\(\int (e x)^{-1+2 n} (a+b (c+d x^n)^q)^p \, dx\) [382]

Optimal result

Integrand size = 25, antiderivative size = 194 \[ \int (e x)^{-1+2 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=-\frac {c x^{-2 n} (e x)^{2 n} \left (c+d x^n\right ) \left (a+b \left (c+d x^n\right )^q\right )^p \left (1+\frac {b \left (c+d x^n\right )^q}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{q},1+\frac {1}{q},-\frac {b \left (c+d x^n\right )^q}{a}\right )}{d^2 e n}+\frac {x^{-2 n} (e x)^{2 n} \left (c+d x^n\right )^2 \left (a+b \left (c+d x^n\right )^q\right )^p \left (1+\frac {b \left (c+d x^n\right )^q}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {2}{q},\frac {2+q}{q},-\frac {b \left (c+d x^n\right )^q}{a}\right )}{2 d^2 e n} \] Output:

-c*(e*x)^(2*n)*(c+d*x^n)*(a+b*(c+d*x^n)^q)^p*hypergeom([-p, 1/q],[1+1/q],- 
b*(c+d*x^n)^q/a)/d^2/e/n/(x^(2*n))/((1+b*(c+d*x^n)^q/a)^p)+1/2*(e*x)^(2*n) 
*(c+d*x^n)^2*(a+b*(c+d*x^n)^q)^p*hypergeom([-p, 2/q],[(2+q)/q],-b*(c+d*x^n 
)^q/a)/d^2/e/n/(x^(2*n))/((1+b*(c+d*x^n)^q/a)^p)
 

Mathematica [F]

\[ \int (e x)^{-1+2 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\int (e x)^{-1+2 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx \] Input:

Integrate[(e*x)^(-1 + 2*n)*(a + b*(c + d*x^n)^q)^p,x]
 

Output:

Integrate[(e*x)^(-1 + 2*n)*(a + b*(c + d*x^n)^q)^p, x]
 

Rubi [F]

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.

\(\displaystyle \int (e x)^{2 n-1} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int (e x)^{2 n-1} \left (a+b \left (c+d x^n\right )^q\right )^pdx\)

Input:

Int[(e*x)^(-1 + 2*n)*(a + b*(c + d*x^n)^q)^p,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int \left (e x \right )^{-1+2 n} {\left (a +b \left (c +d \,x^{n}\right )^{q}\right )}^{p}d x\]

Input:

int((e*x)^(-1+2*n)*(a+b*(c+d*x^n)^q)^p,x)
 

Output:

int((e*x)^(-1+2*n)*(a+b*(c+d*x^n)^q)^p,x)
 

Fricas [F]

\[ \int (e x)^{-1+2 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{q} b + a\right )}^{p} \left (e x\right )^{2 \, n - 1} \,d x } \] Input:

integrate((e*x)^(-1+2*n)*(a+b*(c+d*x^n)^q)^p,x, algorithm="fricas")
 

Output:

integral(((d*x^n + c)^q*b + a)^p*(e*x)^(2*n - 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int (e x)^{-1+2 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\text {Timed out} \] Input:

integrate((e*x)**(-1+2*n)*(a+b*(c+d*x**n)**q)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (e x)^{-1+2 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{q} b + a\right )}^{p} \left (e x\right )^{2 \, n - 1} \,d x } \] Input:

integrate((e*x)^(-1+2*n)*(a+b*(c+d*x^n)^q)^p,x, algorithm="maxima")
 

Output:

integrate(((d*x^n + c)^q*b + a)^p*(e*x)^(2*n - 1), x)
 

Giac [F]

\[ \int (e x)^{-1+2 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{q} b + a\right )}^{p} \left (e x\right )^{2 \, n - 1} \,d x } \] Input:

integrate((e*x)^(-1+2*n)*(a+b*(c+d*x^n)^q)^p,x, algorithm="giac")
 

Output:

integrate(((d*x^n + c)^q*b + a)^p*(e*x)^(2*n - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1+2 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\int {\left (a+b\,{\left (c+d\,x^n\right )}^q\right )}^p\,{\left (e\,x\right )}^{2\,n-1} \,d x \] Input:

int((a + b*(c + d*x^n)^q)^p*(e*x)^(2*n - 1),x)
 

Output:

int((a + b*(c + d*x^n)^q)^p*(e*x)^(2*n - 1), x)
 

Reduce [F]

\[ \int (e x)^{-1+2 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx =\text {Too large to display} \] Input:

int((e*x)^(-1+2*n)*(a+b*(c+d*x^n)^q)^p,x)
 

Output:

(e**(2*n)*(x**(2*n)*((x**n*d + c)**q*b + a)**p*d**2*p*q + x**(2*n)*((x**n* 
d + c)**q*b + a)**p*d**2 + x**n*((x**n*d + c)**q*b + a)**p*c*d*p*q - ((x** 
n*d + c)**q*b + a)**p*c**2 + int((x**(2*n)*((x**n*d + c)**q*b + a)**p)/((x 
**n*d + c)**q*b*p**2*q**2*x + 3*(x**n*d + c)**q*b*p*q*x + 2*(x**n*d + c)** 
q*b*x + a*p**2*q**2*x + 3*a*p*q*x + 2*a*x),x)*a*d**2*n*p**4*q**4 + 4*int(( 
x**(2*n)*((x**n*d + c)**q*b + a)**p)/((x**n*d + c)**q*b*p**2*q**2*x + 3*(x 
**n*d + c)**q*b*p*q*x + 2*(x**n*d + c)**q*b*x + a*p**2*q**2*x + 3*a*p*q*x 
+ 2*a*x),x)*a*d**2*n*p**3*q**3 + 5*int((x**(2*n)*((x**n*d + c)**q*b + a)** 
p)/((x**n*d + c)**q*b*p**2*q**2*x + 3*(x**n*d + c)**q*b*p*q*x + 2*(x**n*d 
+ c)**q*b*x + a*p**2*q**2*x + 3*a*p*q*x + 2*a*x),x)*a*d**2*n*p**2*q**2 + 2 
*int((x**(2*n)*((x**n*d + c)**q*b + a)**p)/((x**n*d + c)**q*b*p**2*q**2*x 
+ 3*(x**n*d + c)**q*b*p*q*x + 2*(x**n*d + c)**q*b*x + a*p**2*q**2*x + 3*a* 
p*q*x + 2*a*x),x)*a*d**2*n*p*q - int((x**n*((x**n*d + c)**q*b + a)**p)/((x 
**n*d + c)**q*b*p**2*q**2*x + 3*(x**n*d + c)**q*b*p*q*x + 2*(x**n*d + c)** 
q*b*x + a*p**2*q**2*x + 3*a*p*q*x + 2*a*x),x)*a*c*d*n*p**3*q**3 - 3*int((x 
**n*((x**n*d + c)**q*b + a)**p)/((x**n*d + c)**q*b*p**2*q**2*x + 3*(x**n*d 
 + c)**q*b*p*q*x + 2*(x**n*d + c)**q*b*x + a*p**2*q**2*x + 3*a*p*q*x + 2*a 
*x),x)*a*c*d*n*p**2*q**2 - 2*int((x**n*((x**n*d + c)**q*b + a)**p)/((x**n* 
d + c)**q*b*p**2*q**2*x + 3*(x**n*d + c)**q*b*p*q*x + 2*(x**n*d + c)**q*b* 
x + a*p**2*q**2*x + 3*a*p*q*x + 2*a*x),x)*a*c*d*n*p*q))/(d**2*e*n*(p**2...