Integrand size = 18, antiderivative size = 82 \[ \int \frac {1}{(2+3 x)^2 \left (1+3 x+x^2\right )} \, dx=\frac {3}{5 (2+3 x)}+\frac {4 \log \left (3-\sqrt {5}+2 x\right )}{\sqrt {5} \left (5-3 \sqrt {5}\right )^2}-\frac {4 \log \left (3+\sqrt {5}+2 x\right )}{\sqrt {5} \left (5+3 \sqrt {5}\right )^2}-\frac {3}{5} \log (2+3 x) \] Output:
3/(10+15*x)+4/5*ln(3-5^(1/2)+2*x)*5^(1/2)/(5-3*5^(1/2))^2-4/5*ln(3+5^(1/2) +2*x)*5^(1/2)/(5+3*5^(1/2))^2-3/5*ln(2+3*x)
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(2+3 x)^2 \left (1+3 x+x^2\right )} \, dx=\frac {1}{50} \left (\left (15+7 \sqrt {5}\right ) \log \left (3 \left (-3+\sqrt {5}-2 x\right )\right )+\left (15-7 \sqrt {5}\right ) \log \left (3 \left (3+\sqrt {5}+2 x\right )\right )+30 \left (\frac {1}{2+3 x}-\log (2+3 x)\right )\right ) \] Input:
Integrate[1/((2 + 3*x)^2*(1 + 3*x + x^2)),x]
Output:
((15 + 7*Sqrt[5])*Log[3*(-3 + Sqrt[5] - 2*x)] + (15 - 7*Sqrt[5])*Log[3*(3 + Sqrt[5] + 2*x)] + 30*((2 + 3*x)^(-1) - Log[2 + 3*x]))/50
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1141, 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.
\(\displaystyle \int \frac {1}{(3 x+2)^2 \left (x^2+3 x+1\right )} \, dx\) |
\(\Big \downarrow \) 1141 |
\(\displaystyle \int \left (-\frac {8}{\sqrt {5} \left (5+3 \sqrt {5}\right )^2 \left (2 x+\sqrt {5}+3\right )}-\frac {9}{5 (3 x+2)}-\frac {9}{5 (3 x+2)^2}+\frac {8}{\sqrt {5} \left (5-3 \sqrt {5}\right )^2 \left (2 x-\sqrt {5}+3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{5 (3 x+2)}+\frac {4 \log \left (2 x-\sqrt {5}+3\right )}{\sqrt {5} \left (5-3 \sqrt {5}\right )^2}-\frac {4 \log \left (2 x+\sqrt {5}+3\right )}{\sqrt {5} \left (5+3 \sqrt {5}\right )^2}-\frac {3}{5} \log (3 x+2)\) |
Input:
Int[1/((2 + 3*x)^2*(1 + 3*x + x^2)),x]
Output:
3/(5*(2 + 3*x)) + (4*Log[3 - Sqrt[5] + 2*x])/(Sqrt[5]*(5 - 3*Sqrt[5])^2) - (4*Log[3 + Sqrt[5] + 2*x])/(Sqrt[5]*(5 + 3*Sqrt[5])^2) - (3*Log[2 + 3*x]) /5
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[ (d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 1] || !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
Time = 0.76 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.56
method | result | size |
default | \(\frac {3}{5 \left (3 x +2\right )}-\frac {3 \ln \left (3 x +2\right )}{5}+\frac {3 \ln \left (x^{2}+3 x +1\right )}{10}-\frac {7 \,\operatorname {arctanh}\left (\frac {\left (2 x +3\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{25}\) | \(46\) |
