Integrand size = 30, antiderivative size = 98 \[ \int \frac {x^2}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {1}{2} \arcsin (2+x)-\frac {\arctan \left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {\arctan \left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]
1/2*arcsin(2+x)-1/2*arctanh(x/(-x^2-4*x-3)^(1/2))-1/2*arctan(1/2*(1+(-3-x) /(-x^2-4*x-3)^(1/2))*2^(1/2))*2^(1/2)+1/2*arctan(1/2*(1+(3+x)/(-x^2-4*x-3) ^(1/2))*2^(1/2))*2^(1/2)
Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {\arctan \left (\frac {3+2 x}{\sqrt {2} \sqrt {-3-4 x-x^2}}\right )}{\sqrt {2}}-\arctan \left (\frac {\sqrt {-3-4 x-x^2}}{3+x}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]
ArcTan[(3 + 2*x)/(Sqrt[2]*Sqrt[-3 - 4*x - x^2])]/Sqrt[2] - ArcTan[Sqrt[-3 - 4*x - x^2]/(3 + x)] - ArcTanh[x/Sqrt[-3 - 4*x - x^2]]/2
Time = 0.53 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2143, 25, 1090, 223, 1361, 27, 1317, 27, 1359, 27, 1360, 219, 1475, 1083, 217}
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.
| \(\displaystyle \int \frac {x^2}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )} \, dx\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {-x^2-4 x-3}}dx+\frac {1}{2} \int -\frac {4 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {-x^2-4 x-3}}dx-\frac {1}{2} \int \frac {4 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle -\frac {1}{2} \int \frac {4 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\frac {1}{4} \int \frac {1}{\sqrt {1-\frac {1}{4} (-2 x-4)^2}}d(-2 x-4)\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {1}{2} \int \frac {4 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
| \(\Big \downarrow \) 1361 |
| \(\displaystyle \frac {1}{2} \left (3 \int \frac {1}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx+\int -\frac {2 (2 x+3)}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\right )-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (3 \int \frac {1}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-2 \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\right )-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
| \(\Big \downarrow \) 1317 |
| \(\displaystyle \frac {1}{2} \left (3 \left (\frac {1}{6} \int -\frac {4 x}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\frac {1}{6} \int -\frac {2 (2 x+3)}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\right )-2 \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\right )-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (3 \left (\frac {1}{3} \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\frac {2}{3} \int \frac {x}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\right )-2 \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\right )-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
| \(\Big \downarrow \) 1359 |
| \(\displaystyle \frac {1}{2} \left (3 \left (\frac {1}{3} \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\frac {16}{3} \int -\frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{4 \left (\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1\right )}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\right )-2 \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\right )-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (3 \left (\frac {1}{3} \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx+\frac {4}{3} \int \frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\right )-2 \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\right )-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
| \(\Big \downarrow \) 1360 |
| \(\displaystyle \frac {1}{2} \left (3 \left (\int \frac {1}{3-\frac {3 x^2}{-x^2-4 x-3}}d\frac {x}{\sqrt {-x^2-4 x-3}}+\frac {4}{3} \int \frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\right )-6 \int \frac {1}{3-\frac {3 x^2}{-x^2-4 x-3}}d\frac {x}{\sqrt {-x^2-4 x-3}}\right )-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (3 \left (\frac {4}{3} \int \frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}+\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\right )-2 \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\right )-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
| \(\Big \downarrow \) 1475 |
| \(\displaystyle \frac {1}{2} \left (3 \left (\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-\frac {4}{3} \left (-\frac {1}{6} \int \frac {1}{\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}-\frac {2 (x+3)}{9 \sqrt {-x^2-4 x-3}}+\frac {1}{3}}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}-\frac {1}{6} \int \frac {1}{\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}+\frac {2 (x+3)}{9 \sqrt {-x^2-4 x-3}}+\frac {1}{3}}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\right )\right )-2 \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\right )-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (3 \left (\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-\frac {4}{3} \left (\frac {1}{3} \int \frac {1}{-\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}-\frac {8}{9}}d\left (\frac {2 (x+3)}{3 \sqrt {-x^2-4 x-3}}-\frac {2}{3}\right )+\frac {1}{3} \int \frac {1}{-\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}-\frac {8}{9}}d\left (\frac {2 (x+3)}{3 \sqrt {-x^2-4 x-3}}+\frac {2}{3}\right )\right )\right )-2 \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\right )-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (3 \left (\frac {2}{3} \sqrt {2} \arctan \left (\frac {x+3}{2 \sqrt {2} \sqrt {-x^2-4 x-3}}\right )+\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\right )-2 \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\right )-\frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )\) |
