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

Optimal. Leaf size=12 \[ 2 x+11 \log (3-x) \]

[Out]

2*x + 11*Log[3 - x]

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Rubi [A]  time = 0.0063038, antiderivative size = 12, normalized size of antiderivative = 1., 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} \[ 2 x+11 \log (3-x) \]

Antiderivative was successfully verified.

[In]

Int[(5 + 2*x)/(-3 + x),x]

[Out]

2*x + 11*Log[3 - 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{5+2 x}{-3+x} \, dx &=\int \left (2+\frac{11}{-3+x}\right ) \, dx\\ &=2 x+11 \log (3-x)\\ \end{align*}

Mathematica [A]  time = 0.0031168, size = 12, normalized size = 1. \[ 2 (x-3)+11 \log (x-3) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + 2*x)/(-3 + x),x]

[Out]

2*(-3 + x) + 11*Log[-3 + x]

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Maple [A]  time = 0.003, size = 11, normalized size = 0.9 \begin{align*} 2\,x+11\,\ln \left ( -3+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x+11*ln(-3+x)

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Maxima [A]  time = 0.958498, size = 14, normalized size = 1.17 \begin{align*} 2 \, x + 11 \, \log \left (x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + 11*log(x - 3)

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Fricas [A]  time = 1.79746, size = 28, normalized size = 2.33 \begin{align*} 2 \, x + 11 \, \log \left (x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + 11*log(x - 3)

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Sympy [A]  time = 0.068852, size = 8, normalized size = 0.67 \begin{align*} 2 x + 11 \log{\left (x - 3 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + 11*log(x - 3)

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Giac [A]  time = 1.046, size = 15, normalized size = 1.25 \begin{align*} 2 \, x + 11 \, \log \left ({\left | x - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + 11*log(abs(x - 3))