Integrand size = 30, antiderivative size = 140 \[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {5}{2} \sqrt {-3-4 x-x^2}-\frac {1}{4} x \sqrt {-3-4 x-x^2}+\frac {11}{2} \arcsin (2+x)+\frac {\arctan \left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\arctan \left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {5}{4} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]
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)
Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.71 \[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {1}{4} \left (-\left ((-10+x) \sqrt {-3-4 x-x^2}\right )-\sqrt {2} \arctan \left (\frac {3+2 x}{\sqrt {2} \sqrt {-3-4 x-x^2}}\right )-44 \arctan \left (\frac {\sqrt {-3-4 x-x^2}}{3+x}\right )-5 \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )\right ) \]
(-((-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
Time = 0.61 (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 |
| \(\displaystyle \frac {11}{2} \arcsin (x+2)+\frac {\arctan \left (\frac {1-\frac {x+3}{\sqrt {-x^2-4 x-3}}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\arctan \left (\frac {\frac {x+3}{\sqrt {-x^2-4 x-3}}+1}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {5}{4} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-\frac {1}{4} \sqrt {-x^2-4 x-3} x+\frac {5}{2} \sqrt {-x^2-4 x-3}\) |
(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
3.2.26.3.1 Defintions of rubi rules used
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]
Time = 0.98 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.11
| method | result | size |
| risch | \(\frac {\left (x -10\right ) \left (x^{2}+4 x +3\right )}{4 \sqrt {-x^{2}-4 x -3}}+\frac {11 \arcsin \left (2+x \right )}{2}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )+5 \,\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{24 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\) | \(155\) |
| default | \(\frac {5 \sqrt {-x^{2}-4 x -3}}{2}+\frac {11 \arcsin \left (2+x \right )}{2}-\frac {x \sqrt {-x^{2}-4 x -3}}{4}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )+5 \,\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{24 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\) | \(159\) |
| trager | \(\left (-\frac {x}{4}+\frac {5}{2}\right ) \sqrt {-x^{2}-4 x -3}-\frac {5 \ln \left (\frac {12 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )^{2} x +28 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x +12 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )-2 \sqrt {-x^{2}-4 x -3}+15 x +10}{2 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x +x -1}\right )}{4}-\frac {3 \ln \left (\frac {12 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )^{2} x +28 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x +12 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )-2 \sqrt {-x^{2}-4 x -3}+15 x +10}{2 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x +x -1}\right ) \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )}{4}+\frac {3 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) \ln \left (\frac {36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )^{2} x +36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x -36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right )-6 \sqrt {-x^{2}-4 x -3}+5 x -30}{6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+20 \textit {\_Z} +9\right ) x +7 x +3}\right )}{4}+\frac {11 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-4 x -3}\right )}{2}\) | \(338\) |
1/4*(x-10)*(x^2+4*x+3)/(-x^2-4*x-3)^(1/2)+11/2*arcsin(2+x)+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))
Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.27 \[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {1}{4} \, \sqrt {-x^{2} - 4 \, x - 3} {\left (x - 10\right )} + \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + \frac {1}{8} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - \frac {11}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x - 3} {\left (x + 2\right )}}{x^{2} + 4 \, x + 3}\right ) + \frac {5}{16} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - \frac {5}{16} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \]
-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)
\[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {x^{4}}{\sqrt {- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \]
\[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int { \frac {x^{4}}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt {-x^{2} - 4 \, x - 3}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.34 \[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {1}{4} \, \sqrt {-x^{2} - 4 \, x - 3} {\left (x - 10\right )} + \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) + \frac {11}{2} \, \arcsin \left (x + 2\right ) - \frac {5}{8} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) + \frac {5}{8} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]
-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)
Timed out. \[ \int \frac {x^4}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {x^4}{\sqrt {-x^2-4\,x-3}\,\left (2\,x^2+4\,x+3\right )} \,d x \]