\(\int x (a+b \log (c x))^p \, dx\) [177]

Optimal result

Integrand size = 12, antiderivative size = 63 \[ \int x (a+b \log (c x))^p \, dx=\frac {2^{-1-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c^2} \]

[Out]

2^(-1-p)*GAMMA(p+1,-2*(a+b*ln(c*x))/b)*(a+b*ln(c*x))^p/c^2/exp(2*a/b)/(((-a-b*ln(c*x))/b)^p)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2346, 2212} \[ \int x (a+b \log (c x))^p \, dx=\frac {2^{-p-1} e^{-\frac {2 a}{b}} (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 (a+b \log (c x))}{b}\right )}{c^2} \]

[In]

Int[x*(a + b*Log[c*x])^p,x]

[Out]

(2^(-1 - p)*Gamma[1 + p, (-2*(a + b*Log[c*x]))/b]*(a + b*Log[c*x])^p)/(c^2*E^((2*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 2346

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log (c x)\right )}{c^2} \\ & = \frac {2^{-1-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int x (a+b \log (c x))^p \, dx=\frac {2^{-1-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c^2} \]

[In]

Integrate[x*(a + b*Log[c*x])^p,x]

[Out]

(2^(-1 - p)*Gamma[1 + p, (-2*(a + b*Log[c*x]))/b]*(a + b*Log[c*x])^p)/(c^2*E^((2*a)/b)*(-((a + b*Log[c*x])/b))
^p)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*log(c*x) + a)^p*x, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x*(a + b*log(c*x))**p, x)

Maxima [A] (verification not implemented)

none

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int x (a+b \log (c x))^p \, dx=-\frac {{\left (b \log \left (c x\right ) + a\right )}^{p + 1} e^{\left (-\frac {2 \, a}{b}\right )} E_{-p}\left (-\frac {2 \, {\left (b \log \left (c x\right ) + a\right )}}{b}\right )}{b c^{2}} \]

[In]

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

[Out]

-(b*log(c*x) + a)^(p + 1)*e^(-2*a/b)*exp_integral_e(-p, -2*(b*log(c*x) + a)/b)/(b*c^2)

Giac [F]

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

[In]

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

[Out]

integrate((b*log(c*x) + a)^p*x, x)

Mupad [F(-1)]

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

[In]

int(x*(a + b*log(c*x))^p,x)

[Out]

int(x*(a + b*log(c*x))^p, x)