Integrand size = 18, antiderivative size = 93 \[ \int \frac {1}{(2+3 x)^3 \left (1+3 x+x^2\right )} \, dx=\frac {3}{10 (2+3 x)^2}+\frac {3}{5 (2+3 x)}-\frac {8 \log \left (3-\sqrt {5}+2 x\right )}{\sqrt {5} \left (5-3 \sqrt {5}\right )^3}+\frac {8 \log \left (3+\sqrt {5}+2 x\right )}{\sqrt {5} \left (5+3 \sqrt {5}\right )^3}-\frac {18}{25} \log (2+3 x) \] Output:
3/10/(2+3*x)^2+3/(10+15*x)-8/5*ln(3-5^(1/2)+2*x)*5^(1/2)/(5-3*5^(1/2))^3+8 /5*ln(3+5^(1/2)+2*x)*5^(1/2)/(5+3*5^(1/2))^3-18/25*ln(2+3*x)
Time = 0.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(2+3 x)^3 \left (1+3 x+x^2\right )} \, dx=\frac {1}{50} \left (\frac {15}{(2+3 x)^2}+\frac {30}{2+3 x}+2 \left (9+4 \sqrt {5}\right ) \log \left (3 \left (-3+\sqrt {5}-2 x\right )\right )+\left (18-8 \sqrt {5}\right ) \log \left (3 \left (9-4 \sqrt {5}\right ) \left (3+\sqrt {5}+2 x\right )\right )-36 \log (2+3 x)\right ) \] Input:
Integrate[1/((2 + 3*x)^3*(1 + 3*x + x^2)),x]
Output:
(15/(2 + 3*x)^2 + 30/(2 + 3*x) + 2*(9 + 4*Sqrt[5])*Log[3*(-3 + Sqrt[5] - 2 *x)] + (18 - 8*Sqrt[5])*Log[3*(9 - 4*Sqrt[5])*(3 + Sqrt[5] + 2*x)] - 36*Lo g[2 + 3*x])/50
Time = 0.28 (sec) , antiderivative size = 93, 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.
\(\displaystyle \int \frac {1}{(3 x+2)^3 \left (x^2+3 x+1\right )} \, dx\) |
\(\Big \downarrow \) 1141 |
\(\displaystyle \int \left (\frac {16}{\sqrt {5} \left (5+3 \sqrt {5}\right )^3 \left (2 x+\sqrt {5}+3\right )}-\frac {54}{25 (3 x+2)}-\frac {9}{5 (3 x+2)^2}-\frac {9}{5 (3 x+2)^3}-\frac {16}{\sqrt {5} \left (5-3 \sqrt {5}\right )^3 \left (2 x-\sqrt {5}+3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{5 (3 x+2)}+\frac {3}{10 (3 x+2)^2}-\frac {8 \log \left (2 x-\sqrt {5}+3\right )}{\sqrt {5} \left (5-3 \sqrt {5}\right )^3}+\frac {8 \log \left (2 x+\sqrt {5}+3\right )}{\sqrt {5} \left (5+3 \sqrt {5}\right )^3}-\frac {18}{25} \log (3 x+2)\) |
Input:
Int[1/((2 + 3*x)^3*(1 + 3*x + x^2)),x]
Output:
3/(10*(2 + 3*x)^2) + 3/(5*(2 + 3*x)) - (8*Log[3 - Sqrt[5] + 2*x])/(Sqrt[5] *(5 - 3*Sqrt[5])^3) + (8*Log[3 + Sqrt[5] + 2*x])/(Sqrt[5]*(5 + 3*Sqrt[5])^ 3) - (18*Log[2 + 3*x])/25
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 = 0.79 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.59
method | result | size |
default | \(\frac {3}{10 \left (3 x +2\right )^{2}}+\frac {3}{5 \left (3 x +2\right )}-\frac {18 \ln \left (3 x +2\right )}{25}+\frac {9 \ln \left (x^{2}+3 x +1\right )}{25}-\frac {8 \,\operatorname {arctanh}\left (\frac {\left (2 x +3\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{25}\) | \(55\) |
risch | \(\frac {\frac {9 x}{5}+\frac {3}{2}}{\left (3 x +2\right )^{2}}-\frac {18 \ln \left (3 x +2\right )}{25}+\frac {9 \ln \left (8 x +12-4 \sqrt {5}\right )}{25}+\frac {4 \ln \left (8 x +12-4 \sqrt {5}\right ) \sqrt {5}}{25}+\frac {9 \ln \left (8 x +12+4 \sqrt {5}\right )}{25}-\frac {4 \ln \left (8 x +12+4 \sqrt {5}\right ) \sqrt {5}}{25}\) | \(82\) |
