\(\int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx\) [146]

Optimal result

Integrand size = 27, antiderivative size = 98 \[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=-\frac {1}{2} \arcsin (2+x)-\frac {\arctan \left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {\arctan \left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \] Output:

-1/2*arcsin(2+x)-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)-1/2*arcta 
nh(x/(-x^2-4*x-3)^(1/2))
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=\frac {\arctan \left (\frac {3+2 x}{\sqrt {2} \sqrt {-3-4 x-x^2}}\right )}{\sqrt {2}}+\arctan \left (\frac {\sqrt {-3-4 x-x^2}}{3+x}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \] Input:

Integrate[Sqrt[-3 - 4*x - x^2]/(3 + 4*x + 2*x^2),x]
 

Output:

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
 

Rubi [A] (warning: unable to verify)

Time = 0.48 (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

\(\displaystyle \frac {1}{2} \arcsin \left (\frac {1}{2} (-2 x-4)\right )+\frac {1}{2} \left (3 \left (\frac {2}{3} \sqrt {2} \arctan \left (\frac {x+3}{2 \sqrt {2} \sqrt {-x^2-4 x-3}}\right )+\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\right )-2 \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\right )\)

Input:

Int[Sqrt[-3 - 4*x - x^2]/(3 + 4*x + 2*x^2),x]
 

Output:

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
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 219

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])
 

rule 223

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]
 

rule 1083

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]
 

rule 1090

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]
 

rule 1317

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]
 

rule 1320

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]
 

rule 1359

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]
 

rule 1360

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]
 

rule 1361

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]
 

rule 1475

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]))
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.24 (sec) , antiderivative size = 325, normalized size of antiderivative = 3.32

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-4 x -3}\right )}{2}+\operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) \ln \left (-\frac {-16 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )^{2} x -24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x +6 \sqrt {-x^{2}-4 x -3}-24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )-5 x -6}{4 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x -x -3}\right )-\frac {\ln \left (-\frac {-16 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )^{2} x +8 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x +6 \sqrt {-x^{2}-4 x -3}+24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )+3 x +6}{4 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x +3 x +3}\right )}{2}-\ln \left (-\frac {-16 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )^{2} x +8 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x +6 \sqrt {-x^{2}-4 x -3}+24 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )+3 x +6}{4 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right ) x +3 x +3}\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{2}+8 \textit {\_Z} +3\right )\)	\(325\)
default	\(-\frac {\arcsin \left (2+x \right )}{2}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )-\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{12 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}-\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )}{3 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )+\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{6 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\)	\(341\)

Input:

int((-x^2-4*x-3)^(1/2)/(2*x^2+4*x+3),x,method=_RETURNVERBOSE)
 

Output:

1/2*RootOf(_Z^2+1)*ln(RootOf(_Z^2+1)*x+2*RootOf(_Z^2+1)+(-x^2-4*x-3)^(1/2) 
)+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+6*(-x^2-4*x-3)^(1/2)-24*RootOf(16*_Z^2+8*_Z+3)-5*x-6)/(4*Ro 
otOf(16*_Z^2+8*_Z+3)*x-x-3))-1/2*ln(-(-16*RootOf(16*_Z^2+8*_Z+3)^2*x+8*Roo 
tOf(16*_Z^2+8*_Z+3)*x+6*(-x^2-4*x-3)^(1/2)+24*RootOf(16*_Z^2+8*_Z+3)+3*x+6 
)/(4*RootOf(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+6*(-x^2-4*x-3)^(1/2)+24*RootOf(16*_Z^2+8*_Z+3)+ 
3*x+6)/(4*RootOf(16*_Z^2+8*_Z+3)*x+3*x+3))*RootOf(16*_Z^2+8*_Z+3)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x - 3} {\left (x + 2\right )}}{x^{2} + 4 \, x + 3}\right ) + \frac {1}{8} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - \frac {1}{8} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \] Input:

integrate((-x^2-4*x-3)^(1/2)/(2*x^2+4*x+3),x, algorithm="fricas")
 

Output:

-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)
 

Sympy [F]

\[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=\int \frac {\sqrt {- \left (x + 1\right ) \left (x + 3\right )}}{2 x^{2} + 4 x + 3}\, dx \] Input:

integrate((-x**2-4*x-3)**(1/2)/(2*x**2+4*x+3),x)
 

Output:

Integral(sqrt(-(x + 1)*(x + 3))/(2*x**2 + 4*x + 3), x)
 

Maxima [F]

\[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=\int { \frac {\sqrt {-x^{2} - 4 \, x - 3}}{2 \, x^{2} + 4 \, x + 3} \,d x } \] Input:

integrate((-x^2-4*x-3)^(1/2)/(2*x^2+4*x+3),x, algorithm="maxima")
 

Output:

integrate(sqrt(-x^2 - 4*x - 3)/(2*x^2 + 4*x + 3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (82) = 164\).

Time = 0.12 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.74 \[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) - \frac {1}{2} \, \arcsin \left (x + 2\right ) - \frac {1}{4} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) + \frac {1}{4} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \] Input:

integrate((-x^2-4*x-3)^(1/2)/(2*x^2+4*x+3),x, algorithm="giac")
 

Output:

-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)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=\int \frac {\sqrt {-x^2-4\,x-3}}{2\,x^2+4\,x+3} \,d x \] Input:

int((- 4*x - x^2 - 3)^(1/2)/(4*x + 2*x^2 + 3),x)
 

Output:

int((- 4*x - x^2 - 3)^(1/2)/(4*x + 2*x^2 + 3), x)
 

Reduce [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx=-\frac {\mathit {asin} \left (x +2\right )}{2}+\frac {\sqrt {2}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )-1}{\sqrt {2}}\right )}{2}+\frac {\sqrt {2}\, \mathit {atan} \left (\frac {3 \tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )-1}{\sqrt {2}}\right )}{2}+\frac {\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )^{2}-2 \tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )+3\right )}{4}-\frac {\mathrm {log}\left (3 \tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )^{2}-2 \tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )+1\right )}{4} \] Input:

int((-x^2-4*x-3)^(1/2)/(2*x^2+4*x+3),x)
 

Output:

( - 2*asin(x + 2) + 2*sqrt(2)*atan((tan(asin(x + 2)/2) - 1)/sqrt(2)) + 2*s 
qrt(2)*atan((3*tan(asin(x + 2)/2) - 1)/sqrt(2)) + log(tan(asin(x + 2)/2)** 
2 - 2*tan(asin(x + 2)/2) + 3) - log(3*tan(asin(x + 2)/2)**2 - 2*tan(asin(x 
 + 2)/2) + 1))/4