Integrand size = 27, antiderivative size = 98 \[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=-\frac {1}{2} \arcsin (2+x)-\frac {\arctan \left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {\arctan \left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]
-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 = 75, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=\frac {\arctan \left (\frac {3+2 x}{\sqrt {2} \sqrt {-3-4 x-x^2}}\right )}{\sqrt {2}}+\arctan \left (\frac {\sqrt {-3-4 x-x^2}}{3+x}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]
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 = 0.47 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {1320, 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.
\(\displaystyle \int \frac {\sqrt {-x^2-4 x-3}}{2 x^2+4 x+3} \, dx\) |
\(\Big \downarrow \) 1320 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{\sqrt {-x^2-4 x-3}}dx-\frac {1}{2} \int \frac {4 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{4} \int \frac {1}{\sqrt {1-\frac {1}{4} (-2 x-4)^2}}d(-2 x-4)-\frac {1}{2} \int \frac {4 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )-\frac {1}{2} \int \frac {4 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\) |
\(\Big \downarrow \) 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 |
\(\displaystyle \frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )+\frac {1}{2} \left (3 \left (\frac {2}{3} \sqrt {2} \arctan \left (\frac {x+3}{2 \sqrt {2} \sqrt {-x^2-4 x-3}}\right )+\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\right )-2 \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\right )\) |
ArcSin[(-4 - 2*x)/2]/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/Sqrt[ -3 - 4*x - x^2]])/2
3.1.14.3.1 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[Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]/((d_) + (e_.)*(x_) + (f_.)*(x_)^ 2), x_Symbol] :> Simp[c/f Int[1/Sqrt[a + b*x + c*x^2], x], x] - Simp[1/f Int[(c*d - a*f + (c*e - b*f)*x)/(Sqrt[a + b*x + c*x^2]*(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]
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]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.14 (sec) , antiderivative size = 322, normalized size of antiderivative = 3.29
method | result | size |
trager | \(\operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) \ln \left (\frac {16 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )^{2} x +24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x +24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )-6 \sqrt {-x^{2}-4 x -3}+5 x +6}{4 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x -x -3}\right )-\frac {\ln \left (\frac {16 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )^{2} x -8 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x -24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )-6 \sqrt {-x^{2}-4 x -3}-3 x -6}{4 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x +3 x +3}\right )}{2}-\ln \left (\frac {16 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )^{2} x -8 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x -24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )-6 \sqrt {-x^{2}-4 x -3}-3 x -6}{4 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x +3 x +3}\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-4 x -3}\right )}{2}\) | \(322\) |
default | \(-\frac {\arcsin \left (2+x \right )}{2}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )-\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{12 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}-\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )}{3 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )+\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{6 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\) | \(341\) |
RootOf(16*_Z^2+8*_Z+3)*ln((16*RootOf(16*_Z^2+8*_Z+3)^2*x+24*RootOf(16*_Z^2 +8*_Z+3)*x+24*RootOf(16*_Z^2+8*_Z+3)-6*(-x^2-4*x-3)^(1/2)+5*x+6)/(4*RootOf (16*_Z^2+8*_Z+3)*x-x-3))-1/2*ln((16*RootOf(16*_Z^2+8*_Z+3)^2*x-8*RootOf(16 *_Z^2+8*_Z+3)*x-24*RootOf(16*_Z^2+8*_Z+3)-6*(-x^2-4*x-3)^(1/2)-3*x-6)/(4*R ootOf(16*_Z^2+8*_Z+3)*x+3*x+3))-ln((16*RootOf(16*_Z^2+8*_Z+3)^2*x-8*RootOf (16*_Z^2+8*_Z+3)*x-24*RootOf(16*_Z^2+8*_Z+3)-6*(-x^2-4*x-3)^(1/2)-3*x-6)/( 4*RootOf(16*_Z^2+8*_Z+3)*x+3*x+3))*RootOf(16*_Z^2+8*_Z+3)+1/2*RootOf(_Z^2+ 1)*ln(RootOf(_Z^2+1)*x+2*RootOf(_Z^2+1)+(-x^2-4*x-3)^(1/2))
Time = 0.30 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x - 3} {\left (x + 2\right )}}{x^{2} + 4 \, x + 3}\right ) + \frac {1}{8} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - \frac {1}{8} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \]
-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)
\[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=\int \frac {\sqrt {- \left (x + 1\right ) \left (x + 3\right )}}{2 x^{2} + 4 x + 3}\, dx \]
\[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=\int { \frac {\sqrt {-x^{2} - 4 \, x - 3}}{2 \, x^{2} + 4 \, x + 3} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (82) = 164\).
Time = 0.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.74 \[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) - \frac {1}{2} \, \arcsin \left (x + 2\right ) - \frac {1}{4} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) + \frac {1}{4} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]
-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. \[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=\int \frac {\sqrt {-x^2-4\,x-3}}{2\,x^2+4\,x+3} \,d x \]