Integrand size = 30, antiderivative size = 197 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=\frac {2 (21+22 x)}{15 \left (1+3 x+2 x^2\right )^{3/2}}+\frac {2 (273+230 x)}{15 \sqrt {1+3 x+2 x^2}}-\frac {1}{50} \sqrt {\frac {1}{3} \left (4885115+1544809 \sqrt {10}\right )} \text {arctanh}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (17-4 \sqrt {10}\right ) x}{2 \sqrt {55-17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right )+\frac {1}{50} \sqrt {\frac {1}{3} \left (4885115-1544809 \sqrt {10}\right )} \text {arctanh}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (17+4 \sqrt {10}\right ) x}{2 \sqrt {55+17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right ) \]
2/15*(21+22*x)/(2*x^2+3*x+1)^(3/2)+2/15*(273+230*x)/(2*x^2+3*x+1)^(1/2)+1/ 150*arctanh(1/2*(12+3*10^(1/2)+x*(17+4*10^(1/2)))/(2*x^2+3*x+1)^(1/2)/(55+ 17*10^(1/2))^(1/2))*(14655345-4634427*10^(1/2))^(1/2)-1/150*arctanh(1/2*(x *(17-4*10^(1/2))+12-3*10^(1/2))/(2*x^2+3*x+1)^(1/2)/(55-17*10^(1/2))^(1/2) )*(14655345+4634427*10^(1/2))^(1/2)
Time = 0.91 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.78 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {1+3 x+2 x^2} \left (294+1071 x+1236 x^2+460 x^3\right )}{15 (1+x)^2 (1+2 x)^2}-\frac {1}{75} \sqrt {14655345+4634427 \sqrt {10}} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right )+\frac {81 \text {arctanh}\left (\frac {\sqrt {1+\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right )}{5 \sqrt {24425575+7724045 \sqrt {10}}} \]
(2*Sqrt[1 + 3*x + 2*x^2]*(294 + 1071*x + 1236*x^2 + 460*x^3))/(15*(1 + x)^ 2*(1 + 2*x)^2) - (Sqrt[14655345 + 4634427*Sqrt[10]]*ArcTanh[(Sqrt[1 - Sqrt [2/5]]*Sqrt[1 + 3*x + 2*x^2])/(1 + 2*x)])/75 + (81*ArcTanh[(Sqrt[1 + Sqrt[ 2/5]]*Sqrt[1 + 3*x + 2*x^2])/(1 + 2*x)])/(5*Sqrt[24425575 + 7724045*Sqrt[1 0]])
Time = 0.44 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1349, 27, 2135, 27, 1365, 27, 1154, 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 |
| \(\displaystyle \frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}-\frac {2}{45} \int -\frac {3 \left (-264 x^2+271 x+320\right )}{2 \left (-3 x^2+4 x+2\right ) \left (2 x^2+3 x+1\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \int \frac {-264 x^2+271 x+320}{\left (-3 x^2+4 x+2\right ) \left (2 x^2+3 x+1\right )^{3/2}}dx+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}-\frac {2}{15} \int -\frac {45 (346-201 x)}{2 \left (-3 x^2+4 x+2\right ) \sqrt {2 x^2+3 x+1}}dx\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (3 \int \frac {346-201 x}{\left (-3 x^2+4 x+2\right ) \sqrt {2 x^2+3 x+1}}dx+\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {1}{15} \left (3 \left (-\frac {3}{5} \left (335+106 \sqrt {10}\right ) \int \frac {1}{2 \left (-3 x-\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx-\frac {3}{5} \left (335-106 \sqrt {10}\right ) \int \frac {1}{2 \left (-3 x+\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx\right )+\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (3 \left (-\frac {3}{10} \left (335+106 \sqrt {10}\right ) \int \frac {1}{\left (-3 x-\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx-\frac {3}{10} \left (335-106 \sqrt {10}\right ) \int \frac {1}{\left (-3 x+\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx\right )+\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{15} \left (3 \left (\frac {3}{5} \left (335+106 \sqrt {10}\right ) \int \frac {1}{4 \left (55-17 \sqrt {10}\right )-\frac {\left (\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )\right )^2}{2 x^2+3 x+1}}d\left (-\frac {\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{\sqrt {2 x^2+3 x+1}}\right )+\frac {3}{5} \left (335-106 \sqrt {10}\right ) \int \frac {1}{4 \left (55+17 \sqrt {10}\right )-\frac {\left (\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )\right )^2}{2 x^2+3 x+1}}d\left (-\frac {\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{\sqrt {2 x^2+3 x+1}}\right )\right )+\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{15} \left (3 \left (-\frac {3 \left (335+106 \sqrt {10}\right ) \text {arctanh}\left (\frac {\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {55-17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right )}{10 \sqrt {55-17 \sqrt {10}}}-\frac {3 \left (335-106 \sqrt {10}\right ) \text {arctanh}\left (\frac {\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {55+17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right )}{10 \sqrt {55+17 \sqrt {10}}}\right )+\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\) |
