3.2.26 \(\int \frac {x^4}{\sqrt {-3-4 x-x^2} (3+4 x+2 x^2)} \, dx\) [126]

Optimal. Leaf size=140 \[ \frac {5}{2} \sqrt {-3-4 x-x^2}-\frac {1}{4} x \sqrt {-3-4 x-x^2}+\frac {11}{2} \sin ^{-1}(2+x)+\frac {\tan ^{-1}\left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {5}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]

[Out]

11/2*arcsin(2+x)-5/4*arctanh(x/(-x^2-4*x-3)^(1/2))+1/4*arctan(1/2*(1+(-3-x)/(-x^2-4*x-3)^(1/2))*2^(1/2))*2^(1/
2)-1/4*arctan(1/2*(1+(3+x)/(-x^2-4*x-3)^(1/2))*2^(1/2))*2^(1/2)+5/2*(-x^2-4*x-3)^(1/2)-1/4*x*(-x^2-4*x-3)^(1/2
)

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Rubi [A]
time = 0.31, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 14, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {6860, 633, 222, 654, 756, 1042, 1000, 12, 1040, 1175, 632, 210, 1041, 212} \begin {gather*} \frac {11}{2} \text {ArcSin}(x+2)+\frac {\text {ArcTan}\left (\frac {1-\frac {x+3}{\sqrt {-x^2-4 x-3}}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\text {ArcTan}\left (\frac {\frac {x+3}{\sqrt {-x^2-4 x-3}}+1}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {1}{4} \sqrt {-x^2-4 x-3} x+\frac {5}{2} \sqrt {-x^2-4 x-3}-\frac {5}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[-3 - 4*x - x^2])/2 - (x*Sqrt[-3 - 4*x - x^2])/4 + (11*ArcSin[2 + x])/2 + ArcTan[(1 - (3 + x)/Sqrt[-3 -
4*x - x^2])/Sqrt[2]]/(2*Sqrt[2]) - ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/(2*Sqrt[2]) - (5*ArcTan
h[x/Sqrt[-3 - 4*x - x^2]])/4

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 222

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 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 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[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 654

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

Rule 756

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)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(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]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

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*} \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx &=\int \left (\frac {5}{4 \sqrt {-3-4 x-x^2}}-\frac {x}{\sqrt {-3-4 x-x^2}}+\frac {x^2}{2 \sqrt {-3-4 x-x^2}}-\frac {15+8 x}{4 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {15+8 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx\right )+\frac {1}{2} \int \frac {x^2}{\sqrt {-3-4 x-x^2}} \, dx+\frac {5}{4} \int \frac {1}{\sqrt {-3-4 x-x^2}} \, dx-\int \frac {x}{\sqrt {-3-4 x-x^2}} \, dx\\ &=\sqrt {-3-4 x-x^2}-\frac {1}{4} x \sqrt {-3-4 x-x^2}-\frac {1}{4} \int \frac {3+6 x}{\sqrt {-3-4 x-x^2}} \, dx+\frac {1}{2} \int \frac {-6-4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {5}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,-4-2 x\right )-\frac {3}{4} \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx+2 \int \frac {1}{\sqrt {-3-4 x-x^2}} \, dx\\ &=\frac {5}{2} \sqrt {-3-4 x-x^2}-\frac {1}{4} x \sqrt {-3-4 x-x^2}+\frac {5}{4} \sin ^{-1}(2+x)+\frac {1}{8} \int \frac {-6-4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {1}{8} \int -\frac {4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {-3-4 x-x^2}} \, dx-3 \text {Subst}\left (\int \frac {1}{3-3 x^2} \, dx,x,\frac {x}{\sqrt {-3-4 x-x^2}}\right )-\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,-4-2 x\right )\\ &=\frac {5}{2} \sqrt {-3-4 x-x^2}-\frac {1}{4} x \sqrt {-3-4 x-x^2}+\frac {13}{4} \sin ^{-1}(2+x)-\tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {1}{2} \int \frac {x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {3}{4} \text {Subst}\left (\int \frac {1}{3-3 x^2} \, dx,x,\frac {x}{\sqrt {-3-4 x-x^2}}\right )-\frac {9}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,-4-2 x\right )\\ &=\frac {5}{2} \sqrt {-3-4 x-x^2}-\frac {1}{4} x \sqrt {-3-4 x-x^2}+\frac {11}{2} \sin ^{-1}(2+x)-\frac {5}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+4 \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 {5}{2} \sqrt {-3-4 x-x^2}-\frac {1}{4} x \sqrt {-3-4 x-x^2}+\frac {11}{2} \sin ^{-1}(2+x)-\frac {5}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )-\frac {1}{6} \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 {1}{6} \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 {5}{2} \sqrt {-3-4 x-x^2}-\frac {1}{4} x \sqrt {-3-4 x-x^2}+\frac {11}{2} \sin ^{-1}(2+x)-\frac {5}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {1}{3} \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 {1}{3} \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 {5}{2} \sqrt {-3-4 x-x^2}-\frac {1}{4} x \sqrt {-3-4 x-x^2}+\frac {11}{2} \sin ^{-1}(2+x)+\frac {\tan ^{-1}\left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {5}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 99, normalized size = 0.71 \begin {gather*} \frac {1}{4} \left (-\left ((-10+x) \sqrt {-3-4 x-x^2}\right )-\sqrt {2} \tan ^{-1}\left (\frac {3+2 x}{\sqrt {2} \sqrt {-3-4 x-x^2}}\right )-44 \tan ^{-1}\left (\frac {\sqrt {-3-4 x-x^2}}{3+x}\right )-5 \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((-10 + x)*Sqrt[-3 - 4*x - x^2]) - Sqrt[2]*ArcTan[(3 + 2*x)/(Sqrt[2]*Sqrt[-3 - 4*x - x^2])] - 44*ArcTan[Sqrt
[-3 - 4*x - x^2]/(3 + x)] - 5*ArcTanh[x/Sqrt[-3 - 4*x - x^2]])/4