risch | \(\frac {1}{5 x +\frac {10}{3}}+\frac {3 \ln \left (3-\sqrt {5}+2 x \right )}{10}+\frac {7 \ln \left (3-\sqrt {5}+2 x \right ) \sqrt {5}}{50}+\frac {3 \ln \left (3+\sqrt {5}+2 x \right )}{10}-\frac {7 \ln \left (3+\sqrt {5}+2 x \right ) \sqrt {5}}{50}-\frac {3 \ln \left (3 x +2\right )}{5}\) | \(71\) |
Input:
int(1/(3*x+2)^2/(x^2+3*x+1),x,method=_RETURNVERBOSE)
Output:
3/5/(3*x+2)-3/5*ln(3*x+2)+3/10*ln(x^2+3*x+1)-7/25*arctanh(1/5*(2*x+3)*5^(1 /2))*5^(1/2)
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(2+3 x)^2 \left (1+3 x+x^2\right )} \, dx=\frac {7 \, \sqrt {5} {\left (3 \, x + 2\right )} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (2 \, x + 3\right )} + 6 \, x + 7}{x^{2} + 3 \, x + 1}\right ) + 15 \, {\left (3 \, x + 2\right )} \log \left (x^{2} + 3 \, x + 1\right ) - 30 \, {\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) + 30}{50 \, {\left (3 \, x + 2\right )}} \] Input:
integrate(1/(2+3*x)^2/(x^2+3*x+1),x, algorithm="fricas")
Output:
1/50*(7*sqrt(5)*(3*x + 2)*log((2*x^2 - sqrt(5)*(2*x + 3) + 6*x + 7)/(x^2 + 3*x + 1)) + 15*(3*x + 2)*log(x^2 + 3*x + 1) - 30*(3*x + 2)*log(3*x + 2) + 30)/(3*x + 2)
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(2+3 x)^2 \left (1+3 x+x^2\right )} \, dx=- \frac {3 \log {\left (x + \frac {2}{3} \right )}}{5} + \left (\frac {3}{10} - \frac {7 \sqrt {5}}{50}\right ) \log {\left (x - \frac {10625 \left (\frac {3}{10} - \frac {7 \sqrt {5}}{50}\right )^{2}}{1869} + \frac {2 \sqrt {5}}{89} + \frac {4801}{1869} \right )} + \left (\frac {3}{10} + \frac {7 \sqrt {5}}{50}\right ) \log {\left (x - \frac {10625 \left (\frac {3}{10} + \frac {7 \sqrt {5}}{50}\right )^{2}}{1869} - \frac {2 \sqrt {5}}{89} + \frac {4801}{1869} \right )} + \frac {3}{15 x + 10} \] Input:
integrate(1/(2+3*x)**2/(x**2+3*x+1),x)
Output:
-3*log(x + 2/3)/5 + (3/10 - 7*sqrt(5)/50)*log(x - 10625*(3/10 - 7*sqrt(5)/ 50)**2/1869 + 2*sqrt(5)/89 + 4801/1869) + (3/10 + 7*sqrt(5)/50)*log(x - 10 625*(3/10 + 7*sqrt(5)/50)**2/1869 - 2*sqrt(5)/89 + 4801/1869) + 3/(15*x + 10)
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(2+3 x)^2 \left (1+3 x+x^2\right )} \, dx=\frac {7}{50} \, \sqrt {5} \log \left (\frac {2 \, x - \sqrt {5} + 3}{2 \, x + \sqrt {5} + 3}\right ) + \frac {3}{5 \, {\left (3 \, x + 2\right )}} + \frac {3}{10} \, \log \left (x^{2} + 3 \, x + 1\right ) - \frac {3}{5} \, \log \left (3 \, x + 2\right ) \] Input:
integrate(1/(2+3*x)^2/(x^2+3*x+1),x, algorithm="maxima")
Output:
