\(\int \frac {1}{x^2 \sqrt {-3-4 x-x^2} (3+4 x+2 x^2)} \, dx\) [132]

Optimal result

Integrand size = 30, antiderivative size = 151 \[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \arctan \left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {2}{27} \sqrt {2} \arctan \left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )-\frac {2}{27} \sqrt {2} \arctan \left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )+\frac {10}{27} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]

[Out]

10/27*arctanh(x/(-x^2-4*x-3)^(1/2))+2/27*arctan(1/2*(1+(-3-x)/(-x^2-4*x-3)^(1/2))*2^(1/2))*2^(1/2)-2/27*arctan
(1/2*(1+(3+x)/(-x^2-4*x-3)^(1/2))*2^(1/2))*2^(1/2)+2/9*arctan(1/3*(3+2*x)*3^(1/2)/(-x^2-4*x-3)^(1/2))*3^(1/2)+
1/9*(-x^2-4*x-3)^(1/2)/x

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6860, 744, 738, 210, 1042, 1000, 12, 1040, 1175, 632, 1041, 212} \[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {2 \arctan \left (\frac {2 x+3}{\sqrt {3} \sqrt {-x^2-4 x-3}}\right )}{3 \sqrt {3}}+\frac {2}{27} \sqrt {2} \arctan \left (\frac {1-\frac {x+3}{\sqrt {-x^2-4 x-3}}}{\sqrt {2}}\right )-\frac {2}{27} \sqrt {2} \arctan \left (\frac {\frac {x+3}{\sqrt {-x^2-4 x-3}}+1}{\sqrt {2}}\right )+\frac {10}{27} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )+\frac {\sqrt {-x^2-4 x-3}}{9 x} \]

[In]

Int[1/(x^2*Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]

[Out]

Sqrt[-3 - 4*x - x^2]/(9*x) + (2*ArcTan[(3 + 2*x)/(Sqrt[3]*Sqrt[-3 - 4*x - x^2])])/(3*Sqrt[3]) + (2*Sqrt[2]*Arc
Tan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]])/27 - (2*Sqrt[2]*ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqr
t[2]])/27 + (10*ArcTanh[x/Sqrt[-3 - 4*x - x^2]])/27

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 210

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 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-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] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 744

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*
((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e
^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c,
0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0]

Rule 1000

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]}, Dist[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] - Dist[1/(2*q), Int[(c*d - a*f - q + (c*e - b*f)*x)/((a + b*x + c*x^2)*Sq
rt[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 1040

Int[(x_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-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] && N
eQ[e^2 - 4*d*f, 0] && EqQ[b*d - a*e, 0]

Rule 1041

Int[((g_) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol]
:> Dist[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 1042

Int[((g_) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol]
:> Dist[-(2*h*d - g*e)/e, Int[1/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[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 1175

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e) - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /
; FreeQ[{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] || ( !Lt
Q[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{3 x^2 \sqrt {-3-4 x-x^2}}-\frac {4}{9 x \sqrt {-3-4 x-x^2}}+\frac {2 (5+4 x)}{9 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )}\right ) \, dx \\ & = \frac {2}{9} \int \frac {5+4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx+\frac {1}{3} \int \frac {1}{x^2 \sqrt {-3-4 x-x^2}} \, dx-\frac {4}{9} \int \frac {1}{x \sqrt {-3-4 x-x^2}} \, dx \\ & = \frac {\sqrt {-3-4 x-x^2}}{9 x}-\frac {2}{9} \int \frac {1}{x \sqrt {-3-4 x-x^2}} \, dx-\frac {2}{9} \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {2}{9} \int \frac {-6-4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx+\frac {8}{9} \text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,\frac {-6-4 x}{\sqrt {-3-4 x-x^2}}\right ) \\ & = \frac {\sqrt {-3-4 x-x^2}}{9 x}+\frac {4 \tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{9 \sqrt {3}}+\frac {1}{27} \int \frac {-6-4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {1}{27} \int -\frac {4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx+\frac {4}{9} \text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,\frac {-6-4 x}{\sqrt {-3-4 x-x^2}}\right )+\frac {4}{3} \text {Subst}\left (\int \frac {1}{3-3 x^2} \, dx,x,\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \\ & = \frac {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {4}{9} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {4}{27} \int \frac {x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {2}{9} \text {Subst}\left (\int \frac {1}{3-3 x^2} \, dx,x,\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \\ & = \frac {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {10}{27} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {32}{27} \text {Subst}\left (\int \frac {1+3 x^2}{-4-8 x^2-36 x^4} \, dx,x,\frac {1+\frac {x}{3}}{\sqrt {-3-4 x-x^2}}\right ) \\ & = \frac {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {10}{27} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )-\frac {4}{81} \text {Subst}\left (\int \frac {1}{\frac {1}{3}-\frac {2 x}{3}+x^2} \, dx,x,\frac {1+\frac {x}{3}}{\sqrt {-3-4 x-x^2}}\right )-\frac {4}{81} \text {Subst}\left (\int \frac {1}{\frac {1}{3}+\frac {2 x}{3}+x^2} \, dx,x,\frac {1+\frac {x}{3}}{\sqrt {-3-4 x-x^2}}\right ) \\ & = \frac {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {10}{27} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {8}{81} \text {Subst}\left (\int \frac {1}{-\frac {8}{9}-x^2} \, dx,x,\frac {2}{3} \left (-1+\frac {3+x}{\sqrt {-3-4 x-x^2}}\right )\right )+\frac {8}{81} \text {Subst}\left (\int \frac {1}{-\frac {8}{9}-x^2} \, dx,x,\frac {2}{3} \left (1+\frac {3+x}{\sqrt {-3-4 x-x^2}}\right )\right ) \\ & = \frac {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {2}{27} \sqrt {2} \tan ^{-1}\left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )-\frac {2}{27} \sqrt {2} \tan ^{-1}\left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )+\frac {10}{27} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {-2 \sqrt {2} x \arctan \left (\frac {3+2 x}{\sqrt {2} \sqrt {-3-4 x-x^2}}\right )+3 \left (\sqrt {-3-4 x-x^2}-4 \sqrt {3} x \arctan \left (\frac {\sqrt {3} \sqrt {-3-4 x-x^2}}{3+x}\right )\right )+10 x \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )}{27 x} \]

