Integrand size = 28, antiderivative size = 76 \[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=-\frac {(3-x) \sqrt {4+2 x+x^2}}{4 \left (3+2 x+x^2\right )}-\frac {\arctan \left (\frac {1+x}{\sqrt {2} \sqrt {4+2 x+x^2}}\right )}{4 \sqrt {2}}+\text {arctanh}\left (\sqrt {4+2 x+x^2}\right ) \]
arctanh((x^2+2*x+4)^(1/2))-1/8*arctan(1/2*(1+x)*2^(1/2)/(x^2+2*x+4)^(1/2)) *2^(1/2)-1/4*(3-x)*(x^2+2*x+4)^(1/2)/(x^2+2*x+3)
Time = 0.61 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.11 \[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\frac {1}{8} \left (\frac {2 (-3+x) \sqrt {4+2 x+x^2}}{3+2 x+x^2}+\sqrt {2} \arctan \left (\frac {3+2 x+x^2-(1+x) \sqrt {4+2 x+x^2}}{\sqrt {2}}\right )\right )+\text {arctanh}\left (\sqrt {4+2 x+x^2}\right ) \]
((2*(-3 + x)*Sqrt[4 + 2*x + x^2])/(3 + 2*x + x^2) + Sqrt[2]*ArcTan[(3 + 2* x + x^2 - (1 + x)*Sqrt[4 + 2*x + x^2])/Sqrt[2]])/8 + ArcTanh[Sqrt[4 + 2*x + x^2]]
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1349, 27, 1358, 27, 1313, 217, 1357, 219}
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.
| \(\displaystyle \int \frac {2 x+3}{\left (x^2+2 x+3\right )^2 \sqrt {x^2+2 x+4}} \, dx\) |
| \(\Big \downarrow \) 1349 |
| \(\displaystyle \frac {1}{8} \int -\frac {2 (4 x+5)}{\left (x^2+2 x+3\right ) \sqrt {x^2+2 x+4}}dx-\frac {(3-x) \sqrt {x^2+2 x+4}}{4 \left (x^2+2 x+3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{4} \int \frac {4 x+5}{\left (x^2+2 x+3\right ) \sqrt {x^2+2 x+4}}dx-\frac {\sqrt {x^2+2 x+4} (3-x)}{4 \left (x^2+2 x+3\right )}\) |
| \(\Big \downarrow \) 1358 |
| \(\displaystyle \frac {1}{4} \left (-\int \frac {1}{\left (x^2+2 x+3\right ) \sqrt {x^2+2 x+4}}dx-2 \int \frac {2 (x+1)}{\left (x^2+2 x+3\right ) \sqrt {x^2+2 x+4}}dx\right )-\frac {(3-x) \sqrt {x^2+2 x+4}}{4 \left (x^2+2 x+3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (-\int \frac {1}{\left (x^2+2 x+3\right ) \sqrt {x^2+2 x+4}}dx-4 \int \frac {x+1}{\left (x^2+2 x+3\right ) \sqrt {x^2+2 x+4}}dx\right )-\frac {(3-x) \sqrt {x^2+2 x+4}}{4 \left (x^2+2 x+3\right )}\) |
| \(\Big \downarrow \) 1313 |
| \(\displaystyle \frac {1}{4} \left (4 \int \frac {1}{-\frac {8 (x+1)^2}{x^2+2 x+4}-16}d\frac {2 (x+1)}{\sqrt {x^2+2 x+4}}-4 \int \frac {x+1}{\left (x^2+2 x+3\right ) \sqrt {x^2+2 x+4}}dx\right )-\frac {(3-x) \sqrt {x^2+2 x+4}}{4 \left (x^2+2 x+3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{4} \left (-4 \int \frac {x+1}{\left (x^2+2 x+3\right ) \sqrt {x^2+2 x+4}}dx-\frac {\arctan \left (\frac {x+1}{\sqrt {2} \sqrt {x^2+2 x+4}}\right )}{\sqrt {2}}\right )-\frac {(3-x) \sqrt {x^2+2 x+4}}{4 \left (x^2+2 x+3\right )}\) |
| \(\Big \downarrow \) 1357 |
| \(\displaystyle \frac {1}{4} \left (8 \int \frac {1}{2-2 \left (x^2+2 x+4\right )}d\sqrt {x^2+2 x+4}-\frac {\arctan \left (\frac {x+1}{\sqrt {2} \sqrt {x^2+2 x+4}}\right )}{\sqrt {2}}\right )-\frac {(3-x) \sqrt {x^2+2 x+4}}{4 \left (x^2+2 x+3\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} \left (4 \text {arctanh}\left (\sqrt {x^2+2 x+4}\right )-\frac {\arctan \left (\frac {x+1}{\sqrt {2} \sqrt {x^2+2 x+4}}\right )}{\sqrt {2}}\right )-\frac {(3-x) \sqrt {x^2+2 x+4}}{4 \left (x^2+2 x+3\right )}\) |
-1/4*((3 - x)*Sqrt[4 + 2*x + x^2])/(3 + 2*x + x^2) + (-(ArcTan[(1 + x)/(Sq rt[2]*Sqrt[4 + 2*x + x^2])]/Sqrt[2]) + 4*ArcTanh[Sqrt[4 + 2*x + x^2]])/4
3.3.81.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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])
Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*( x_)^2]), x_Symbol] :> Simp[-2*e Subst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e )*x^2), x], x, (e + 2*f*x)/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[c*e - b*f, 0]
