3.51.11 \(\int \frac {x+(9-x) \log (9-x)}{-9 x^2+x^3} \, dx\)

Optimal. Leaf size=10 \[ \frac {\log (9-x)}{x} \]

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Rubi [A]  time = 0.12, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1593, 6742, 36, 31, 29, 2395} \begin {gather*} \frac {\log (9-x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + (9 - x)*Log[9 - x])/(-9*x^2 + x^3),x]

[Out]

Log[9 - x]/x

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x+(9-x) \log (9-x)}{(-9+x) x^2} \, dx\\ &=\int \left (\frac {1}{(-9+x) x}-\frac {\log (9-x)}{x^2}\right ) \, dx\\ &=\int \frac {1}{(-9+x) x} \, dx-\int \frac {\log (9-x)}{x^2} \, dx\\ &=\frac {\log (9-x)}{x}+\frac {1}{9} \int \frac {1}{-9+x} \, dx-\frac {1}{9} \int \frac {1}{x} \, dx+\int \frac {1}{(9-x) x} \, dx\\ &=\frac {1}{9} \log (9-x)+\frac {\log (9-x)}{x}-\frac {\log (x)}{9}+\frac {1}{9} \int \frac {1}{9-x} \, dx+\frac {1}{9} \int \frac {1}{x} \, dx\\ &=\frac {\log (9-x)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log (9-x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + (9 - x)*Log[9 - x])/(-9*x^2 + x^3),x]

[Out]

Log[9 - x]/x

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fricas [A]  time = 0.83, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log \left (-x + 9\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9-x)*log(9-x)+x)/(x^3-9*x^2),x, algorithm="fricas")

[Out]

log(-x + 9)/x

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giac [A]  time = 0.14, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log \left (-x + 9\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9-x)*log(9-x)+x)/(x^3-9*x^2),x, algorithm="giac")

[Out]

log(-x + 9)/x

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maple [A]  time = 0.12, size = 11, normalized size = 1.10

method	result	size
norman	\(\frac {\ln \left (9-x \right )}{x}\)	\(11\)
risch	\(\frac {\ln \left (9-x \right )}{x}\)	\(11\)
derivativedivides	\(\frac {\ln \left (9-x \right )}{9}+\frac {\ln \left (9-x \right ) \left (9-x \right )}{9 x}\)	\(26\)
default	\(\frac {\ln \left (9-x \right )}{9}+\frac {\ln \left (9-x \right ) \left (9-x \right )}{9 x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9-x)*ln(9-x)+x)/(x^3-9*x^2),x,method=_RETURNVERBOSE)

[Out]

ln(9-x)/x

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maxima [A]  time = 0.40, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log \left (-x + 9\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9-x)*log(9-x)+x)/(x^3-9*x^2),x, algorithm="maxima")

[Out]

log(-x + 9)/x

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mupad [B]  time = 3.29, size = 10, normalized size = 1.00 \begin {gather*} \frac {\ln \left (9-x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - log(9 - x)*(x - 9))/(9*x^2 - x^3),x)

[Out]

log(9 - x)/x

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sympy [A]  time = 0.10, size = 5, normalized size = 0.50 \begin {gather*} \frac {\log {\left (9 - x \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9-x)*ln(9-x)+x)/(x**3-9*x**2),x)

[Out]

log(9 - x)/x

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