3.26 \(\int \frac{-1+2 x}{3+2 x} \, dx\)

Optimal. Leaf size=10 \[ x-2 \log (2 x+3) \]

[Out]

x - 2*Log[3 + 2*x]

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Rubi [A]  time = 0.0050317, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ x-2 \log (2 x+3) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 2*x)/(3 + 2*x),x]

[Out]

x - 2*Log[3 + 2*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{-1+2 x}{3+2 x} \, dx &=\int \left (1-\frac{4}{3+2 x}\right ) \, dx\\ &=x-2 \log (3+2 x)\\ \end{align*}

Mathematica [A]  time = 0.0017653, size = 10, normalized size = 1. \[ x-2 \log (2 x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 2*x)/(3 + 2*x),x]

[Out]

x - 2*Log[3 + 2*x]

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Maple [A]  time = 0.001, size = 11, normalized size = 1.1 \begin{align*} x-2\,\ln \left ( 3+2\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.941657, size = 14, normalized size = 1.4 \begin{align*} x - 2 \, \log \left (2 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(2*x + 3)

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Fricas [A]  time = 2.07079, size = 27, normalized size = 2.7 \begin{align*} x - 2 \, \log \left (2 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(2*x + 3)

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Sympy [A]  time = 0.072634, size = 8, normalized size = 0.8 \begin{align*} x - 2 \log{\left (2 x + 3 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(2*x + 3)

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Giac [A]  time = 1.04953, size = 15, normalized size = 1.5 \begin{align*} x - 2 \, \log \left ({\left | 2 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(abs(2*x + 3))