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e _.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(a + b*x + c*x^2)^(p + 1)* ((d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b* c*d - 2*a*c*e + a*b*f))*x), x] + Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b* d - a*e)*(c*e - b*f))*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f *x^2)^q*Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1 ) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*((-h)*c *e)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g *b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2 *a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))* (p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a *((-h)*c*e)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h* c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c* e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1])
Int[((g_) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e _.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-2*g Subst[Int[1/(b*d - a*e - b*x^2), 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[c*e - b*f, 0] & & EqQ[h*e - 2*g*f, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-(h*e - 2*g*f)/(2*f) Int[1/ ((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[h/(2*f) Int[(e + 2*f*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[c *e - b*f, 0] && NeQ[h*e - 2*g*f, 0]
Time = 0.84 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(\frac {\left (-3+x \right ) \sqrt {x^{2}+2 x +4}}{4 x^{2}+8 x +12}+\operatorname {arctanh}\left (\sqrt {x^{2}+2 x +4}\right )-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (2 x +2\right )}{4 \sqrt {x^{2}+2 x +4}}\right )}{8}\) | \(64\) |
| default | \(-\frac {1}{2 \left (\sqrt {x^{2}+2 x +4}+1\right )}+\frac {\ln \left (\sqrt {x^{2}+2 x +4}+1\right )}{2}-\frac {1}{2 \left (\sqrt {x^{2}+2 x +4}-1\right )}-\frac {\ln \left (\sqrt {x^{2}+2 x +4}-1\right )}{2}+\frac {\frac {3}{4}+\frac {3 x}{4}}{\sqrt {x^{2}+2 x +4}\, \left (\frac {\left (1+x \right )^{2}}{x^{2}+2 x +4}+2\right )}-\frac {\arctan \left (\frac {\left (1+x \right ) \sqrt {2}}{2 \sqrt {x^{2}+2 x +4}}\right ) \sqrt {2}}{8}\) | \(123\) |
| trager | \(\frac {\left (-3+x \right ) \sqrt {x^{2}+2 x +4}}{4 x^{2}+8 x +12}-3 \ln \left (-\frac {48384 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )^{2} x +960 \sqrt {x^{2}+2 x +4}\, \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )+15312 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +143 \sqrt {x^{2}+2 x +4}-3696 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )+1210 x -605}{48 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +7 x -3}\right ) \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )+3 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) \ln \left (\frac {-16128 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )^{2} x +320 \sqrt {x^{2}+2 x +4}\, \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )-5648 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +59 \sqrt {x^{2}+2 x +4}-1232 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )-494 x -209}{16 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +3 x +1}\right )-\ln \left (-\frac {48384 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )^{2} x +960 \sqrt {x^{2}+2 x +4}\, \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )+15312 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +143 \sqrt {x^{2}+2 x +4}-3696 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )+1210 x -605}{48 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +7 x -3}\right )\) | \(375\) |
1/4*(-3+x)/(x^2+2*x+3)*(x^2+2*x+4)^(1/2)+arctanh((x^2+2*x+4)^(1/2))-1/8*2^ (1/2)*arctan(1/4*2^(1/2)/(x^2+2*x+4)^(1/2)*(2*x+2))
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (61) = 122\).
