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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 30

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

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]

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 = 0.01, size = 21, normalized size = 1.00 \[ \frac {(a+b \log (c x))^{p+1}}{b (p+1)} \]

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))

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 26, normalized size = 1.24 \[ \frac {{\left (b \log \left (c x\right ) + a\right )} {\left (b \log \left (c x\right ) + a\right )}^{p}}{b p + b} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.39, size = 21, normalized size = 1.00 \[ \frac {{\left (b \log \left (c x\right ) + a\right )}^{p + 1}}{b {\left (p + 1\right )}} \]

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))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 22, normalized size = 1.05 \[ \frac {\left (b \ln \left (c x \right )+a \right )^{p +1}}{\left (p +1\right ) b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.53, size = 21, normalized size = 1.00 \[ \frac {{\left (b \log \left (c x\right ) + a\right )}^{p + 1}}{b {\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*log(c*x) + a)^(p + 1)/(b*(p + 1))

________________________________________________________________________________________

mupad [B]  time = 3.70, size = 21, normalized size = 1.00 \[ \frac {{\left (a+b\,\ln \left (c\,x\right )\right )}^{p+1}}{b\,\left (p+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x))^p/x,x)

[Out]

(a + b*log(c*x))^(p + 1)/(b*(p + 1))

________________________________________________________________________________________

sympy [A]  time = 1.31, size = 39, normalized size = 1.86 \[ - \begin {cases} - a^{p} \log {\relax (x )} & \text {for}\: b = 0 \\- \frac {\begin {cases} \frac {\left (a + b \log {\left (c x \right )}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b \log {\left (c x \right )} \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________