(2*(21 + 22*x))/(15*(1 + 3*x + 2*x^2)^(3/2)) + ((2*(273 + 230*x))/Sqrt[1 + 3*x + 2*x^2] + 3*((-3*(335 + 106*Sqrt[10])*ArcTanh[(3*(4 - Sqrt[10]) + (1 7 - 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/(10* Sqrt[55 - 17*Sqrt[10]]) - (3*(335 - 106*Sqrt[10])*ArcTanh[(3*(4 + Sqrt[10] ) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])] )/(10*Sqrt[55 + 17*Sqrt[10]])))/15
3.1.30.3.1 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[(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 = 2.52 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.39
| method | result | size |
| trager | \(\frac {\frac {184}{3} x^{3}+\frac {824}{5} x^{2}+\frac {714}{5} x +\frac {196}{5}}{\left (2 x^{2}+3 x +1\right )^{\frac {3}{2}}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right ) \ln \left (\frac {-504352000 x \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right )+20232850257120 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right ) x +1601657971200 \sqrt {2 x^{2}+3 x +1}\, \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}+11686912631520 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right )-1087910594433 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right ) x +3015957496555836 \sqrt {2 x^{2}+3 x +1}-628400494638 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right )}{1200 x \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-7974733 x -3089618}\right )}{150}-\frac {2 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right ) \ln \left (-\frac {-22695840000 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{5} x -540905633498400 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{3} x +1201243478400 \sqrt {2 x^{2}+3 x +1}\, \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-525911068418400 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{3}+5908432074489101275 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right ) x -12042322347390237 \sqrt {2 x^{2}+3 x +1}+4281865136972328150 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )}{1200 x \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-1795497 x +3089618}\right )}{5}\) | \(471\) |
| default | \(\text {Expression too large to display}\) | \(878\) |
2/15*(460*x^3+1236*x^2+1071*x+294)/(2*x^2+3*x+1)^(3/2)+1/150*RootOf(_Z^2+3 600*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2-29310690)*ln((-504352000*x*Ro otOf(96000*_Z^4-781618400*_Z^2+6561)^4*RootOf(_Z^2+3600*RootOf(96000*_Z^4- 781618400*_Z^2+6561)^2-29310690)+20232850257120*RootOf(96000*_Z^4-78161840 0*_Z^2+6561)^2*RootOf(_Z^2+3600*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2-2 9310690)*x+1601657971200*(2*x^2+3*x+1)^(1/2)*RootOf(96000*_Z^4-781618400*_ Z^2+6561)^2+11686912631520*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2*RootOf (_Z^2+3600*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2-29310690)-108791059443 3*RootOf(_Z^2+3600*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2-29310690)*x+30 15957496555836*(2*x^2+3*x+1)^(1/2)-628400494638*RootOf(_Z^2+3600*RootOf(96 000*_Z^4-781618400*_Z^2+6561)^2-29310690))/(1200*x*RootOf(96000*_Z^4-78161 8400*_Z^2+6561)^2-7974733*x-3089618))-2/5*RootOf(96000*_Z^4-781618400*_Z^2 +6561)*ln(-(-22695840000*RootOf(96000*_Z^4-781618400*_Z^2+6561)^5*x-540905 633498400*RootOf(96000*_Z^4-781618400*_Z^2+6561)^3*x+1201243478400*(2*x^2+ 3*x+1)^(1/2)*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2-525911068418400*Root Of(96000*_Z^4-781618400*_Z^2+6561)^3+5908432074489101275*RootOf(96000*_Z^4 -781618400*_Z^2+6561)*x-12042322347390237*(2*x^2+3*x+1)^(1/2)+428186513697 2328150*RootOf(96000*_Z^4-781618400*_Z^2+6561))/(1200*x*RootOf(96000*_Z^4- 781618400*_Z^2+6561)^2-1795497*x+3089618))
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (141) = 282\).
