\(\int \frac {x^4}{\sqrt {-3-4 x-x^2} (3+4 x+2 x^2)} \, dx\) [154]

Optimal result

Integrand size = 30, antiderivative size = 140 \[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {5}{2} \sqrt {-3-4 x-x^2}-\frac {1}{4} x \sqrt {-3-4 x-x^2}+\frac {11}{2} \arcsin (2+x)+\frac {\arctan \left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\arctan \left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {5}{4} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.71 \[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {1}{4} \left (-\left ((-10+x) \sqrt {-3-4 x-x^2}\right )-\sqrt {2} \arctan \left (\frac {3+2 x}{\sqrt {2} \sqrt {-3-4 x-x^2}}\right )-44 \arctan \left (\frac {\sqrt {-3-4 x-x^2}}{3+x}\right )-5 \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7279, 2009}

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.

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

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {x^2}{2 \sqrt {-x^2-4 x-3}}-\frac {x}{\sqrt {-x^2-4 x-3}}+\frac {5}{4 \sqrt {-x^2-4 x-3}}-\frac {8 x+15}{4 \sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {11}{2} \arcsin (x+2)+\frac {\arctan \left (\frac {1-\frac {x+3}{\sqrt {-x^2-4 x-3}}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\arctan \left (\frac {\frac {x+3}{\sqrt {-x^2-4 x-3}}+1}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {5}{4} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-\frac {1}{4} \sqrt {-x^2-4 x-3} x+\frac {5}{2} \sqrt {-x^2-4 x-3}\)

Input:

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

Output:

(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]) - Ar 
cTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/(2*Sqrt[2]) - (5*ArcTanh[ 
x/Sqrt[-3 - 4*x - x^2]])/4
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7279

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] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 3.33 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.11

method	result	size
risch	\(\frac {\left (-10+x \right ) \left (x^{2}+4 x +3\right )}{4 \sqrt {-x^{2}-4 x -3}}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )+5 \,\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{24 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}+\frac {11 \arcsin \left (2+x \right )}{2}\)	\(155\)
default	\(\frac {11 \arcsin \left (2+x \right )}{2}+\frac {5 \sqrt {-x^{2}-4 x -3}}{2}-\frac {x \sqrt {-x^{2}-4 x -3}}{4}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )+5 \,\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{24 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\)	\(159\)
trager	\(\left (\frac {5}{2}-\frac {x}{4}\right ) \sqrt {-x^{2}-4 x -3}+\frac {3 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) \ln \left (\frac {36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )^{2} x +36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x -36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )-6 \sqrt {-x^{2}-4 x -3}+5 x -30}{6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x +7 x +3}\right )}{4}-\frac {5 \ln \left (\frac {12 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )^{2} x +28 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x +12 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )-2 \sqrt {-x^{2}-4 x -3}+15 x +10}{2 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x +x -1}\right )}{4}-\frac {3 \ln \left (\frac {12 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )^{2} x +28 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x +12 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )-2 \sqrt {-x^{2}-4 x -3}+15 x +10}{2 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x +x -1}\right ) \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )}{4}-\frac {11 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-4 x -3}\right )}{2}\)	\(337\)

Input:

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

Output:

1/4*(-10+x)*(x^2+4*x+3)/(-x^2-4*x-3)^(1/2)+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))+11/2*arcsin(2+x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.27 \[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {1}{4} \, \sqrt {-x^{2} - 4 \, x - 3} {\left (x - 10\right )} + \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + \frac {1}{8} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - \frac {11}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x - 3} {\left (x + 2\right )}}{x^{2} + 4 \, x + 3}\right ) + \frac {5}{16} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - \frac {5}{16} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \] Input:

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

Output:

-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)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.34 \[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {1}{4} \, \sqrt {-x^{2} - 4 \, x - 3} {\left (x - 10\right )} + \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) + \frac {11}{2} \, \arcsin \left (x + 2\right ) - \frac {5}{8} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) + \frac {5}{8} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.87 \[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {11 \mathit {asin} \left (x +2\right )}{2}-\frac {\sqrt {2}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )-1}{\sqrt {2}}\right )}{4}-\frac {\sqrt {2}\, \mathit {atan} \left (\frac {3 \tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )-1}{\sqrt {2}}\right )}{4}-\frac {\sqrt {-x^{2}-4 x -3}\, x}{4}+\frac {5 \sqrt {-x^{2}-4 x -3}}{2}+\frac {5 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )^{2}-2 \tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )+3\right )}{8}-\frac {5 \,\mathrm {log}\left (3 \tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )^{2}-2 \tan \left (\frac {\mathit {asin} \left (x +2\right )}{2}\right )+1\right )}{8} \] Input:

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

Output:

(44*asin(x + 2) - 2*sqrt(2)*atan((tan(asin(x + 2)/2) - 1)/sqrt(2)) - 2*sqr 
t(2)*atan((3*tan(asin(x + 2)/2) - 1)/sqrt(2)) - 2*sqrt( - x**2 - 4*x - 3)* 
x + 20*sqrt( - x**2 - 4*x - 3) + 5*log(tan(asin(x + 2)/2)**2 - 2*tan(asin( 
x + 2)/2) + 3) - 5*log(3*tan(asin(x + 2)/2)**2 - 2*tan(asin(x + 2)/2) + 1) 
)/8