Integrand size = 30, antiderivative size = 174 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{3/2}} \, dx=\frac {2 (21+22 x)}{5 \sqrt {1+3 x+2 x^2}}-\frac {1}{10} \sqrt {\frac {3}{5} \left (2065+653 \sqrt {10}\right )} \text {arctanh}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (17-4 \sqrt {10}\right ) x}{2 \sqrt {55-17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right )+\frac {1}{10} \sqrt {\frac {3}{5} \left (2065-653 \sqrt {10}\right )} \text {arctanh}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (17+4 \sqrt {10}\right ) x}{2 \sqrt {55+17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right ) \]
2/5*(21+22*x)/(2*x^2+3*x+1)^(1/2)+1/50*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))*(30975-9795*10^(1/2 ))^(1/2)-1/50*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))*(30975+9795*10^(1/2))^(1/2)
Time = 0.82 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.75 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{3/2}} \, dx=\frac {1}{25} \left (\frac {5 (42+44 x)}{\sqrt {1+3 x+2 x^2}}-\sqrt {30975+9795 \sqrt {10}} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right )+\frac {45 \text {arctanh}\left (\frac {\sqrt {1+\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right )}{\sqrt {2065+653 \sqrt {10}}}\right ) \]
((5*(42 + 44*x))/Sqrt[1 + 3*x + 2*x^2] - Sqrt[30975 + 9795*Sqrt[10]]*ArcTa nh[(Sqrt[1 - Sqrt[2/5]]*Sqrt[1 + 3*x + 2*x^2])/(1 + 2*x)] + (45*ArcTanh[(S qrt[1 + Sqrt[2/5]]*Sqrt[1 + 3*x + 2*x^2])/(1 + 2*x)])/Sqrt[2065 + 653*Sqrt [10]])/25
Time = 0.33 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1349, 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.
| \(\displaystyle \int \frac {x+2}{\left (-3 x^2+4 x+2\right ) \left (2 x^2+3 x+1\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1349 |
| \(\displaystyle \frac {2 (22 x+21)}{5 \sqrt {2 x^2+3 x+1}}-\frac {2}{15} \int -\frac {9 (16-9 x)}{2 \left (-3 x^2+4 x+2\right ) \sqrt {2 x^2+3 x+1}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \int \frac {16-9 x}{\left (-3 x^2+4 x+2\right ) \sqrt {2 x^2+3 x+1}}dx+\frac {2 (22 x+21)}{5 \sqrt {2 x^2+3 x+1}}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {3}{5} \left (-3 \left (3+\sqrt {10}\right ) \int \frac {1}{2 \left (-3 x-\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx-3 \left (3-\sqrt {10}\right ) \int \frac {1}{2 \left (-3 x+\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx\right )+\frac {2 (22 x+21)}{5 \sqrt {2 x^2+3 x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \left (-\frac {3}{2} \left (3+\sqrt {10}\right ) \int \frac {1}{\left (-3 x-\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx-\frac {3}{2} \left (3-\sqrt {10}\right ) \int \frac {1}{\left (-3 x+\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx\right )+\frac {2 (22 x+21)}{5 \sqrt {2 x^2+3 x+1}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {3}{5} \left (3 \left (3+\sqrt {10}\right ) \int \frac {1}{4 \left (55-17 \sqrt {10}\right )-\frac {\left (\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )\right )^2}{2 x^2+3 x+1}}d\left (-\frac {\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{\sqrt {2 x^2+3 x+1}}\right )+3 \left (3-\sqrt {10}\right ) \int \frac {1}{4 \left (55+17 \sqrt {10}\right )-\frac {\left (\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )\right )^2}{2 x^2+3 x+1}}d\left (-\frac {\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{\sqrt {2 x^2+3 x+1}}\right )\right )+\frac {2 (22 x+21)}{5 \sqrt {2 x^2+3 x+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{5} \left (-\frac {3 \left (3+\sqrt {10}\right ) \text {arctanh}\left (\frac {\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {55-17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right )}{2 \sqrt {55-17 \sqrt {10}}}-\frac {3 \left (3-\sqrt {10}\right ) \text {arctanh}\left (\frac {\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {55+17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right )}{2 \sqrt {55+17 \sqrt {10}}}\right )+\frac {2 (22 x+21)}{5 \sqrt {2 x^2+3 x+1}}\) |
(2*(21 + 22*x))/(5*Sqrt[1 + 3*x + 2*x^2]) + (3*((-3*(3 + 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])])/(2*Sqrt[55 - 17*Sqrt[10]]) - (3*(3 - 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])])/(2*Sqrt[55 + 17*Sqrt[10]])))/5
3.1.29.3.1 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]
Time = 2.46 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.20
| method | result | size |
| risch | \(\frac {\frac {42}{5}+\frac {44 x}{5}}{\sqrt {2 x^{2}+3 x +1}}-\frac {9 \left (10+3 \sqrt {10}\right ) \sqrt {10}\, \operatorname {arctanh}\left (\frac {55-17 \sqrt {10}+\frac {9 \left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {55-17 \sqrt {10}}\, \sqrt {18 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )+55-17 \sqrt {10}}}\right )}{100 \sqrt {55-17 \sqrt {10}}}-\frac {9 \left (-10+3 \sqrt {10}\right ) \sqrt {10}\, \operatorname {arctanh}\left (\frac {55+17 \sqrt {10}+\frac {9 \left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {55+17 \sqrt {10}}\, \sqrt {18 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )+55+17 \sqrt {10}}}\right )}{100 \sqrt {55+17 \sqrt {10}}}\) | \(209\) |
