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

Optimal. Leaf size=17 \[ -5+3 (5+x)-\log \left (\frac {1+x}{3}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 0.59, 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} \begin {gather*} 3 x-\log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)/(1 + x),x]

[Out]

3*x - 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 {gather*} \begin {aligned} \text {integral} &=\int \left (3+\frac {1}{-1-x}\right ) \, dx\\ &=3 x-\log (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.71 \begin {gather*} 3 (1+x)-\log (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)/(1 + x),x]

[Out]

3*(1 + x) - Log[1 + x]

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fricas [A]  time = 0.64, size = 10, normalized size = 0.59 \begin {gather*} 3 \, x - \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x - log(x + 1)

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giac [A]  time = 0.16, size = 11, normalized size = 0.65 \begin {gather*} 3 \, x - \log \left ({\left | x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x - log(abs(x + 1))

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maple [A]  time = 0.07, size = 11, normalized size = 0.65

method	result	size
default	\(3 x -\ln \left (x +1\right )\)	\(11\)
norman	\(3 x -\ln \left (x +1\right )\)	\(11\)
meijerg	\(3 x -\ln \left (x +1\right )\)	\(11\)
risch	\(3 x -\ln \left (x +1\right )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)/(x+1),x,method=_RETURNVERBOSE)

[Out]

3*x-ln(x+1)

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maxima [A]  time = 0.39, size = 10, normalized size = 0.59 \begin {gather*} 3 \, x - \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x - log(x + 1)

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mupad [B]  time = 3.93, size = 10, normalized size = 0.59 \begin {gather*} 3\,x-\ln \left (x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)/(x + 1),x)

[Out]

3*x - log(x + 1)

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sympy [A]  time = 0.06, size = 7, normalized size = 0.41 \begin {gather*} 3 x - \log {\left (x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x - log(x + 1)

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