Integrand size = 18, antiderivative size = 61 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=\frac {5+4 x}{4 \left (3+6 x+2 x^2\right )^2}-\frac {3+2 x}{2 \left (3+6 x+2 x^2\right )}+\frac {\text {arctanh}\left (\frac {3+2 x}{\sqrt {3}}\right )}{\sqrt {3}} \] Output:
1/4*(5+4*x)/(2*x^2+6*x+3)^2+1/2*(-3-2*x)/(2*x^2+6*x+3)+1/3*arctanh(1/3*(3+ 2*x)*3^(1/2))*3^(1/2)
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=\frac {1}{12} \left (-\frac {3 \left (13+44 x+36 x^2+8 x^3\right )}{\left (3+6 x+2 x^2\right )^2}-2 \sqrt {3} \log \left (-3+\sqrt {3}-2 x\right )+2 \sqrt {3} \log \left (3+\sqrt {3}+2 x\right )\right ) \] Input:
Integrate[(-3 + 2*x)/(3 + 6*x + 2*x^2)^3,x]
Output:
((-3*(13 + 44*x + 36*x^2 + 8*x^3))/(3 + 6*x + 2*x^2)^2 - 2*Sqrt[3]*Log[-3 + Sqrt[3] - 2*x] + 2*Sqrt[3]*Log[3 + Sqrt[3] + 2*x])/12
Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.52, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1159, 1086, 1081, 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.
\(\displaystyle \int \frac {2 x-3}{\left (2 x^2+6 x+3\right )^3} \, dx\) |
\(\Big \downarrow \) 1159 |
\(\displaystyle 3 \int \frac {1}{\left (2 x^2+6 x+3\right )^2}dx+\frac {4 x+5}{4 \left (2 x^2+6 x+3\right )^2}\) |
\(\Big \downarrow \) 1086 |
\(\displaystyle 3 \left (-\frac {1}{3} \int \frac {1}{2 x^2+6 x+3}dx-\frac {2 x+3}{6 \left (2 x^2+6 x+3\right )}\right )+\frac {4 x+5}{4 \left (2 x^2+6 x+3\right )^2}\) |
\(\Big \downarrow \) 1081 |
\(\displaystyle 3 \left (-\frac {2}{3} \int \left (\frac {1}{2 \sqrt {3} \left (2 x-\sqrt {3}+3\right )}-\frac {1}{2 \sqrt {3} \left (2 x+\sqrt {3}+3\right )}\right )dx-\frac {2 x+3}{6 \left (2 x^2+6 x+3\right )}\right )+\frac {4 x+5}{4 \left (2 x^2+6 x+3\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 x+5}{4 \left (2 x^2+6 x+3\right )^2}+3 \left (-\frac {2 x+3}{6 \left (2 x^2+6 x+3\right )}-\frac {2}{3} \left (\frac {\log \left (2 x-\sqrt {3}+3\right )}{4 \sqrt {3}}-\frac {\log \left (2 x+\sqrt {3}+3\right )}{4 \sqrt {3}}\right )\right )\) |
Input:
Int[(-3 + 2*x)/(3 + 6*x + 2*x^2)^3,x]
Output:
(5 + 4*x)/(4*(3 + 6*x + 2*x^2)^2) + 3*(-1/6*(3 + 2*x)/(3 + 6*x + 2*x^2) - (2*(Log[3 - Sqrt[3] + 2*x]/(4*Sqrt[3]) - Log[3 + Sqrt[3] + 2*x]/(4*Sqrt[3] )))/3)
