### 3.184 $$\int \frac{3+2 x}{(1+x)^2} \, dx$$

Optimal. Leaf size=14 $2 \log (x+1)-\frac{1}{x+1}$

[Out]

-(1 + x)^(-1) + 2*Log[1 + x]

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Rubi [A]  time = 0.0053683, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {43} $2 \log (x+1)-\frac{1}{x+1}$

Antiderivative was successfully veriﬁed.

[In]

Int[(3 + 2*x)/(1 + x)^2,x]

[Out]

-(1 + x)^(-1) + 2*Log[1 + x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0036457, size = 14, normalized size = 1. $2 \log (x+1)-\frac{1}{x+1}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 + 2*x)/(1 + x)^2,x]

[Out]

-(1 + x)^(-1) + 2*Log[1 + x]

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Maple [A]  time = 0.004, size = 15, normalized size = 1.1 \begin{align*} - \left ( 1+x \right ) ^{-1}+2\,\ln \left ( 1+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.930263, size = 19, normalized size = 1.36 \begin{align*} -\frac{1}{x + 1} + 2 \, \log \left (x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(x + 1) + 2*log(x + 1)

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Fricas [A]  time = 1.87043, size = 49, normalized size = 3.5 \begin{align*} \frac{2 \,{\left (x + 1\right )} \log \left (x + 1\right ) - 1}{x + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(x + 1)*log(x + 1) - 1)/(x + 1)

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Sympy [A]  time = 0.075777, size = 10, normalized size = 0.71 \begin{align*} 2 \log{\left (x + 1 \right )} - \frac{1}{x + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x + 1) - 1/(x + 1)

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Giac [A]  time = 1.04253, size = 20, normalized size = 1.43 \begin{align*} -\frac{1}{x + 1} + 2 \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(x + 1) + 2*log(abs(x + 1))