### 3.185 $$\int \frac{1}{x (1+x) (3+2 x)} \, dx$$

Optimal. Leaf size=23 $\frac{\log (x)}{3}-\log (x+1)+\frac{2}{3} \log (2 x+3)$

[Out]

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

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Rubi [A]  time = 0.0080618, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.062, Rules used = {72} $\frac{\log (x)}{3}-\log (x+1)+\frac{2}{3} \log (2 x+3)$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x*(1 + x)*(3 + 2*x)),x]

[Out]

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

Rule 72

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*} \int \frac{1}{x (1+x) (3+2 x)} \, dx &=\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]  time = 0.0044875, size = 23, normalized size = 1. $\frac{\log (x)}{3}-\log (x+1)+\frac{2}{3} \log (2 x+3)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x*(1 + x)*(3 + 2*x)),x]

[Out]

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

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Maple [A]  time = 0.006, size = 20, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{3}}-\ln \left ( 1+x \right ) +{\frac{2\,\ln \left ( 3+2\,x \right ) }{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.934752, size = 26, normalized size = 1.13 \begin{align*} \frac{2}{3} \, \log \left (2 \, x + 3\right ) - \log \left (x + 1\right ) + \frac{1}{3} \, \log \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.90344, size = 59, normalized size = 2.57 \begin{align*} \frac{2}{3} \, \log \left (2 \, x + 3\right ) - \log \left (x + 1\right ) + \frac{1}{3} \, \log \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.117938, size = 19, normalized size = 0.83 \begin{align*} \frac{\log{\left (x \right )}}{3} - \log{\left (x + 1 \right )} + \frac{2 \log{\left (x + \frac{3}{2} \right )}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05534, size = 30, normalized size = 1.3 \begin{align*} \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 ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))