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c Int[ExpandIntegrand[1/((b/2 - q/2 + c*x)*(b/2 + q/2 + c*x)), x], x], x]] /; FreeQ[{a, b, c}, x] && NiceSqrtQ[b^2 - 4*a*c]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {-24 x -30}{24 \left (2 x^{2}+6 x +3\right )^{2}}-\frac {4 x +6}{4 \left (2 x^{2}+6 x +3\right )}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (4 x +6\right ) \sqrt {3}}{6}\right )}{3}\) | \(56\) |
risch | \(\frac {-2 x^{3}-9 x^{2}-11 x -\frac {13}{4}}{\left (2 x^{2}+6 x +3\right )^{2}}+\frac {\ln \left (3+2 x +\sqrt {3}\right ) \sqrt {3}}{6}-\frac {\ln \left (3+2 x -\sqrt {3}\right ) \sqrt {3}}{6}\) | \(61\) |
Input:
int((2*x-3)/(2*x^2+6*x+3)^3,x,method=_RETURNVERBOSE)
Output:
-1/24*(-24*x-30)/(2*x^2+6*x+3)^2-1/4*(4*x+6)/(2*x^2+6*x+3)+1/3*3^(1/2)*arc tanh(1/6*(4*x+6)*3^(1/2))
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.59 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=-\frac {24 \, x^{3} - 2 \, \sqrt {3} {\left (4 \, x^{4} + 24 \, x^{3} + 48 \, x^{2} + 36 \, x + 9\right )} \log \left (\frac {2 \, x^{2} + \sqrt {3} {\left (2 \, x + 3\right )} + 6 \, x + 6}{2 \, x^{2} + 6 \, x + 3}\right ) + 108 \, x^{2} + 132 \, x + 39}{12 \, {\left (4 \, x^{4} + 24 \, x^{3} + 48 \, x^{2} + 36 \, x + 9\right )}} \] Input:
integrate((-3+2*x)/(2*x^2+6*x+3)^3,x, algorithm="fricas")
Output:
-1/12*(24*x^3 - 2*sqrt(3)*(4*x^4 + 24*x^3 + 48*x^2 + 36*x + 9)*log((2*x^2 + sqrt(3)*(2*x + 3) + 6*x + 6)/(2*x^2 + 6*x + 3)) + 108*x^2 + 132*x + 39)/ (4*x^4 + 24*x^3 + 48*x^2 + 36*x + 9)
Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.25 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=\frac {- 8 x^{3} - 36 x^{2} - 44 x - 13}{16 x^{4} + 96 x^{3} + 192 x^{2} + 144 x + 36} - \frac {\sqrt {3} \log {\left (x - \frac {\sqrt {3}}{2} + \frac {3}{2} \right )}}{6} + \frac {\sqrt {3} \log {\left (x + \frac {\sqrt {3}}{2} + \frac {3}{2} \right )}}{6} \] Input:
integrate((-3+2*x)/(2*x**2+6*x+3)**3,x)
Output:
(-8*x**3 - 36*x**2 - 44*x - 13)/(16*x**4 + 96*x**3 + 192*x**2 + 144*x + 36 ) - sqrt(3)*log(x - sqrt(3)/2 + 3/2)/6 + sqrt(3)*log(x + sqrt(3)/2 + 3/2)/ 6
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.10 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\frac {2 \, x - \sqrt {3} + 3}{2 \, x + \sqrt {3} + 3}\right ) - \frac {8 \, x^{3} + 36 \, x^{2} + 44 \, x + 13}{4 \, {\left (4 \, x^{4} + 24 \, x^{3} + 48 \, x^{2} + 36 \, x + 9\right )}} \] Input:
integrate((-3+2*x)/(2*x^2+6*x+3)^3,x, algorithm="maxima")
Output:
