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

3.5.18.1 Optimal result

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

output

(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)
 

3.5.18.2 Mathematica [A] (verified)

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

input

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

output

((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)
 

3.5.18.3 Rubi [A] (warning: unable to verify)

Time = 0.45 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2895, 2836, 2737, 2612}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \, dx\)

\(\Big \downarrow \) 2895

\(\displaystyle \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^pdx\)

\(\Big \downarrow \) 2836

\(\displaystyle \frac {\int \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^pd(e+f x)}{f}\)

\(\Big \downarrow \) 2737

\(\displaystyle \frac {(e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \int \left (c d^n (e+f x)^{m n}\right )^{\frac {1}{m n}} \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^pd\log \left (c d^n (e+f x)^{m n}\right )}{f m n}\)

\(\Big \downarrow \) 2612

\(\displaystyle \frac {(e+f x) e^{-\frac {a}{b m n}} \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^p \left (-\frac {a+b \log \left (c d^n (e+f x)^{m n}\right )}{b m n}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c d^n (e+f x)^{m n}\right )}{b m n}\right )}{f}\)

input

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

output

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

3.5.18.3.1 Defintions of rubi rules used

rule 2612

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 2737

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[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 2836

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : 
> Simp[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 2895

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]]
 

3.5.18.4 Maple [F]

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

input

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

output

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

3.5.18.5 Fricas [A] (verification not implemented)

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

input

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

output

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
 

3.5.18.6 Sympy [F]

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

input

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

output

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

3.5.18.7 Maxima [F(-2)]

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

input

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

output

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

3.5.18.8 Giac [F]

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

input

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

output

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

3.5.18.9 Mupad [F(-1)]

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

input

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

output

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