\(\int \frac {1}{x (1+x) (3+2 x)} \, dx\) [185]

Optimal result

Integrand size = 16, antiderivative size = 23 \[ \int \frac {1}{x (1+x) (3+2 x)} \, dx=\frac {\log (x)}{3}-\log (1+x)+\frac {2}{3} \log (3+2 x) \]

[Out]

1/3*ln(x)-ln(1+x)+2/3*ln(3+2*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {84} \[ \int \frac {1}{x (1+x) (3+2 x)} \, dx=\frac {\log (x)}{3}-\log (x+1)+\frac {2}{3} \log (2 x+3) \]

[In]

Int[1/(x*(1 + x)*(3 + 2*x)),x]

[Out]

Log[x]/3 - Log[1 + x] + (2*Log[3 + 2*x])/3

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{-1-x}+\frac {1}{3 x}+\frac {4}{3 (3+2 x)}\right ) \, dx \\ & = \frac {\log (x)}{3}-\log (1+x)+\frac {2}{3} \log (3+2 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (1+x) (3+2 x)} \, dx=\frac {\log (x)}{3}-\log (1+x)+\frac {2}{3} \log (3+2 x) \]

[In]

Integrate[1/(x*(1 + x)*(3 + 2*x)),x]

[Out]

Log[x]/3 - Log[1 + x] + (2*Log[3 + 2*x])/3

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78

method	result	size
parallelrisch	\(\frac {\ln \left (x \right )}{3}-\ln \left (1+x \right )+\frac {2 \ln \left (\frac {3}{2}+x \right )}{3}\)	\(18\)
default	\(\frac {\ln \left (x \right )}{3}-\ln \left (1+x \right )+\frac {2 \ln \left (3+2 x \right )}{3}\)	\(20\)
norman	\(\frac {\ln \left (x \right )}{3}-\ln \left (1+x \right )+\frac {2 \ln \left (3+2 x \right )}{3}\)	\(20\)
risch	\(\frac {\ln \left (x \right )}{3}-\ln \left (1+x \right )+\frac {2 \ln \left (3+2 x \right )}{3}\)	\(20\)

[In]

int(1/x/(1+x)/(3+2*x),x,method=_RETURNVERBOSE)

[Out]

1/3*ln(x)-ln(1+x)+2/3*ln(3/2+x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x (1+x) (3+2 x)} \, dx=\frac {2}{3} \, \log \left (2 \, x + 3\right ) - \log \left (x + 1\right ) + \frac {1}{3} \, \log \left (x\right ) \]

[In]

integrate(1/x/(1+x)/(3+2*x),x, algorithm="fricas")

[Out]

2/3*log(2*x + 3) - log(x + 1) + 1/3*log(x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x (1+x) (3+2 x)} \, dx=\frac {\log {\left (x \right )}}{3} - \log {\left (x + 1 \right )} + \frac {2 \log {\left (x + \frac {3}{2} \right )}}{3} \]

[In]

integrate(1/x/(1+x)/(3+2*x),x)

[Out]

log(x)/3 - log(x + 1) + 2*log(x + 3/2)/3

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x (1+x) (3+2 x)} \, dx=\frac {2}{3} \, \log \left (2 \, x + 3\right ) - \log \left (x + 1\right ) + \frac {1}{3} \, \log \left (x\right ) \]

[In]

integrate(1/x/(1+x)/(3+2*x),x, algorithm="maxima")

[Out]

2/3*log(2*x + 3) - log(x + 1) + 1/3*log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x (1+x) (3+2 x)} \, dx=\frac {2}{3} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{3} \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate(1/x/(1+x)/(3+2*x),x, algorithm="giac")

[Out]

2/3*log(abs(2*x + 3)) - log(abs(x + 1)) + 1/3*log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x (1+x) (3+2 x)} \, dx=\frac {2\,\ln \left (x+\frac {3}{2}\right )}{3}-\ln \left (x+1\right )+\frac {\ln \left (x\right )}{3} \]

[In]

int(1/(x*(2*x + 3)*(x + 1)),x)

[Out]

(2*log(x + 3/2))/3 - log(x + 1) + log(x)/3