3.177 \(\int \frac{x}{-5+x} \, dx\)

Optimal. Leaf size=10 \[ x+5 \log (5-x) \]

[Out]

x + 5*Log[5 - x]

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Rubi [A]  time = 0.0044039, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {43} \[ x+5 \log (5-x) \]

Antiderivative was successfully verified.

[In]

Int[x/(-5 + x),x]

[Out]

x + 5*Log[5 - 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{x}{-5+x} \, dx &=\int \left (1+\frac{5}{-5+x}\right ) \, dx\\ &=x+5 \log (5-x)\\ \end{align*}

Mathematica [A]  time = 0.0014074, size = 8, normalized size = 0.8 \[ x+5 \log (x-5) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(-5 + x),x]

[Out]

x + 5*Log[-5 + x]

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Maple [A]  time = 0.002, size = 9, normalized size = 0.9 \begin{align*} x+5\,\ln \left ( -5+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-5+x),x)

[Out]

x+5*ln(-5+x)

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Maxima [A]  time = 0.927806, size = 11, normalized size = 1.1 \begin{align*} x + 5 \, \log \left (x - 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 5*log(x - 5)

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Fricas [A]  time = 1.78642, size = 24, normalized size = 2.4 \begin{align*} x + 5 \, \log \left (x - 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 5*log(x - 5)

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Sympy [A]  time = 0.067517, size = 7, normalized size = 0.7 \begin{align*} x + 5 \log{\left (x - 5 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-5+x),x)

[Out]

x + 5*log(x - 5)

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Giac [A]  time = 1.05003, size = 12, normalized size = 1.2 \begin{align*} x + 5 \, \log \left ({\left | x - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 5*log(abs(x - 5))