-1/6*sqrt(3)*log((2*x - sqrt(3) + 3)/(2*x + sqrt(3) + 3)) - 1/4*(8*x^3 + 3 6*x^2 + 44*x + 13)/(4*x^4 + 24*x^3 + 48*x^2 + 36*x + 9)
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 4 \, x - 2 \, \sqrt {3} + 6 \right |}}{{\left | 4 \, x + 2 \, \sqrt {3} + 6 \right |}}\right ) - \frac {8 \, x^{3} + 36 \, x^{2} + 44 \, x + 13}{4 \, {\left (2 \, x^{2} + 6 \, x + 3\right )}^{2}} \] Input:
integrate((-3+2*x)/(2*x^2+6*x+3)^3,x, algorithm="giac")
Output:
-1/6*sqrt(3)*log(abs(4*x - 2*sqrt(3) + 6)/abs(4*x + 2*sqrt(3) + 6)) - 1/4* (8*x^3 + 36*x^2 + 44*x + 13)/(2*x^2 + 6*x + 3)^2
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=\frac {\sqrt {3}\,\mathrm {atanh}\left (\sqrt {3}\,\left (\frac {2\,x}{3}+1\right )\right )}{3}-\frac {\frac {x^3}{2}+\frac {9\,x^2}{4}+\frac {11\,x}{4}+\frac {13}{16}}{x^4+6\,x^3+12\,x^2+9\,x+\frac {9}{4}} \] Input:
int((2*x - 3)/(6*x + 2*x^2 + 3)^3,x)
Output:
(3^(1/2)*atanh(3^(1/2)*((2*x)/3 + 1)))/3 - ((11*x)/4 + (9*x^2)/4 + x^3/2 + 13/16)/(9*x + 12*x^2 + 6*x^3 + x^4 + 9/4)
Time = 0.14 (sec) , antiderivative size = 188, normalized size of antiderivative = 3.08 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=\frac {-4 \sqrt {3}\, \mathrm {log}\left (-\sqrt {3}+2 x +3\right ) x^{4}-24 \sqrt {3}\, \mathrm {log}\left (-\sqrt {3}+2 x +3\right ) x^{3}-48 \sqrt {3}\, \mathrm {log}\left (-\sqrt {3}+2 x +3\right ) x^{2}-36 \sqrt {3}\, \mathrm {log}\left (-\sqrt {3}+2 x +3\right ) x -9 \sqrt {3}\, \mathrm {log}\left (-\sqrt {3}+2 x +3\right )+4 \sqrt {3}\, \mathrm {log}\left (\sqrt {3}+2 x +3\right ) x^{4}+24 \sqrt {3}\, \mathrm {log}\left (\sqrt {3}+2 x +3\right ) x^{3}+48 \sqrt {3}\, \mathrm {log}\left (\sqrt {3}+2 x +3\right ) x^{2}+36 \sqrt {3}\, \mathrm {log}\left (\sqrt {3}+2 x +3\right ) x +9 \sqrt {3}\, \mathrm {log}\left (\sqrt {3}+2 x +3\right )+2 x^{4}-30 x^{2}-48 x -15}{24 x^{4}+144 x^{3}+288 x^{2}+216 x +54} \] Input:
int((-3+2*x)/(2*x^2+6*x+3)^3,x)
Output:
( - 4*sqrt(3)*log( - sqrt(3) + 2*x + 3)*x**4 - 24*sqrt(3)*log( - sqrt(3) + 2*x + 3)*x**3 - 48*sqrt(3)*log( - sqrt(3) + 2*x + 3)*x**2 - 36*sqrt(3)*lo g( - sqrt(3) + 2*x + 3)*x - 9*sqrt(3)*log( - sqrt(3) + 2*x + 3) + 4*sqrt(3 )*log(sqrt(3) + 2*x + 3)*x**4 + 24*sqrt(3)*log(sqrt(3) + 2*x + 3)*x**3 + 4 8*sqrt(3)*log(sqrt(3) + 2*x + 3)*x**2 + 36*sqrt(3)*log(sqrt(3) + 2*x + 3)* x + 9*sqrt(3)*log(sqrt(3) + 2*x + 3) + 2*x**4 - 30*x**2 - 48*x - 15)/(6*(4 *x**4 + 24*x**3 + 48*x**2 + 36*x + 9))