\(\int (a+b \log (c (d (e+f x)^m)^n))^p \, dx\) [437]

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)
 

Mathematica [A] (verified)

Time = 0.21 (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)
 

Rubi [A] (warning: unable to verify)

Time = 0.79 (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)
 

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]]
 

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)
 

Fricas [A] (verification not implemented)

Time = 0.08 (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
 

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)
 

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
 

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)
                                                                                    
                                                                                    
 

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)
 

Reduce [F]

\[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \, dx =\text {Too large to display} \] Input:

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

Output:

((log(d**n*(e + f*x)**(m*n)*c)*b + a)**p*log(d**n*(e + f*x)**(m*n)*c)*b*e* 
p + (log(d**n*(e + f*x)**(m*n)*c)*b + a)**p*a*e*p + (log(d**n*(e + f*x)**( 
m*n)*c)*b + a)**p*a*f*p*x + (log(d**n*(e + f*x)**(m*n)*c)*b + a)**p*a*f*x 
+ int(((log(d**n*(e + f*x)**(m*n)*c)*b + a)**p*log(d**n*(e + f*x)**(m*n)*c 
)*x)/(log(d**n*(e + f*x)**(m*n)*c)*a*b*e + log(d**n*(e + f*x)**(m*n)*c)*a* 
b*f*x + log(d**n*(e + f*x)**(m*n)*c)*b**2*e*m*n*p + log(d**n*(e + f*x)**(m 
*n)*c)*b**2*f*m*n*p*x + a**2*e + a**2*f*x + a*b*e*m*n*p + a*b*f*m*n*p*x),x 
)*a*b**2*f**2*m*n*p**2 + int(((log(d**n*(e + f*x)**(m*n)*c)*b + a)**p*log( 
d**n*(e + f*x)**(m*n)*c)*x)/(log(d**n*(e + f*x)**(m*n)*c)*a*b*e + log(d**n 
*(e + f*x)**(m*n)*c)*a*b*f*x + log(d**n*(e + f*x)**(m*n)*c)*b**2*e*m*n*p + 
 log(d**n*(e + f*x)**(m*n)*c)*b**2*f*m*n*p*x + a**2*e + a**2*f*x + a*b*e*m 
*n*p + a*b*f*m*n*p*x),x)*a*b**2*f**2*m*n*p + int(((log(d**n*(e + f*x)**(m* 
n)*c)*b + a)**p*log(d**n*(e + f*x)**(m*n)*c)*x)/(log(d**n*(e + f*x)**(m*n) 
*c)*a*b*e + log(d**n*(e + f*x)**(m*n)*c)*a*b*f*x + log(d**n*(e + f*x)**(m* 
n)*c)*b**2*e*m*n*p + log(d**n*(e + f*x)**(m*n)*c)*b**2*f*m*n*p*x + a**2*e 
+ a**2*f*x + a*b*e*m*n*p + a*b*f*m*n*p*x),x)*b**3*f**2*m**2*n**2*p**3 + in 
t(((log(d**n*(e + f*x)**(m*n)*c)*b + a)**p*log(d**n*(e + f*x)**(m*n)*c)*x) 
/(log(d**n*(e + f*x)**(m*n)*c)*a*b*e + log(d**n*(e + f*x)**(m*n)*c)*a*b*f* 
x + log(d**n*(e + f*x)**(m*n)*c)*b**2*e*m*n*p + log(d**n*(e + f*x)**(m*n)* 
c)*b**2*f*m*n*p*x + a**2*e + a**2*f*x + a*b*e*m*n*p + a*b*f*m*n*p*x),x)...