| trager | \(\frac {\frac {42}{5}+\frac {44 x}{5}}{\sqrt {2 x^{2}+3 x +1}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+144 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2}-2478\right ) \ln \left (\frac {928000 x \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+144 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2}-2478\right )-78692160 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+144 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2}-2478\right ) x -45448800 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+144 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2}-2478\right )+5015040 \sqrt {2 x^{2}+3 x +1}\, \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2}+793611 \operatorname {RootOf}\left (\textit {\_Z}^{2}+144 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2}-2478\right ) x +458406 \operatorname {RootOf}\left (\textit {\_Z}^{2}+144 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2}-2478\right )+19958292 \sqrt {2 x^{2}+3 x +1}}{240 x \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2}-3371 x -1306}\right )}{10}+\frac {6 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right ) \ln \left (-\frac {8352000 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{5} x +420781440 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{3} x +3761280 \sqrt {2 x^{2}+3 x +1}\, \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2}+409039200 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{3}-9707055281 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right ) x -79694079 \sqrt {2 x^{2}+3 x +1}-7034757246 \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )}{240 x \operatorname {RootOf}\left (3840 \textit {\_Z}^{4}-66080 \textit {\_Z}^{2}+9\right )^{2}-759 x +1306}\right )}{5}\) | \(461\) |
| default | \(-\frac {\left (8+\sqrt {10}\right ) \sqrt {10}\, \left (\frac {1}{3 \left (\frac {55}{9}+\frac {17 \sqrt {10}}{9}\right ) \sqrt {2 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+\left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )+\frac {55}{9}+\frac {17 \sqrt {10}}{9}}}-\frac {\left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (3+4 x \right )}{3 \left (\frac {55}{9}+\frac {17 \sqrt {10}}{9}\right ) \left (\frac {440}{9}+\frac {136 \sqrt {10}}{9}-\left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right )^{2}\right ) \sqrt {2 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+\left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )+\frac {55}{9}+\frac {17 \sqrt {10}}{9}}}-\frac {\operatorname {arctanh}\left (\frac {55+17 \sqrt {10}+\frac {9 \left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {55+17 \sqrt {10}}\, \sqrt {18 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )+55+17 \sqrt {10}}}\right )}{\left (\frac {55}{9}+\frac {17 \sqrt {10}}{9}\right ) \sqrt {55+17 \sqrt {10}}}\right )}{20}-\frac {\left (-8+\sqrt {10}\right ) \sqrt {10}\, \left (\frac {1}{3 \left (\frac {55}{9}-\frac {17 \sqrt {10}}{9}\right ) \sqrt {2 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+\left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )+\frac {55}{9}-\frac {17 \sqrt {10}}{9}}}-\frac {\left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (3+4 x \right )}{3 \left (\frac {55}{9}-\frac {17 \sqrt {10}}{9}\right ) \left (\frac {440}{9}-\frac {136 \sqrt {10}}{9}-\left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right )^{2}\right ) \sqrt {2 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+\left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )+\frac {55}{9}-\frac {17 \sqrt {10}}{9}}}-\frac {\operatorname {arctanh}\left (\frac {55-17 \sqrt {10}+\frac {9 \left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {55-17 \sqrt {10}}\, \sqrt {18 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )+55-17 \sqrt {10}}}\right )}{\left (\frac {55}{9}-\frac {17 \sqrt {10}}{9}\right ) \sqrt {55-17 \sqrt {10}}}\right )}{20}\) | \(466\) |
2/5*(21+22*x)/(2*x^2+3*x+1)^(1/2)-9/100*(10+3*10^(1/2))*10^(1/2)/(55-17*10 ^(1/2))^(1/2)*arctanh(9/2*(110/9-34/9*10^(1/2)+(17/3-4/3*10^(1/2))*(x-2/3+ 1/3*10^(1/2)))/(55-17*10^(1/2))^(1/2)/(18*(x-2/3+1/3*10^(1/2))^2+9*(17/3-4 /3*10^(1/2))*(x-2/3+1/3*10^(1/2))+55-17*10^(1/2))^(1/2))-9/100*(-10+3*10^( 1/2))*10^(1/2)/(55+17*10^(1/2))^(1/2)*arctanh(9/2*(110/9+34/9*10^(1/2)+(17 /3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(55+17*10^(1/2))^(1/2)/(18*(x-2/3-1 /3*10^(1/2))^2+9*(17/3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))+55+17*10^(1/2))^ (1/2))
Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (122) = 244\).