Time = 0.30 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.21 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=\frac {23520 \, x^{4} + 70560 \, x^{3} + \sqrt {3} {\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt {1544809 \, \sqrt {10} + 4885115} \log \left (-\frac {243 \, \sqrt {10} x + {\left (893 \, \sqrt {10} \sqrt {3} x - 2824 \, \sqrt {3} x\right )} \sqrt {1544809 \, \sqrt {10} + 4885115} + 486 \, x - 486 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 486}{x}\right ) - \sqrt {3} {\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt {1544809 \, \sqrt {10} + 4885115} \log \left (-\frac {243 \, \sqrt {10} x - {\left (893 \, \sqrt {10} \sqrt {3} x - 2824 \, \sqrt {3} x\right )} \sqrt {1544809 \, \sqrt {10} + 4885115} + 486 \, x - 486 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 486}{x}\right ) + \sqrt {3} {\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt {-1544809 \, \sqrt {10} + 4885115} \log \left (\frac {243 \, \sqrt {10} x + {\left (893 \, \sqrt {10} \sqrt {3} x + 2824 \, \sqrt {3} x\right )} \sqrt {-1544809 \, \sqrt {10} + 4885115} - 486 \, x + 486 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 486}{x}\right ) - \sqrt {3} {\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt {-1544809 \, \sqrt {10} + 4885115} \log \left (\frac {243 \, \sqrt {10} x - {\left (893 \, \sqrt {10} \sqrt {3} x + 2824 \, \sqrt {3} x\right )} \sqrt {-1544809 \, \sqrt {10} + 4885115} - 486 \, x + 486 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 486}{x}\right ) + 76440 \, x^{2} + 20 \, {\left (460 \, x^{3} + 1236 \, x^{2} + 1071 \, x + 294\right )} \sqrt {2 \, x^{2} + 3 \, x + 1} + 35280 \, x + 5880}{150 \, {\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )}} \]
1/150*(23520*x^4 + 70560*x^3 + sqrt(3)*(4*x^4 + 12*x^3 + 13*x^2 + 6*x + 1) *sqrt(1544809*sqrt(10) + 4885115)*log(-(243*sqrt(10)*x + (893*sqrt(10)*sqr t(3)*x - 2824*sqrt(3)*x)*sqrt(1544809*sqrt(10) + 4885115) + 486*x - 486*sq rt(2*x^2 + 3*x + 1) + 486)/x) - sqrt(3)*(4*x^4 + 12*x^3 + 13*x^2 + 6*x + 1 )*sqrt(1544809*sqrt(10) + 4885115)*log(-(243*sqrt(10)*x - (893*sqrt(10)*sq rt(3)*x - 2824*sqrt(3)*x)*sqrt(1544809*sqrt(10) + 4885115) + 486*x - 486*s qrt(2*x^2 + 3*x + 1) + 486)/x) + sqrt(3)*(4*x^4 + 12*x^3 + 13*x^2 + 6*x + 1)*sqrt(-1544809*sqrt(10) + 4885115)*log((243*sqrt(10)*x + (893*sqrt(10)*s qrt(3)*x + 2824*sqrt(3)*x)*sqrt(-1544809*sqrt(10) + 4885115) - 486*x + 486 *sqrt(2*x^2 + 3*x + 1) - 486)/x) - sqrt(3)*(4*x^4 + 12*x^3 + 13*x^2 + 6*x + 1)*sqrt(-1544809*sqrt(10) + 4885115)*log((243*sqrt(10)*x - (893*sqrt(10) *sqrt(3)*x + 2824*sqrt(3)*x)*sqrt(-1544809*sqrt(10) + 4885115) - 486*x + 4 86*sqrt(2*x^2 + 3*x + 1) - 486)/x) + 76440*x^2 + 20*(460*x^3 + 1236*x^2 + 1071*x + 294)*sqrt(2*x^2 + 3*x + 1) + 35280*x + 5880)/(4*x^4 + 12*x^3 + 13 *x^2 + 6*x + 1)
\[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=- \int \frac {x}{12 x^{6} \sqrt {2 x^{2} + 3 x + 1} + 20 x^{5} \sqrt {2 x^{2} + 3 x + 1} - 17 x^{4} \sqrt {2 x^{2} + 3 x + 1} - 58 x^{3} \sqrt {2 x^{2} + 3 x + 1} - 47 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 16 x \sqrt {2 x^{2} + 3 x + 1} - 2 \sqrt {2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{12 x^{6} \sqrt {2 x^{2} + 3 x + 1} + 20 x^{5} \sqrt {2 x^{2} + 3 x + 1} - 17 x^{4} \sqrt {2 x^{2} + 3 x + 1} - 58 x^{3} \sqrt {2 x^{2} + 3 x + 1} - 47 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 16 x \sqrt {2 x^{2} + 3 x + 1} - 2 \sqrt {2 x^{2} + 3 x + 1}}\, dx \]
