3.13.3 \(\int \frac {-5-2 x}{3+2 x} \, dx\)

Optimal. Leaf size=24 \[ \log \left (-\frac {e^{2-x}}{\left (2+e^5\right ) (3+2 x)}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 0.50, 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} \begin {gather*} -x-\log (2 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 - 2*x)/(3 + 2*x),x]

[Out]

-x - 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 {gather*} \begin {aligned} \text {integral} &=\int \left (-1-\frac {2}{3+2 x}\right ) \, dx\\ &=-x-\log (3+2 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.50 \begin {gather*} -x-\log (3+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 - 2*x)/(3 + 2*x),x]

[Out]

-x - Log[3 + 2*x]

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fricas [A]  time = 0.81, size = 12, normalized size = 0.50 \begin {gather*} -x - \log \left (2 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-5)/(2*x+3),x, algorithm="fricas")

[Out]

-x - log(2*x + 3)

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giac [A]  time = 0.26, size = 13, normalized size = 0.54 \begin {gather*} -x - \log \left ({\left | 2 \, x + 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-5)/(2*x+3),x, algorithm="giac")

[Out]

-x - log(abs(2*x + 3))

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maple [A]  time = 0.20, size = 13, normalized size = 0.54

method	result	size
default	\(-x -\ln \left (2 x +3\right )\)	\(13\)
norman	\(-x -\ln \left (2 x +3\right )\)	\(13\)
meijerg	\(-\ln \left (1+\frac {2 x}{3}\right )-x\)	\(13\)
risch	\(-x -\ln \left (2 x +3\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x-5)/(2*x+3),x,method=_RETURNVERBOSE)

[Out]

-x-ln(2*x+3)

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maxima [A]  time = 0.38, size = 12, normalized size = 0.50 \begin {gather*} -x - \log \left (2 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-5)/(2*x+3),x, algorithm="maxima")

[Out]

-x - log(2*x + 3)

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mupad [B]  time = 0.07, size = 10, normalized size = 0.42 \begin {gather*} -x-\ln \left (x+\frac {3}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + 5)/(2*x + 3),x)

[Out]

- x - log(x + 3/2)

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sympy [A]  time = 0.06, size = 8, normalized size = 0.33 \begin {gather*} - x - \log {\left (2 x + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-5)/(2*x+3),x)

[Out]

-x - log(2*x + 3)

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