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

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ 2 \log (x+1)-\frac {1}{x+1} \]

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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fricas [A]  time = 0.39, size = 17, normalized size = 1.21 \[ \frac {2 \, {\left (x + 1\right )} \log \left (x + 1\right ) - 1}{x + 1} \]

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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giac [A]  time = 1.09, size = 15, normalized size = 1.07 \[ -\frac {1}{x + 1} + 2 \, \log \left ({\left | x + 1 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 15, normalized size = 1.07 \[ 2 \ln \left (x +1\right )-\frac {1}{x +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.49, size = 14, normalized size = 1.00 \[ -\frac {1}{x + 1} + 2 \, \log \left (x + 1\right ) \]

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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mupad [B]  time = 0.03, size = 14, normalized size = 1.00 \[ 2\,\ln \left (x+1\right )-\frac {1}{x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 3)/(x + 1)^2,x)

[Out]

2*log(x + 1) - 1/(x + 1)

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sympy [A]  time = 0.09, size = 10, normalized size = 0.71 \[ 2 \log {\left (x + 1 \right )} - \frac {1}{x + 1} \]

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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