7/50*sqrt(5)*log((2*x - sqrt(5) + 3)/(2*x + sqrt(5) + 3)) + 3/5/(3*x + 2) + 3/10*log(x^2 + 3*x + 1) - 3/5*log(3*x + 2)
Time = 0.33 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(2+3 x)^2 \left (1+3 x+x^2\right )} \, dx=-\frac {7}{50} \, \sqrt {5} \log \left (\frac {{\left | -9 \, \sqrt {5} - \frac {30}{3 \, x + 2} + 15 \right |}}{{\left | 9 \, \sqrt {5} - \frac {30}{3 \, x + 2} + 15 \right |}}\right ) + \frac {3}{5 \, {\left (3 \, x + 2\right )}} + \frac {3}{10} \, \log \left ({\left | -\frac {5}{3 \, x + 2} + \frac {5}{{\left (3 \, x + 2\right )}^{2}} - 1 \right |}\right ) \] Input:
integrate(1/(2+3*x)^2/(x^2+3*x+1),x, algorithm="giac")
Output:
-7/50*sqrt(5)*log(abs(-9*sqrt(5) - 30/(3*x + 2) + 15)/abs(9*sqrt(5) - 30/( 3*x + 2) + 15)) + 3/5/(3*x + 2) + 3/10*log(abs(-5/(3*x + 2) + 5/(3*x + 2)^ 2 - 1))
Time = 5.69 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(2+3 x)^2 \left (1+3 x+x^2\right )} \, dx=\frac {1}{5\,\left (x+\frac {2}{3}\right )}-\frac {3\,\ln \left (x+\frac {2}{3}\right )}{5}+\ln \left (x-\frac {\sqrt {5}}{2}+\frac {3}{2}\right )\,\left (\frac {7\,\sqrt {5}}{50}+\frac {3}{10}\right )-\ln \left (x+\frac {\sqrt {5}}{2}+\frac {3}{2}\right )\,\left (\frac {7\,\sqrt {5}}{50}-\frac {3}{10}\right ) \] Input:
int(1/((3*x + 2)^2*(3*x + x^2 + 1)),x)
Output:
1/(5*(x + 2/3)) - (3*log(x + 2/3))/5 + log(x - 5^(1/2)/2 + 3/2)*((7*5^(1/2 ))/50 + 3/10) - log(x + 5^(1/2)/2 + 3/2)*((7*5^(1/2))/50 - 3/10)
Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.57 \[ \int \frac {1}{(2+3 x)^2 \left (1+3 x+x^2\right )} \, dx=\frac {21 \sqrt {5}\, \mathrm {log}\left (-\sqrt {5}+2 x +3\right ) x +14 \sqrt {5}\, \mathrm {log}\left (-\sqrt {5}+2 x +3\right )-21 \sqrt {5}\, \mathrm {log}\left (\sqrt {5}+2 x +3\right ) x -14 \sqrt {5}\, \mathrm {log}\left (\sqrt {5}+2 x +3\right )+45 \,\mathrm {log}\left (-\sqrt {5}+2 x +3\right ) x +30 \,\mathrm {log}\left (-\sqrt {5}+2 x +3\right )+45 \,\mathrm {log}\left (\sqrt {5}+2 x +3\right ) x +30 \,\mathrm {log}\left (\sqrt {5}+2 x +3\right )-90 \,\mathrm {log}\left (3 x +2\right ) x -60 \,\mathrm {log}\left (3 x +2\right )-45 x}{150 x +100} \] Input:
int(1/(2+3*x)^2/(x^2+3*x+1),x)
Output:
(21*sqrt(5)*log( - sqrt(5) + 2*x + 3)*x + 14*sqrt(5)*log( - sqrt(5) + 2*x + 3) - 21*sqrt(5)*log(sqrt(5) + 2*x + 3)*x - 14*sqrt(5)*log(sqrt(5) + 2*x + 3) + 45*log( - sqrt(5) + 2*x + 3)*x + 30*log( - sqrt(5) + 2*x + 3) + 45* log(sqrt(5) + 2*x + 3)*x + 30*log(sqrt(5) + 2*x + 3) - 90*log(3*x + 2)*x - 60*log(3*x + 2) - 45*x)/(50*(3*x + 2))