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

Optimal. Leaf size=26 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^{p+1}}{b n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2302, 30} \[ \frac {\left (a+b \log \left (c x^n\right )\right )^{p+1}}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^p/x,x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.00 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^{p+1}}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^p/x,x]

[Out]

(a + b*Log[c*x^n])^(1 + p)/(b*n*(1 + p))

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fricas [A]  time = 0.42, size = 35, normalized size = 1.35 \[ \frac {{\left (b n \log \relax (x) + b \log \relax (c) + a\right )} {\left (b n \log \relax (x) + b \log \relax (c) + a\right )}^{p}}{b n p + b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.25, size = 27, normalized size = 1.04 \[ \frac {{\left (b n \log \relax (x) + b \log \relax (c) + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 27, normalized size = 1.04 \[ \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{p +1}}{\left (p +1\right ) b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.71, size = 26, normalized size = 1.00 \[ \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.67, size = 26, normalized size = 1.00 \[ \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^{p+1}}{b\,n\,\left (p+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))^p/x,x)

[Out]

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

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sympy [A]  time = 1.86, size = 56, normalized size = 2.15 \[ - \begin {cases} - a^{p} \log {\relax (x )} & \text {for}\: b = 0 \\- \left (a + b \log {\relax (c )}\right )^{p} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {\begin {cases} \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b \log {\left (c x^{n} \right )} \right )} & \text {otherwise} \end {cases}}{b n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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