3.179 \(\int \frac{(a+b \log (c x))^p}{x} \, dx\)

Optimal. Leaf size=21 \[ \frac{(a+b \log (c x))^{p+1}}{b (p+1)} \]

[Out]

(a + b*Log[c*x])^(1 + p)/(b*(1 + p))

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Rubi [A]  time = 0.0276097, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2302, 30} \[ \frac{(a+b \log (c x))^{p+1}}{b (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x])^p/x,x]

[Out]

(a + b*Log[c*x])^(1 + p)/(b*(1 + p))

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

Mathematica [A]  time = 0.0062163, size = 21, normalized size = 1. \[ \frac{(a+b \log (c x))^{p+1}}{b (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x])^p/x,x]

[Out]

(a + b*Log[c*x])^(1 + p)/(b*(1 + p))

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Maple [A]  time = 0.037, size = 22, normalized size = 1.1 \begin{align*}{\frac{ \left ( a+b\ln \left ( cx \right ) \right ) ^{1+p}}{b \left ( 1+p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a+b*ln(c*x))^(1+p)/b/(1+p)

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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(c*x))^p/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.03041, size = 63, normalized size = 3. \begin{align*} \frac{{\left (b \log \left (c x\right ) + a\right )}{\left (b \log \left (c x\right ) + a\right )}^{p}}{b p + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*log(c*x) + a)*(b*log(c*x) + a)^p/(b*p + b)

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Sympy [A]  time = 1.33317, size = 39, normalized size = 1.86 \begin{align*} - \begin{cases} - a^{p} \log{\left (x \right )} & \text{for}\: b = 0 \\- \frac{\begin{cases} \frac{\left (a + b \log{\left (c x \right )}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + b \log{\left (c 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(c*x))**p/x,x)

[Out]

-Piecewise((-a**p*log(x), Eq(b, 0)), (-Piecewise(((a + b*log(c*x))**(p + 1)/(p + 1), Ne(p, -1)), (log(a + b*lo
g(c*x)), True))/b, True))

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Giac [A]  time = 1.28614, size = 28, normalized size = 1.33 \begin{align*} \frac{{\left (b \log \left (c x\right ) + a\right )}^{p + 1}}{b{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*log(c*x) + a)^(p + 1)/(b*(p + 1))