Time = 0.24 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.29 \[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\frac {\sqrt {2} {\left (x^{2} + 2 \, x + 3\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x + 2\right )} + \frac {1}{2} \, \sqrt {2} \sqrt {x^{2} + 2 \, x + 4}\right ) - \sqrt {2} {\left (x^{2} + 2 \, x + 3\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} x + \frac {1}{2} \, \sqrt {2} \sqrt {x^{2} + 2 \, x + 4}\right ) + 2 \, x^{2} - 4 \, {\left (x^{2} + 2 \, x + 3\right )} \log \left (x^{2} - \sqrt {x^{2} + 2 \, x + 4} {\left (x + 2\right )} + 3 \, x + 5\right ) + 4 \, {\left (x^{2} + 2 \, x + 3\right )} \log \left (x^{2} - \sqrt {x^{2} + 2 \, x + 4} x + x + 3\right ) + 2 \, \sqrt {x^{2} + 2 \, x + 4} {\left (x - 3\right )} + 4 \, x + 6}{8 \, {\left (x^{2} + 2 \, x + 3\right )}} \]
1/8*(sqrt(2)*(x^2 + 2*x + 3)*arctan(-1/2*sqrt(2)*(x + 2) + 1/2*sqrt(2)*sqr t(x^2 + 2*x + 4)) - sqrt(2)*(x^2 + 2*x + 3)*arctan(-1/2*sqrt(2)*x + 1/2*sq rt(2)*sqrt(x^2 + 2*x + 4)) + 2*x^2 - 4*(x^2 + 2*x + 3)*log(x^2 - sqrt(x^2 + 2*x + 4)*(x + 2) + 3*x + 5) + 4*(x^2 + 2*x + 3)*log(x^2 - sqrt(x^2 + 2*x + 4)*x + x + 3) + 2*sqrt(x^2 + 2*x + 4)*(x - 3) + 4*x + 6)/(x^2 + 2*x + 3 )
\[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\int \frac {2 x + 3}{\left (x^{2} + 2 x + 3\right )^{2} \sqrt {x^{2} + 2 x + 4}}\, dx \]
\[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\int { \frac {2 \, x + 3}{\sqrt {x^{2} + 2 \, x + 4} {\left (x^{2} + 2 \, x + 3\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (61) = 122\).
Time = 0.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.09 \[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\frac {1}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x - \sqrt {x^{2} + 2 \, x + 4} + 2\right )}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}\right ) + \frac {4 \, {\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{3} + 13 \, {\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{2} + 26 \, x - 26 \, \sqrt {x^{2} + 2 \, x + 4} + 26}{2 \, {\left ({\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{4} + 4 \, {\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{3} + 8 \, {\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{2} + 8 \, x - 8 \, \sqrt {x^{2} + 2 \, x + 4} + 12\right )}} - \frac {1}{2} \, \log \left ({\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{2} + 4 \, x - 4 \, \sqrt {x^{2} + 2 \, x + 4} + 6\right ) + \frac {1}{2} \, \log \left ({\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{2} + 2\right ) \]
1/8*sqrt(2)*arctan(-1/2*sqrt(2)*(x - sqrt(x^2 + 2*x + 4) + 2)) - 1/8*sqrt( 2)*arctan(-1/2*sqrt(2)*(x - sqrt(x^2 + 2*x + 4))) + 1/2*(4*(x - sqrt(x^2 + 2*x + 4))^3 + 13*(x - sqrt(x^2 + 2*x + 4))^2 + 26*x - 26*sqrt(x^2 + 2*x + 4) + 26)/((x - sqrt(x^2 + 2*x + 4))^4 + 4*(x - sqrt(x^2 + 2*x + 4))^3 + 8 *(x - sqrt(x^2 + 2*x + 4))^2 + 8*x - 8*sqrt(x^2 + 2*x + 4) + 12) - 1/2*log ((x - sqrt(x^2 + 2*x + 4))^2 + 4*x - 4*sqrt(x^2 + 2*x + 4) + 6) + 1/2*log( (x - sqrt(x^2 + 2*x + 4))^2 + 2)
Timed out. \[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\int \frac {2\,x+3}{{\left (x^2+2\,x+3\right )}^2\,\sqrt {x^2+2\,x+4}} \,d x \]