\(\int \frac {(a+b \log (c x^n))^p}{x} \, dx\) [171]

Optimal result

Integrand size = 16, antiderivative size = 26 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^{1+p}}{b n (1+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2339, 30} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^{p+1}}{b n (p+1)} \]

[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 2339

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*} \text {integral}& = \frac {\text {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*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^{1+p}}{b n (1+p)} \]

[In]

Integrate[(a + b*Log[c*x^n])^p/x,x]

[Out]

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(\frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p +1}}{b n \left (p +1\right )}\)	\(27\)
default	\(\frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p +1}}{b n \left (p +1\right )}\)	\(27\)
parallelrisch	\(-\frac {-\ln \left (c \,x^{n}\right ) {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p} b^{2}-{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p} a b}{b^{2} n \left (p +1\right )}\)	\(54\)
risch	\(\frac {{\left (\ln \left (x^{n}\right ) b +a +b \left (\ln \left (c \right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )\right )}^{p +1}}{n b \left (p +1\right )}\)	\(76\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} {\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}^{p}}{b n p + b n} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=- \begin {cases} - a^{p} \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (a + b \log {\left (c \right )}\right )^{p} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\begin {cases} \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b \log {\left (c x^{n} \right )} \right )} & \text {otherwise} \end {cases}}{b n} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]

[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))

Mupad [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^{p+1}}{b\,n\,\left (p+1\right )} \]

[In]

int((a + b*log(c*x^n))^p/x,x)

[Out]

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