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

Optimal. Leaf size=16 \[ \frac{2 x}{3}-\frac{1}{9} \log (3 x+2) \]

[Out]

(2*x)/3 - Log[2 + 3*x]/9

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Rubi [A]  time = 0.0063008, antiderivative size = 16, 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} \[ \frac{2 x}{3}-\frac{1}{9} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x)/(2 + 3*x),x]

[Out]

(2*x)/3 - Log[2 + 3*x]/9

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}{2+3 x} \, dx &=\int \left (\frac{2}{3}-\frac{1}{3 (2+3 x)}\right ) \, dx\\ &=\frac{2 x}{3}-\frac{1}{9} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0027664, size = 17, normalized size = 1.06 \[ \frac{1}{9} (6 x-\log (3 x+2)+4) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x)/(2 + 3*x),x]

[Out]

(4 + 6*x - Log[2 + 3*x])/9

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Maple [A]  time = 0.001, size = 13, normalized size = 0.8 \begin{align*}{\frac{2\,x}{3}}-{\frac{\ln \left ( 2+3\,x \right ) }{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x-1/9*ln(2+3*x)

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Maxima [A]  time = 0.947144, size = 16, normalized size = 1. \begin{align*} \frac{2}{3} \, x - \frac{1}{9} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x - 1/9*log(3*x + 2)

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Fricas [A]  time = 1.51052, size = 35, normalized size = 2.19 \begin{align*} \frac{2}{3} \, x - \frac{1}{9} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x - 1/9*log(3*x + 2)

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Sympy [A]  time = 0.069165, size = 12, normalized size = 0.75 \begin{align*} \frac{2 x}{3} - \frac{\log{\left (3 x + 2 \right )}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x/3 - log(3*x + 2)/9

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Giac [A]  time = 1.07865, size = 18, normalized size = 1.12 \begin{align*} \frac{2}{3} \, x - \frac{1}{9} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x - 1/9*log(abs(3*x + 2))