\(\int (a+b \log (c x))^p \, dx\) [178]

Optimal result

Integrand size = 10, antiderivative size = 56 \[ \int (a+b \log (c x))^p \, dx=\frac {e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log (c x)}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c} \]

[Out]

GAMMA(p+1,(-a-b*ln(c*x))/b)*(a+b*ln(c*x))^p/c/exp(a/b)/(((-a-b*ln(c*x))/b)^p)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2336, 2212} \[ \int (a+b \log (c x))^p \, dx=\frac {e^{-\frac {a}{b}} (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log (c x)}{b}\right )}{c} \]

[In]

Int[(a + b*Log[c*x])^p,x]

[Out]

(Gamma[1 + p, -((a + b*Log[c*x])/b)]*(a + b*Log[c*x])^p)/(c*E^(a/b)*(-((a + b*Log[c*x])/b))^p)

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2336

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log (c x)\right )}{c} \\ & = \frac {e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log (c x)}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int (a+b \log (c x))^p \, dx=\frac {e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log (c x)}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c} \]

[In]

Integrate[(a + b*Log[c*x])^p,x]

[Out]

(Gamma[1 + p, -((a + b*Log[c*x])/b)]*(a + b*Log[c*x])^p)/(c*E^(a/b)*(-((a + b*Log[c*x])/b))^p)

Maple [F]

\[\int \left (a +b \ln \left (x c \right )\right )^{p}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.68 \[ \int (a+b \log (c x))^p \, dx=\frac {e^{\left (-\frac {b p \log \left (-\frac {1}{b}\right ) + a}{b}\right )} \Gamma \left (p + 1, -\frac {b \log \left (c x\right ) + a}{b}\right )}{c} \]

[In]

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

[Out]

e^(-(b*p*log(-1/b) + a)/b)*gamma(p + 1, -(b*log(c*x) + a)/b)/c

Sympy [F]

\[ \int (a+b \log (c x))^p \, dx=\int \left (a + b \log {\left (c x \right )}\right )^{p}\, dx \]

[In]

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

[Out]

Integral((a + b*log(c*x))**p, x)

Maxima [A] (verification not implemented)

none

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int (a+b \log (c x))^p \, dx=-\frac {{\left (b \log \left (c x\right ) + a\right )}^{p + 1} e^{\left (-\frac {a}{b}\right )} E_{-p}\left (-\frac {b \log \left (c x\right ) + a}{b}\right )}{b c} \]

[In]

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

[Out]

-(b*log(c*x) + a)^(p + 1)*e^(-a/b)*exp_integral_e(-p, -(b*log(c*x) + a)/b)/(b*c)

Giac [F]

\[ \int (a+b \log (c x))^p \, dx=\int { {\left (b \log \left (c x\right ) + a\right )}^{p} \,d x } \]

[In]

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

[Out]

integrate((b*log(c*x) + a)^p, x)

Mupad [F(-1)]

Timed out. \[ \int (a+b \log (c x))^p \, dx=\int {\left (a+b\,\ln \left (c\,x\right )\right )}^p \,d x \]

[In]

int((a + b*log(c*x))^p,x)

[Out]

int((a + b*log(c*x))^p, x)