3.1.38 \(\int \frac {3+4 x}{\sqrt {-3-4 x-x^2} (3+4 x+2 x^2)} \, dx\) [38]

3.1.38.1 Optimal result

Integrand size = 32, antiderivative size = 86 \[ \int \frac {3+4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\sqrt {2} \arctan \left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )-\sqrt {2} \arctan \left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )+\text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]

output

arctanh(x/(-x^2-4*x-3)^(1/2))+arctan(1/2*(1+(-3-x)/(-x^2-4*x-3)^(1/2))*2^( 
1/2))*2^(1/2)-arctan(1/2*(1+(3+x)/(-x^2-4*x-3)^(1/2))*2^(1/2))*2^(1/2)
 

3.1.38.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.59 \[ \int \frac {3+4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\sqrt {2} \arctan \left (\frac {3+2 x}{\sqrt {2} \sqrt {-3-4 x-x^2}}\right )+\text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]

input

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

output

-(Sqrt[2]*ArcTan[(3 + 2*x)/(Sqrt[2]*Sqrt[-3 - 4*x - x^2])]) + ArcTanh[x/Sq 
rt[-3 - 4*x - x^2]]
 

3.1.38.3 Rubi [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {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.

\(\displaystyle \int \frac {4 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )} \, dx\)

\(\Big \downarrow \) 1361

\(\displaystyle -3 \int \frac {1}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\int -\frac {2 (2 x+3)}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-3 \int \frac {1}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\)

\(\Big \downarrow \) 1317

\(\displaystyle 2 \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-3 \left (\frac {1}{6} \int -\frac {4 x}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\frac {1}{6} \int -\frac {2 (2 x+3)}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-3 \left (\frac {1}{3} \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\frac {2}{3} \int \frac {x}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\right )\)

\(\Big \downarrow \) 1359

\(\displaystyle 2 \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-3 \left (\frac {1}{3} \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\frac {16}{3} \int -\frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{4 \left (\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1\right )}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-3 \left (\frac {1}{3} \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx+\frac {4}{3} \int \frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\right )\)

\(\Big \downarrow \) 1360

\(\displaystyle 6 \int \frac {1}{3-\frac {3 x^2}{-x^2-4 x-3}}d\frac {x}{\sqrt {-x^2-4 x-3}}-3 \left (\int \frac {1}{3-\frac {3 x^2}{-x^2-4 x-3}}d\frac {x}{\sqrt {-x^2-4 x-3}}+\frac {4}{3} \int \frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-3 \left (\frac {4}{3} \int \frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}+\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\right )\)

\(\Big \downarrow \) 1475

\(\displaystyle 2 \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-3 \left (\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-\frac {4}{3} \left (-\frac {1}{6} \int \frac {1}{\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}-\frac {2 (x+3)}{9 \sqrt {-x^2-4 x-3}}+\frac {1}{3}}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}-\frac {1}{6} \int \frac {1}{\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}+\frac {2 (x+3)}{9 \sqrt {-x^2-4 x-3}}+\frac {1}{3}}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\right )\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle 2 \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-3 \left (\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-\frac {4}{3} \left (\frac {1}{3} \int \frac {1}{-\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}-\frac {8}{9}}d\left (\frac {2 (x+3)}{3 \sqrt {-x^2-4 x-3}}-\frac {2}{3}\right )+\frac {1}{3} \int \frac {1}{-\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}-\frac {8}{9}}d\left (\frac {2 (x+3)}{3 \sqrt {-x^2-4 x-3}}+\frac {2}{3}\right )\right )\right )\)

\(\Big \downarrow \) 217

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

input

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

output

-3*((2*Sqrt[2]*ArcTan[(3 + x)/(2*Sqrt[2]*Sqrt[-3 - 4*x - x^2])])/3 + ArcTa 
nh[x/Sqrt[-3 - 4*x - x^2]]/3) + 2*ArcTanh[x/Sqrt[-3 - 4*x - x^2]]
 

3.1.38.3.1 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 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 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 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]))
 

3.1.38.4 Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.43

method	result	size
default	\(\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )-\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{6 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\)	\(123\)
trager	\(-\ln \left (\frac {4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )^{2} x +4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) x +12 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )-6 \sqrt {-x^{2}-4 x -3}-3 x -6}{2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) x -3 x -3}\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )+\operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) \ln \left (\frac {4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )^{2} x -12 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) x -12 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )-6 \sqrt {-x^{2}-4 x -3}+5 x +6}{2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) x +x +3}\right )+\ln \left (\frac {4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )^{2} x +4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) x +12 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )-6 \sqrt {-x^{2}-4 x -3}-3 x -6}{2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) x -3 x -3}\right )\)	\(280\)

input

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

output

1/6*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))
 

3.1.38.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.53 \[ \int \frac {3+4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - \frac {1}{4} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \]

input

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

output

1/2*sqrt(2)*arctan(1/2*(sqrt(2)*x + 3*sqrt(2)*sqrt(-x^2 - 4*x - 3))/(2*x + 
 3)) + 1/2*sqrt(2)*arctan(-1/2*(sqrt(2)*x - 3*sqrt(2)*sqrt(-x^2 - 4*x - 3) 
)/(2*x + 3)) - 1/4*log(-(2*sqrt(-x^2 - 4*x - 3)*x + 4*x + 3)/x^2) + 1/4*lo 
g((2*sqrt(-x^2 - 4*x - 3)*x - 4*x - 3)/x^2)
 

3.1.38.6 Sympy [F]

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

input

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

output

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

3.1.38.7 Maxima [F]

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

input

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

output

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

3.1.38.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (73) = 146\).

Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.90 \[ \int \frac {3+4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) + \frac {1}{2} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) - \frac {1}{2} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]

input

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

output

sqrt(2)*arctan(1/2*sqrt(2)*(3*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) + s 
qrt(2)*arctan(1/2*sqrt(2)*((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) + 1/2* 
log(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 3*(sqrt(-x^2 - 4*x - 3) - 1)^2/ 
(x + 2)^2 + 1) - 1/2*log(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + (sqrt(-x^2 
 - 4*x - 3) - 1)^2/(x + 2)^2 + 3)
 

3.1.38.9 Mupad [F(-1)]

Timed out. \[ \int \frac {3+4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {4\,x+3}{\sqrt {-x^2-4\,x-3}\,\left (2\,x^2+4\,x+3\right )} \,d x \]

input

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

output

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