Input:
int(1/(3*x+2)^3/(x^2+3*x+1),x,method=_RETURNVERBOSE)
Output:
3/10/(3*x+2)^2+3/5/(3*x+2)-18/25*ln(3*x+2)+9/25*ln(x^2+3*x+1)-8/25*arctanh (1/5*(2*x+3)*5^(1/2))*5^(1/2)
Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(2+3 x)^3 \left (1+3 x+x^2\right )} \, dx=\frac {8 \, \sqrt {5} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (2 \, x + 3\right )} + 6 \, x + 7}{x^{2} + 3 \, x + 1}\right ) + 18 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (x^{2} + 3 \, x + 1\right ) - 36 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 90 \, x + 75}{50 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \] Input:
integrate(1/(2+3*x)^3/(x^2+3*x+1),x, algorithm="fricas")
Output:
1/50*(8*sqrt(5)*(9*x^2 + 12*x + 4)*log((2*x^2 - sqrt(5)*(2*x + 3) + 6*x + 7)/(x^2 + 3*x + 1)) + 18*(9*x^2 + 12*x + 4)*log(x^2 + 3*x + 1) - 36*(9*x^2 + 12*x + 4)*log(3*x + 2) + 90*x + 75)/(9*x^2 + 12*x + 4)
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(2+3 x)^3 \left (1+3 x+x^2\right )} \, dx=\frac {18 x + 15}{90 x^{2} + 120 x + 40} - \frac {18 \log {\left (x + \frac {2}{3} \right )}}{25} + \left (\frac {9}{25} - \frac {4 \sqrt {5}}{25}\right ) \log {\left (x - \frac {63125 \left (\frac {9}{25} - \frac {4 \sqrt {5}}{25}\right )^{2}}{15576} + \frac {43 \sqrt {5}}{1298} + \frac {39625}{15576} \right )} + \left (\frac {4 \sqrt {5}}{25} + \frac {9}{25}\right ) \log {\left (x - \frac {63125 \left (\frac {4 \sqrt {5}}{25} + \frac {9}{25}\right )^{2}}{15576} - \frac {43 \sqrt {5}}{1298} + \frac {39625}{15576} \right )} \] Input:
integrate(1/(2+3*x)**3/(x**2+3*x+1),x)
Output:
(18*x + 15)/(90*x**2 + 120*x + 40) - 18*log(x + 2/3)/25 + (9/25 - 4*sqrt(5 )/25)*log(x - 63125*(9/25 - 4*sqrt(5)/25)**2/15576 + 43*sqrt(5)/1298 + 396 25/15576) + (4*sqrt(5)/25 + 9/25)*log(x - 63125*(4*sqrt(5)/25 + 9/25)**2/1 5576 - 43*sqrt(5)/1298 + 39625/15576)
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(2+3 x)^3 \left (1+3 x+x^2\right )} \, dx=\frac {4}{25} \, \sqrt {5} \log \left (\frac {2 \, x - \sqrt {5} + 3}{2 \, x + \sqrt {5} + 3}\right ) + \frac {3 \, {\left (6 \, x + 5\right )}}{10 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {9}{25} \, \log \left (x^{2} + 3 \, x + 1\right ) - \frac {18}{25} \, \log \left (3 \, x + 2\right ) \] Input:
integrate(1/(2+3*x)^3/(x^2+3*x+1),x, algorithm="maxima")
Output:
4/25*sqrt(5)*log((2*x - sqrt(5) + 3)/(2*x + sqrt(5) + 3)) + 3/10*(6*x + 5) /(9*x^2 + 12*x + 4) + 9/25*log(x^2 + 3*x + 1) - 18/25*log(3*x + 2)
Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(2+3 x)^3 \left (1+3 x+x^2\right )} \, dx=\frac {4}{25} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x - \sqrt {5} + 3 \right |}}{{\left | 2 \, x + \sqrt {5} + 3 \right |}}\right ) + \frac {3 \, {\left (6 \, x + 5\right )}}{10 \, {\left (3 \, x + 2\right )}^{2}} + \frac {9}{25} \, \log \left ({\left | x^{2} + 3 \, x + 1 \right |}\right ) - \frac {18}{25} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \] Input:
integrate(1/(2+3*x)^3/(x^2+3*x+1),x, algorithm="giac")
Output:
4/25*sqrt(5)*log(abs(2*x - sqrt(5) + 3)/abs(2*x + sqrt(5) + 3)) + 3/10*(6* x + 5)/(3*x + 2)^2 + 9/25*log(abs(x^2 + 3*x + 1)) - 18/25*log(abs(3*x + 2) )
Time = 5.66 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(2+3 x)^3 \left (1+3 x+x^2\right )} \, dx=\frac {\frac {x}{5}+\frac {1}{6}}{x^2+\frac {4\,x}{3}+\frac {4}{9}}-\frac {18\,\ln \left (x+\frac {2}{3}\right )}{25}+\ln \left (x-\frac {\sqrt {5}}{2}+\frac {3}{2}\right )\,\left (\frac {4\,\sqrt {5}}{25}+\frac {9}{25}\right )-\ln \left (x+\frac {\sqrt {5}}{2}+\frac {3}{2}\right )\,\left (\frac {4\,\sqrt {5}}{25}-\frac {9}{25}\right ) \] Input:
int(1/((3*x + 2)^3*(3*x + x^2 + 1)),x)
Output:
(x/5 + 1/6)/((4*x)/3 + x^2 + 4/9) - (18*log(x + 2/3))/25 + log(x - 5^(1/2) /2 + 3/2)*((4*5^(1/2))/25 + 9/25) - log(x + 5^(1/2)/2 + 3/2)*((4*5^(1/2))/ 25 - 9/25)
Time = 0.22 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.24 \[ \int \frac {1}{(2+3 x)^3 \left (1+3 x+x^2\right )} \, dx=\frac {144 \sqrt {5}\, \mathrm {log}\left (-\sqrt {5}+2 x +3\right ) x^{2}+192 \sqrt {5}\, \mathrm {log}\left (-\sqrt {5}+2 x +3\right ) x +64 \sqrt {5}\, \mathrm {log}\left (-\sqrt {5}+2 x +3\right )-144 \sqrt {5}\, \mathrm {log}\left (\sqrt {5}+2 x +3\right ) x^{2}-192 \sqrt {5}\, \mathrm {log}\left (\sqrt {5}+2 x +3\right ) x -64 \sqrt {5}\, \mathrm {log}\left (\sqrt {5}+2 x +3\right )+324 \,\mathrm {log}\left (-\sqrt {5}+2 x +3\right ) x^{2}+432 \,\mathrm {log}\left (-\sqrt {5}+2 x +3\right ) x +144 \,\mathrm {log}\left (-\sqrt {5}+2 x +3\right )+324 \,\mathrm {log}\left (\sqrt {5}+2 x +3\right ) x^{2}+432 \,\mathrm {log}\left (\sqrt {5}+2 x +3\right ) x +144 \,\mathrm {log}\left (\sqrt {5}+2 x +3\right )-648 \,\mathrm {log}\left (3 x +2\right ) x^{2}-864 \,\mathrm {log}\left (3 x +2\right ) x -288 \,\mathrm {log}\left (3 x +2\right )-135 x^{2}+90}{900 x^{2}+1200 x +400} \] Input:
int(1/(2+3*x)^3/(x^2+3*x+1),x)
Output:
(144*sqrt(5)*log( - sqrt(5) + 2*x + 3)*x**2 + 192*sqrt(5)*log( - sqrt(5) + 2*x + 3)*x + 64*sqrt(5)*log( - sqrt(5) + 2*x + 3) - 144*sqrt(5)*log(sqrt( 5) + 2*x + 3)*x**2 - 192*sqrt(5)*log(sqrt(5) + 2*x + 3)*x - 64*sqrt(5)*log (sqrt(5) + 2*x + 3) + 324*log( - sqrt(5) + 2*x + 3)*x**2 + 432*log( - sqrt (5) + 2*x + 3)*x + 144*log( - sqrt(5) + 2*x + 3) + 324*log(sqrt(5) + 2*x + 3)*x**2 + 432*log(sqrt(5) + 2*x + 3)*x + 144*log(sqrt(5) + 2*x + 3) - 648 *log(3*x + 2)*x**2 - 864*log(3*x + 2)*x - 288*log(3*x + 2) - 135*x**2 + 90 )/(100*(9*x**2 + 12*x + 4))