\(\int (a+b \log (c (d (e+f x)^m)^n))^p \, dx\) [418]

Optimal result

Integrand size = 20, antiderivative size = 131 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \, dx=\frac {e^{-\frac {a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right ) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \left (-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )^{-p}}{f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2436, 2337, 2212, 2495} \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \, dx=\frac {(e+f x) e^{-\frac {a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \left (-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{f} \]

[In]

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

[Out]

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

Rule 2495

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \, dx=\frac {e^{-\frac {a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right ) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \left (-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )^{-p}}{f} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.61 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \, dx=\frac {e^{\left (-\frac {b m n p \log \left (-\frac {1}{b m n}\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )} \Gamma \left (p + 1, -\frac {b m n \log \left (f x + e\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}{f} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((a+b*log(c*(d*(f*x+e)^m)^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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*log(c*(d*(e + f*x)^m)^n))^p,x)

[Out]

int((a + b*log(c*(d*(e + f*x)^m)^n))^p, x)