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Maple [A]
time = 0.45, size = 159, normalized size = 1.14

method	result	size
risch	\(\frac {\left (x -10\right ) \left (x^{2}+4 x +3\right )}{4 \sqrt {-x^{2}-4 x -3}}+\frac {11 \arcsin \left (x +2\right )}{2}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )+5 \arctanh \left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{24 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\)	\(155\)
default	\(-\frac {x \sqrt {-x^{2}-4 x -3}}{4}+\frac {5 \sqrt {-x^{2}-4 x -3}}{2}+\frac {11 \arcsin \left (x +2\right )}{2}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )+5 \arctanh \left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{24 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\)	\(159\)
trager	\(\left (-\frac {x}{4}+\frac {5}{2}\right ) \sqrt {-x^{2}-4 x -3}+\frac {11 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-x \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-4 x -3}\right )}{2}+\frac {5 \ln \left (-\frac {12 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right )^{2} x -28 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right ) x +2 \sqrt {-x^{2}-4 x -3}-12 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right )+15 x +10}{2 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right ) x -x +1}\right )}{4}-\frac {3 \ln \left (-\frac {12 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right )^{2} x -28 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right ) x +2 \sqrt {-x^{2}-4 x -3}-12 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right )+15 x +10}{2 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right ) x -x +1}\right ) \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right )}{4}+\frac {3 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right ) \ln \left (-\frac {36 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right )^{2} x -36 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right ) x +6 \sqrt {-x^{2}-4 x -3}+36 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right )+5 x -30}{6 \RootOf \left (12 \textit {\_Z}^{2}-20 \textit {\_Z} +9\right ) x -7 x -3}\right )}{4}\)	\(345\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x*(-x^2-4*x-3)^(1/2)+5/2*(-x^2-4*x-3)^(1/2)+11/2*arcsin(x+2)+1/24*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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 178, normalized size = 1.27 \begin {gather*} -\frac {1}{4} \, \sqrt {-x^{2} - 4 \, x - 3} {\left (x - 10\right )} + \frac {1}{8} \, \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}{8} \, \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 {11}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x - 3} {\left (x + 2\right )}}{x^{2} + 4 \, x + 3}\right ) + \frac {5}{16} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - \frac {5}{16} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(-x^2 - 4*x - 3)*(x - 10) + 1/8*sqrt(2)*arctan(1/2*(sqrt(2)*x + 3*sqrt(2)*sqrt(-x^2 - 4*x - 3))/(2*x
+ 3)) + 1/8*sqrt(2)*arctan(-1/2*(sqrt(2)*x - 3*sqrt(2)*sqrt(-x^2 - 4*x - 3))/(2*x + 3)) - 11/2*arctan(sqrt(-x^
2 - 4*x - 3)*(x + 2)/(x^2 + 4*x + 3)) + 5/16*log(-(2*sqrt(-x^2 - 4*x - 3)*x + 4*x + 3)/x^2) - 5/16*log((2*sqrt
(-x^2 - 4*x - 3)*x - 4*x - 3)/x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\sqrt {- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 5.90, size = 188, normalized size = 1.34 \begin {gather*} -\frac {1}{4} \, \sqrt {-x^{2} - 4 \, x - 3} {\left (x - 10\right )} + \frac {1}{4} \, \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}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) + \frac {11}{2} \, \arcsin \left (x + 2\right ) - \frac {5}{8} \, \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}{8} \, \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(-x^2 - 4*x - 3)*(x - 10) + 1/4*sqrt(2)*arctan(1/2*sqrt(2)*(3*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)
) + 1/4*sqrt(2)*arctan(1/2*sqrt(2)*((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) + 11/2*arcsin(x + 2) - 5/8*log(2*
(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 3*(sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 1) + 5/8*log(2*(sqrt(-x^2 - 4*
x - 3) - 1)/(x + 2) + (sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{\sqrt {-x^2-4\,x-3}\,\left (2\,x^2+4\,x+3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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