3.2.80 \(\int \frac {-5+6 x}{3+2 x} \, dx\) [180]

Optimal. Leaf size=12 \[ 3 x-7 \log (3+2 x) \]

[Out]

3*x-7*ln(3+2*x)

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} 3 x-7 \log (2 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + 6*x)/(3 + 2*x),x]

[Out]

3*x - 7*Log[3 + 2*x]

Rule 45

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 {-5+6 x}{3+2 x} \, dx &=\int \left (3-\frac {14}{3+2 x}\right ) \, dx\\ &=3 x-7 \log (3+2 x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.25 \begin {gather*} \frac {9}{2}+3 x-7 \log (3+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + 6*x)/(3 + 2*x),x]

[Out]

9/2 + 3*x - 7*Log[3 + 2*x]

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Mathics [A]
time = 1.69, size = 12, normalized size = 1.00 \begin {gather*} 3 x-7 \text {Log}\left [3+2 x\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(-5 + 6*x)/(3 + 2*x),x]')

[Out]

3 x - 7 Log[3 + 2 x]

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Maple [A]
time = 0.05, size = 13, normalized size = 1.08

method	result	size
default	\(3 x -7 \ln \left (3+2 x \right )\)	\(13\)
norman	\(3 x -7 \ln \left (3+2 x \right )\)	\(13\)
meijerg	\(-7 \ln \left (1+\frac {2 x}{3}\right )+3 x\)	\(13\)
risch	\(3 x -7 \ln \left (3+2 x \right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5+6*x)/(3+2*x),x,method=_RETURNVERBOSE)

[Out]

3*x-7*ln(3+2*x)

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Maxima [A]
time = 0.27, size = 12, normalized size = 1.00 \begin {gather*} 3 \, x - 7 \, \log \left (2 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+6*x)/(3+2*x),x, algorithm="maxima")

[Out]

3*x - 7*log(2*x + 3)

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Fricas [A]
time = 0.32, size = 12, normalized size = 1.00 \begin {gather*} 3 \, x - 7 \, \log \left (2 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+6*x)/(3+2*x),x, algorithm="fricas")

[Out]

3*x - 7*log(2*x + 3)

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Sympy [A]
time = 0.04, size = 10, normalized size = 0.83 \begin {gather*} 3 x - 7 \log {\left (2 x + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+6*x)/(3+2*x),x)

[Out]

3*x - 7*log(2*x + 3)

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Giac [A]
time = 0.00, size = 16, normalized size = 1.33 \begin {gather*} \frac {6}{2} x-7 \ln \left |2 x+3\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+6*x)/(3+2*x),x)

[Out]

3*x - 7*log(abs(2*x + 3))

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Mupad [B]
time = 0.16, size = 10, normalized size = 0.83 \begin {gather*} 3\,x-7\,\ln \left (x+\frac {3}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x - 5)/(2*x + 3),x)

[Out]

3*x - 7*log(x + 3/2)

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