[In]

Integrate[1/(x^2*Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]

[Out]

(-2*Sqrt[2]*x*ArcTan[(3 + 2*x)/(Sqrt[2]*Sqrt[-3 - 4*x - x^2])] + 3*(Sqrt[-3 - 4*x - x^2] - 4*Sqrt[3]*x*ArcTan[
(Sqrt[3]*Sqrt[-3 - 4*x - x^2])/(3 + x)]) + 10*x*ArcTanh[x/Sqrt[-3 - 4*x - x^2]])/(27*x)

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.12

method	result	size
default	\(-\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (-6-4 x \right ) \sqrt {3}}{6 \sqrt {-x^{2}-4 x -3}}\right )}{9}+\frac {\sqrt {-x^{2}-4 x -3}}{9 x}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )-5 \,\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{81 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\)	\(169\)
risch	\(-\frac {x^{2}+4 x +3}{9 x \sqrt {-x^{2}-4 x -3}}-\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (-6-4 x \right ) \sqrt {3}}{6 \sqrt {-x^{2}-4 x -3}}\right )}{9}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )-5 \,\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{81 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\)	\(177\)
trager	\(\frac {\sqrt {-x^{2}-4 x -3}}{9 x}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x +3 \sqrt {-x^{2}-4 x -3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{x}\right )}{9}+\frac {10 \ln \left (-\frac {-96000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2} x +4160 \sqrt {-x^{2}-4 x -3}\, \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )+96000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2}+7232 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x +9 \sqrt {-x^{2}-4 x -3}-41232 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )+468 x +3510}{16 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x -16 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )-3 x}\right )}{27}-\frac {16 \ln \left (-\frac {-96000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2} x +4160 \sqrt {-x^{2}-4 x -3}\, \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )+96000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2}+7232 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x +9 \sqrt {-x^{2}-4 x -3}-41232 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )+468 x +3510}{16 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x -16 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )-3 x}\right ) \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )}{9}+\frac {16 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) \ln \left (\frac {288000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2} x +12480 \sqrt {-x^{2}-4 x -3}\, \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )-288000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2}-98304 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x -2627 \sqrt {-x^{2}-4 x -3}-3696 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )+6576 x +2740}{48 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x -48 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )-x +10}\right )}{9}\)	\(506\)

[In]

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

[Out]

-2/9*3^(1/2)*arctan(1/6*(-6-4*x)*3^(1/2)/(-x^2-4*x-3)^(1/2))+1/9*(-x^2-4*x-3)^(1/2)/x+1/81*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))-5*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))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {12 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} \sqrt {-x^{2} - 4 \, x - 3} {\left (2 \, x + 3\right )}}{3 \, {\left (x^{2} + 4 \, x + 3\right )}}\right ) - 2 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - 2 \, \sqrt {2} x \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + 5 \, x \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - 5 \, x \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) - 6 \, \sqrt {-x^{2} - 4 \, x - 3}}{54 \, x} \]

[In]

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

[Out]

-1/54*(12*sqrt(3)*x*arctan(1/3*sqrt(3)*sqrt(-x^2 - 4*x - 3)*(2*x + 3)/(x^2 + 4*x + 3)) - 2*sqrt(2)*x*arctan(1/
2*(sqrt(2)*x + 3*sqrt(2)*sqrt(-x^2 - 4*x - 3))/(2*x + 3)) - 2*sqrt(2)*x*arctan(-1/2*(sqrt(2)*x - 3*sqrt(2)*sqr
t(-x^2 - 4*x - 3))/(2*x + 3)) + 5*x*log(-(2*sqrt(-x^2 - 4*x - 3)*x + 4*x + 3)/x^2) - 5*x*log((2*sqrt(-x^2 - 4*
x - 3)*x - 4*x - 3)/x^2) - 6*sqrt(-x^2 - 4*x - 3))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (121) = 242\).

Time = 0.31 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.78 \[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {2}{27} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) - \frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac {2}{27} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) - \frac {\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 2}{18 \, {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right )}} + \frac {5}{27} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) - \frac {5}{27} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]

[In]

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

[Out]

2/27*sqrt(2)*arctan(1/2*sqrt(2)*(3*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) - 4/9*sqrt(3)*arctan(1/3*sqrt(3)*(
2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) + 2/27*sqrt(2)*arctan(1/2*sqrt(2)*((sqrt(-x^2 - 4*x - 3) - 1)/(x +
2) + 1)) - 1/18*((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 2)/((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + (sqrt(-x^2 - 4*
x - 3) - 1)^2/(x + 2)^2 + 1) + 5/27*log(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 3*(sqrt(-x^2 - 4*x - 3) - 1)^2/
(x + 2)^2 + 1) - 5/27*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. \[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {1}{x^2\,\sqrt {-x^2-4\,x-3}\,\left (2\,x^2+4\,x+3\right )} \,d x \]

[In]

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

[Out]

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