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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ 2 (x-3)+11 \log (x-3) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + 2*x)/(-3 + x),x]

[Out]

2*(-3 + x) + 11*Log[-3 + x]

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fricas [A]  time = 0.44, size = 10, normalized size = 0.83 \[ 2 \, x + 11 \, \log \left (x - 3\right ) \]

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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giac [A]  time = 1.04, size = 11, normalized size = 0.92 \[ 2 \, x + 11 \, \log \left ({\left | x - 3 \right |}\right ) \]

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))

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maple [A]  time = 0.00, size = 11, normalized size = 0.92 \[ 2 x +11 \ln \left (x -3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 10, normalized size = 0.83 \[ 2 \, x + 11 \, \log \left (x - 3\right ) \]

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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mupad [B]  time = 0.03, size = 10, normalized size = 0.83 \[ 2\,x+11\,\ln \left (x-3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 5)/(x - 3),x)

[Out]

2*x + 11*log(x - 3)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.67 \[ 2 x + 11 \log {\left (x - 3 \right )} \]

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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