3.3.60 \(\int \frac {5+2 x}{-3+x} \, dx\) [260]

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

[Out]

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

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Rubi [A]
time = 0.00, 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 = {45} \begin {gather*} 2 x+11 \log (3-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 + 2*x)/(-3 + x),x]

[Out]

2*x + 11*Log[3 - 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+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 \begin {gather*} 2 (-3+x)+11 \log (-3+x) \end {gather*}

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 = 0.05, size = 11, normalized size = 0.92

method	result	size
default	\(2 x +11 \ln \left (-3+x \right )\)	\(11\)
norman	\(2 x +11 \ln \left (-3+x \right )\)	\(11\)
risch	\(2 x +11 \ln \left (-3+x \right )\)	\(11\)
meijerg	\(11 \ln \left (1-\frac {x}{3}\right )+2 x\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 2.47, size = 10, normalized size = 0.83 \begin {gather*} 2 \, x + 11 \, \log \left (x - 3\right ) \end {gather*}

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 = 0.44, size = 10, normalized size = 0.83 \begin {gather*} 2 \, x + 11 \, \log \left (x - 3\right ) \end {gather*}

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 = 0.02, size = 8, normalized size = 0.67 \begin {gather*} 2 x + 11 \log {\left (x - 3 \right )} \end {gather*}

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 = 0.44, size = 11, normalized size = 0.92 \begin {gather*} 2 \, x + 11 \, \log \left ({\left | x - 3 \right |}\right ) \end {gather*}

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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Mupad [B]
time = 0.03, size = 10, normalized size = 0.83 \begin {gather*} 2\,x+11\,\ln \left (x-3\right ) \end {gather*}

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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