\(\int (a+b \log (c (d+e x)^n))^p \, dx\) [31]

Optimal result

Integrand size = 16, antiderivative size = 103 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-p}}{e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2436, 2337, 2212} \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^p \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e} \]

[In]

Int[(a + b*Log[c*(d + e*x)^n])^p,x]

[Out]

((d + e*x)*Gamma[1 + p, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(a + b*Log[c*(d + e*x)^n])^p)/(e*E^(a/(b*n))*(c*(
d + e*x)^n)^n^(-1)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^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 2337

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

Rule 2436

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

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-p}}{e} \]

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^p,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.57 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx=\frac {e^{\left (-\frac {b n p \log \left (-\frac {1}{b n}\right ) + b \log \left (c\right ) + a}{b n}\right )} \Gamma \left (p + 1, -\frac {b n \log \left (e x + d\right ) + b \log \left (c\right ) + a}{b n}\right )}{e} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*log(c*(d + e*x)**n))**p, x)

Maxima [F(-2)]

Exception generated. \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

Giac [F]

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

[In]

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

[Out]

integrate((b*log((e*x + d)^n*c) + a)^p, x)

Mupad [F(-1)]

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

[In]

int((a + b*log(c*(d + e*x)^n))^p,x)

[Out]

int((a + b*log(c*(d + e*x)^n))^p, x)