-1/2*ArcSin[(-4 - 2*x)/2] + (3*((2*Sqrt[2]*ArcTan[(3 + x)/(2*Sqrt[2]*Sqrt[ -3 - 4*x - x^2])])/3 + ArcTanh[x/Sqrt[-3 - 4*x - x^2]]/3) - 2*ArcTanh[x/Sq rt[-3 - 4*x - x^2]])/2
3.2.28.3.1 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/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[1/(((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[(c*d - a*f + q + (c*e - b*f)*x)/((a + b*x + c*x^2 )*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[(c*d - a*f - q + (c*e - b*f)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[c*e - b*f, 0] && NegQ[b^2 - 4*a*c]
Int[(x_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (e_.)*(x_) + (f_.) *(x_)^2]), x_Symbol] :> Simp[-2*e Subst[Int[(1 - d*x^2)/(c*e - b*f - e*(2 *c*d - b*e + 2*a*f)*x^2 + d^2*(c*e - b*f)*x^4), x], x, (1 + (e + Sqrt[e^2 - 4*d*f])*(x/(2*d)))/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[b*d - a*e, 0]
Int[((g_) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (e_ .)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[g Subst[Int[1/(a + (c*d - a*f )*x^2), x], 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] && EqQ[b*d - a*e, 0] & & EqQ[2*h*d - g*e, 0]
Int[((g_) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (e_ .)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-(2*h*d - g*e)/e Int[1/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[h/e Int[(2*d + e*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] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[b*d - a*e, 0] && NeQ[2*h*d - g*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^ 2, x], x], x] + Simp[e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; F reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] , 0]))
Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_ .)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[ 1/c Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x ^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
Time = 0.73 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.33
| method | result | size |
| default | \(\frac {\arcsin \left (2+x \right )}{2}-\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )-\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{12 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\) | \(130\) |
| trager | \(\frac {\ln \left (-\frac {16 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right )^{2} x +8 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right ) x +24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right )+6 \sqrt {-x^{2}-4 x -3}-3 x -6}{4 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right ) x -3 x -3}\right )}{2}-\ln \left (-\frac {16 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right )^{2} x +8 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right ) x +24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right )+6 \sqrt {-x^{2}-4 x -3}-3 x -6}{4 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right ) x -3 x -3}\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right )+\operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right ) \ln \left (-\frac {16 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right )^{2} x -24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right ) x -24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right )+6 \sqrt {-x^{2}-4 x -3}+5 x +6}{4 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-8 \textit {\_Z} +3\right ) x +x +3}\right )-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-4 x -3}\right )}{2}\) | \(323\) |
1/2*arcsin(2+x)-1/12*3^(1/2)*4^(1/2)*(3*x^2/(-3/2-x)^2-12)^(1/2)*(2^(1/2)* arctan(1/6*(3*x^2/(-3/2-x)^2-12)^(1/2)*2^(1/2))-arctanh(3*x/(-3/2-x)/(3*x^ 2/(-3/2-x)^2-12)^(1/2)))/((x^2/(-3/2-x)^2-4)/(1+x/(-3/2-x))^2)^(1/2)/(1+x/ (-3/2-x))
Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.64 \[ \int \frac {x^2}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x - 3} {\left (x + 2\right )}}{x^{2} + 4 \, x + 3}\right ) + \frac {1}{8} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - \frac {1}{8} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \]
-1/4*sqrt(2)*arctan(1/2*(sqrt(2)*x + 3*sqrt(2)*sqrt(-x^2 - 4*x - 3))/(2*x + 3)) - 1/4*sqrt(2)*arctan(-1/2*(sqrt(2)*x - 3*sqrt(2)*sqrt(-x^2 - 4*x - 3 ))/(2*x + 3)) - 1/2*arctan(sqrt(-x^2 - 4*x - 3)*(x + 2)/(x^2 + 4*x + 3)) + 1/8*log(-(2*sqrt(-x^2 - 4*x - 3)*x + 4*x + 3)/x^2) - 1/8*log((2*sqrt(-x^2 - 4*x - 3)*x - 4*x - 3)/x^2)
\[ \int \frac {x^2}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {x^{2}}{\sqrt {- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \]
\[ \int \frac {x^2}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int { \frac {x^{2}}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt {-x^{2} - 4 \, x - 3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (82) = 164\).
Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.74 \[ \int \frac {x^2}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) + \frac {1}{2} \, \arcsin \left (x + 2\right ) - \frac {1}{4} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) + \frac {1}{4} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]
-1/2*sqrt(2)*arctan(1/2*sqrt(2)*(3*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1) ) - 1/2*sqrt(2)*arctan(1/2*sqrt(2)*((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1 )) + 1/2*arcsin(x + 2) - 1/4*log(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 3* (sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 1) + 1/4*log(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + (sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 3)
Timed out. \[ \int \frac {x^2}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {x^2}{\sqrt {-x^2-4\,x-3}\,\left (2\,x^2+4\,x+3\right )} \,d x \]