3.619 \(\int \frac{(a+b \log (x))^n}{x} \, dx\)

Optimal. Leaf size=19 \[ \frac{(a+b \log (x))^{n+1}}{b (n+1)} \]

[Out]

(a + b*Log[x])^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.0282467, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2302, 30} \[ \frac{(a+b \log (x))^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[x])^n/x,x]

[Out]

(a + b*Log[x])^(1 + n)/(b*(1 + n))

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b \log (x))^n}{x} \, dx &=\frac{\operatorname{Subst}\left (\int x^n \, dx,x,a+b \log (x)\right )}{b}\\ &=\frac{(a+b \log (x))^{1+n}}{b (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0116802, size = 19, normalized size = 1. \[ \frac{(a+b \log (x))^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[x])^n/x,x]

[Out]

(a + b*Log[x])^(1 + n)/(b*(1 + n))

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Maple [A]  time = 0.001, size = 20, normalized size = 1.1 \begin{align*}{\frac{ \left ( a+b\ln \left ( x \right ) \right ) ^{1+n}}{b \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(x))^n/x,x)

[Out]

(a+b*ln(x))^(1+n)/b/(1+n)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(x))^n/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.36099, size = 58, normalized size = 3.05 \begin{align*} \frac{{\left (b \log \left (x\right ) + a\right )}{\left (b \log \left (x\right ) + a\right )}^{n}}{b n + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(x))^n/x,x, algorithm="fricas")

[Out]

(b*log(x) + a)*(b*log(x) + a)^n/(b*n + b)

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Sympy [A]  time = 1.25934, size = 36, normalized size = 1.89 \begin{align*} - \begin{cases} - a^{n} \log{\left (x \right )} & \text{for}\: b = 0 \\- \frac{\begin{cases} \frac{\left (a + b \log{\left (x \right )}\right )^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (a + b \log{\left (x \right )} \right )} & \text{otherwise} \end{cases}}{b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(x))**n/x,x)

[Out]

-Piecewise((-a**n*log(x), Eq(b, 0)), (-Piecewise(((a + b*log(x))**(n + 1)/(n + 1), Ne(n, -1)), (log(a + b*log(
x)), True))/b, True))

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Giac [A]  time = 1.14566, size = 26, normalized size = 1.37 \begin{align*} \frac{{\left (b \log \left (x\right ) + a\right )}^{n + 1}}{b{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(x))^n/x,x, algorithm="giac")

[Out]

(b*log(x) + a)^(n + 1)/(b*(n + 1))