Time = 0.32 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.10 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {1959 \, \sqrt {10} + 6195} \log \left (-\frac {45 \, \sqrt {10} x + {\left (41 \, \sqrt {10} \sqrt {5} x - 130 \, \sqrt {5} x\right )} \sqrt {1959 \, \sqrt {10} + 6195} + 90 \, x - 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 90}{x}\right ) - \sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {1959 \, \sqrt {10} + 6195} \log \left (-\frac {45 \, \sqrt {10} x - {\left (41 \, \sqrt {10} \sqrt {5} x - 130 \, \sqrt {5} x\right )} \sqrt {1959 \, \sqrt {10} + 6195} + 90 \, x - 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 90}{x}\right ) + \sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {-1959 \, \sqrt {10} + 6195} \log \left (\frac {45 \, \sqrt {10} x + {\left (41 \, \sqrt {10} \sqrt {5} x + 130 \, \sqrt {5} x\right )} \sqrt {-1959 \, \sqrt {10} + 6195} - 90 \, x + 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 90}{x}\right ) - \sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {-1959 \, \sqrt {10} + 6195} \log \left (\frac {45 \, \sqrt {10} x - {\left (41 \, \sqrt {10} \sqrt {5} x + 130 \, \sqrt {5} x\right )} \sqrt {-1959 \, \sqrt {10} + 6195} - 90 \, x + 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 90}{x}\right ) + 840 \, x^{2} + 20 \, \sqrt {2 \, x^{2} + 3 \, x + 1} {\left (22 \, x + 21\right )} + 1260 \, x + 420}{50 \, {\left (2 \, x^{2} + 3 \, x + 1\right )}} \]
1/50*(sqrt(5)*(2*x^2 + 3*x + 1)*sqrt(1959*sqrt(10) + 6195)*log(-(45*sqrt(1 0)*x + (41*sqrt(10)*sqrt(5)*x - 130*sqrt(5)*x)*sqrt(1959*sqrt(10) + 6195) + 90*x - 90*sqrt(2*x^2 + 3*x + 1) + 90)/x) - sqrt(5)*(2*x^2 + 3*x + 1)*sqr t(1959*sqrt(10) + 6195)*log(-(45*sqrt(10)*x - (41*sqrt(10)*sqrt(5)*x - 130 *sqrt(5)*x)*sqrt(1959*sqrt(10) + 6195) + 90*x - 90*sqrt(2*x^2 + 3*x + 1) + 90)/x) + sqrt(5)*(2*x^2 + 3*x + 1)*sqrt(-1959*sqrt(10) + 6195)*log((45*sq rt(10)*x + (41*sqrt(10)*sqrt(5)*x + 130*sqrt(5)*x)*sqrt(-1959*sqrt(10) + 6 195) - 90*x + 90*sqrt(2*x^2 + 3*x + 1) - 90)/x) - sqrt(5)*(2*x^2 + 3*x + 1 )*sqrt(-1959*sqrt(10) + 6195)*log((45*sqrt(10)*x - (41*sqrt(10)*sqrt(5)*x + 130*sqrt(5)*x)*sqrt(-1959*sqrt(10) + 6195) - 90*x + 90*sqrt(2*x^2 + 3*x + 1) - 90)/x) + 840*x^2 + 20*sqrt(2*x^2 + 3*x + 1)*(22*x + 21) + 1260*x + 420)/(2*x^2 + 3*x + 1)
\[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{3/2}} \, dx=- \int \frac {x}{6 x^{4} \sqrt {2 x^{2} + 3 x + 1} + x^{3} \sqrt {2 x^{2} + 3 x + 1} - 13 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 10 x \sqrt {2 x^{2} + 3 x + 1} - 2 \sqrt {2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{6 x^{4} \sqrt {2 x^{2} + 3 x + 1} + x^{3} \sqrt {2 x^{2} + 3 x + 1} - 13 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 10 x \sqrt {2 x^{2} + 3 x + 1} - 2 \sqrt {2 x^{2} + 3 x + 1}}\, dx \]
-Integral(x/(6*x**4*sqrt(2*x**2 + 3*x + 1) + x**3*sqrt(2*x**2 + 3*x + 1) - 13*x**2*sqrt(2*x**2 + 3*x + 1) - 10*x*sqrt(2*x**2 + 3*x + 1) - 2*sqrt(2*x **2 + 3*x + 1)), x) - Integral(2/(6*x**4*sqrt(2*x**2 + 3*x + 1) + x**3*sqr t(2*x**2 + 3*x + 1) - 13*x**2*sqrt(2*x**2 + 3*x + 1) - 10*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. 668 vs. \(2 (122) = 244\).