-Integral(x/(12*x**6*sqrt(2*x**2 + 3*x + 1) + 20*x**5*sqrt(2*x**2 + 3*x + 1) - 17*x**4*sqrt(2*x**2 + 3*x + 1) - 58*x**3*sqrt(2*x**2 + 3*x + 1) - 47* 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) + 20*x**5*sqr t(2*x**2 + 3*x + 1) - 17*x**4*sqrt(2*x**2 + 3*x + 1) - 58*x**3*sqrt(2*x**2 + 3*x + 1) - 47*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. \(2 (141) = 282\).
Time = 0.32 (sec) , antiderivative size = 1276, normalized size of antiderivative = 6.48 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
-1/300*sqrt(10)*(980*sqrt(10)*x/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) + 55* (2*x^2 + 3*x + 1)^(3/2)) - 980*sqrt(10)*x/(17*sqrt(10)*(2*x^2 + 3*x + 1)^( 3/2) - 55*(2*x^2 + 3*x + 1)^(3/2)) + 5292*sqrt(10)*x/(374*sqrt(10)*sqrt(2* x^2 + 3*x + 1) + 1183*sqrt(2*x^2 + 3*x + 1)) - 5292*sqrt(10)*x/(374*sqrt(1 0)*sqrt(2*x^2 + 3*x + 1) - 1183*sqrt(2*x^2 + 3*x + 1)) - 15680*sqrt(10)*x/ (17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) + 15680*sqr t(10)*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) + 3 520*x/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) + 55*(2*x^2 + 3*x + 1)^(3/2)) + 3520*x/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) - 55*(2*x^2 + 3*x + 1)^(3/2)) + 19008*x/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 1183*sqrt(2*x^2 + 3*x + 1 )) + 19008*x/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 1183*sqrt(2*x^2 + 3*x + 1)) - 56320*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) - 56320*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1 )) + 750*sqrt(10)/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) + 55*(2*x^2 + 3*x + 1)^(3/2)) - 750*sqrt(10)/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) - 55*(2*x^2 + 3*x + 1)^(3/2)) + 4050*sqrt(10)/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 1 183*sqrt(2*x^2 + 3*x + 1)) - 4050*sqrt(10)/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 1183*sqrt(2*x^2 + 3*x + 1)) - 11760*sqrt(10)/(17*sqrt(10)*sqrt(2*x^ 2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) + 11760*sqrt(10)/(17*sqrt(10)*sqr t(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) + 2760/(17*sqrt(10)*(2*x...
Time = 0.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.61 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (115 \, x + 309\right )} x + 1071\right )} x + 294\right )}}{15 \, {\left (2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}}} + 0.00115890443050800 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} + 5.90976932712000\right ) - 36.0928986365333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.176527156327000\right ) + 36.0928986365333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.919278730509000\right ) - 0.00115890442528267 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 1.04272727395000\right ) \]
2/15*((4*(115*x + 309)*x + 1071)*x + 294)/(2*x^2 + 3*x + 1)^(3/2) + 0.0011 5890443050800*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) + 5.90976932712000) - 36.0928986365333*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 0.1765271563270 00) + 36.0928986365333*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 0.91927873 0509000) - 0.00115890442528267*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 1. 04272727395000)
Timed out. \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=\int \frac {x+2}{{\left (2\,x^2+3\,x+1\right )}^{5/2}\,\left (-3\,x^2+4\,x+2\right )} \,d x \]