\(\int (a+b \log (c (d (e+f x)^p)^q))^n \, dx\) [535]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\frac {e^{-\frac {a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \Gamma \left (1+n,-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n}}{f} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.77 (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.

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

\(\Big \downarrow \) 2895

\(\displaystyle \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^ndx\)

\(\Big \downarrow \) 2836

\(\displaystyle \frac {\int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^nd(e+f x)}{f}\)

\(\Big \downarrow \) 2737

\(\displaystyle \frac {(e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \int \left (c d^q (e+f x)^{p q}\right )^{\frac {1}{p q}} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^nd\log \left (c d^q (e+f x)^{p q}\right )}{f p q}\)

\(\Big \downarrow \) 2612

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.61 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\frac {e^{\left (-\frac {b n p q \log \left (-\frac {1}{b p q}\right ) + b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )} \Gamma \left (n + 1, -\frac {b p q \log \left (f x + e\right ) + b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )}{f} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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