x + 1) + 55*x**4*sqrt(-2*x**2 + 3*x + 1) + 22*x**3*s qrt(-2*x**2 + 3*x + 1) - 31*x**2*sqrt(-2*x**2 + 3*x + 1) - 16*x*sqrt(-2*x* *2 + 3*x + 1) - 2*sqrt(-2*x**2 + 3*x + 1)), x)
Leaf count of result is larger than twice the leaf count of optimal. 1276 vs. \(2 (137) = 274\).
Time = 0.32 (sec) , antiderivative size = 1276, normalized size of antiderivative = 6.61 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
1/17340*sqrt(10)*(2108*sqrt(10)*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2 *x^2 + 3*x + 1)^(3/2)) - 2108*sqrt(10)*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2 ) - (-2*x^2 + 3*x + 1)^(3/2)) - 56916*sqrt(10)*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) + 56916*sqrt(10)*x/(2*sqrt(10)*sqrt (-2*x^2 + 3*x + 1) - 11*sqrt(-2*x^2 + 3*x + 1)) + 1984*sqrt(10)*x/(sqrt(10 )*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) - 1984*sqrt(10)*x/(sqrt (10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) - 70227*sqrt(10)*arc sin(8/17*sqrt(17)*sqrt(10)*x/abs(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/a bs(6*x + 2*sqrt(10) - 4) - 6/17*sqrt(17)*sqrt(10)/abs(6*x + 2*sqrt(10) - 4 ) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/(2*sqrt(10)*sqrt(sqrt(10) + 1) + 11*sqrt(sqrt(10) + 1)) - 2176*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 + 3*x + 1)^(3/2)) - 2176*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) - (- 2*x^2 + 3*x + 1)^(3/2)) + 58752*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + 11* sqrt(-2*x^2 + 3*x + 1)) + 58752*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11* sqrt(-2*x^2 + 3*x + 1)) - 2048*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(- 2*x^2 + 3*x + 1)) - 2048*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) + 561816*arcsin(8/17*sqrt(17)*sqrt(10)*x/abs(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/abs(6*x + 2*sqrt(10) - 4) - 6/17*sqrt(17)*sqrt(10)/ abs(6*x + 2*sqrt(10) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/(2*s qrt(10)*sqrt(sqrt(10) + 1) + 11*sqrt(sqrt(10) + 1)) - 714*sqrt(10)/(sqr...
Timed out. \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=\int \frac {x+2}{{\left (-2\,x^2+3\,x+1\right )}^{5/2}\,\left (-3\,x^2+4\,x+2\right )} \,d x \]