Time = 0.30 (sec) , antiderivative size = 668, normalized size of antiderivative = 3.84 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{3/2}} \, dx=-\frac {1}{60} \, \sqrt {10} {\left (\frac {588 \, \sqrt {10} x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {588 \, \sqrt {10} x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + \frac {2112 \, x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + \frac {2112 \, x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {27 \, \sqrt {10} \log \left (\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {17 \, \sqrt {10} + 55}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{{\left (17 \, \sqrt {10} + 55\right )}^{\frac {3}{2}}} - \frac {\sqrt {10} \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}}{{\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{{\left (-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}\right )}^{\frac {3}{2}}} + \frac {450 \, \sqrt {10}}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {450 \, \sqrt {10}}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {216 \, \log \left (\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {17 \, \sqrt {10} + 55}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{{\left (17 \, \sqrt {10} + 55\right )}^{\frac {3}{2}}} + \frac {8 \, \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}}{{\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{{\left (-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}\right )}^{\frac {3}{2}}} + \frac {1656}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + \frac {1656}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}}\right )} \]
-1/60*sqrt(10)*(588*sqrt(10)*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqr t(2*x^2 + 3*x + 1)) - 588*sqrt(10)*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) + 2112*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55 *sqrt(2*x^2 + 3*x + 1)) + 2112*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*s qrt(2*x^2 + 3*x + 1)) - 27*sqrt(10)*log(2/9*sqrt(10) + 2/3*sqrt(2*x^2 + 3* x + 1)*sqrt(17*sqrt(10) + 55)/abs(6*x - 2*sqrt(10) - 4) + 34/9*sqrt(10)/ab s(6*x - 2*sqrt(10) - 4) + 110/9/abs(6*x - 2*sqrt(10) - 4) + 17/18)/(17*sqr t(10) + 55)^(3/2) - sqrt(10)*log(-2/9*sqrt(10) + 2*sqrt(2*x^2 + 3*x + 1)*s qrt(-17/9*sqrt(10) + 55/9)/abs(6*x + 2*sqrt(10) - 4) - 34/9*sqrt(10)/abs(6 *x + 2*sqrt(10) - 4) + 110/9/abs(6*x + 2*sqrt(10) - 4) + 17/18)/(-17/9*sqr t(10) + 55/9)^(3/2) + 450*sqrt(10)/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55 *sqrt(2*x^2 + 3*x + 1)) - 450*sqrt(10)/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) - 216*log(2/9*sqrt(10) + 2/3*sqrt(2*x^2 + 3*x + 1)*sqrt(17*sqrt(10) + 55)/abs(6*x - 2*sqrt(10) - 4) + 34/9*sqrt(10)/abs( 6*x - 2*sqrt(10) - 4) + 110/9/abs(6*x - 2*sqrt(10) - 4) + 17/18)/(17*sqrt( 10) + 55)^(3/2) + 8*log(-2/9*sqrt(10) + 2*sqrt(2*x^2 + 3*x + 1)*sqrt(-17/9 *sqrt(10) + 55/9)/abs(6*x + 2*sqrt(10) - 4) - 34/9*sqrt(10)/abs(6*x + 2*sq rt(10) - 4) + 110/9/abs(6*x + 2*sqrt(10) - 4) + 17/18)/(-17/9*sqrt(10) + 5 5/9)^(3/2) + 1656/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) + 1656/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x ...
Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.64 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (22 \, x + 21\right )}}{5 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + 0.0140045514133333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} + 5.90976932712000\right ) - 4.97793168620000 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.176527156327000\right ) + 4.97793168620000 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.919278730509000\right ) - 0.0140045514125333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 1.04272727395000\right ) \]
2/5*(22*x + 21)/sqrt(2*x^2 + 3*x + 1) + 0.0140045514133333*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) + 5.90976932712000) - 4.97793168620000*log(-sqrt(2 )*x + sqrt(2*x^2 + 3*x + 1) - 0.176527156327000) + 4.97793168620000*log(-s qrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 0.919278730509000) - 0.0140045514125333 *log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 1.04272727395000)
Timed out. \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{3/2}} \, dx=\int \frac {x+2}{{\left (2\,x^2+3\,x+1\right )}^{3/2}\,\left (-3\,x^2+4\,x+2\right )} \,d x \]