\(\int (e x)^{-1-n} (a+\frac {b}{c+d x^n})^p \, dx\) [356]

Optimal result

Integrand size = 25, antiderivative size = 79 \[ \int (e x)^{-1-n} \left (a+\frac {b}{c+d x^n}\right )^p \, dx=-\frac {b d x^n (e x)^{-n} \left (a+\frac {b}{c+d x^n}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,\frac {c \left (a+\frac {b}{c+d x^n}\right )}{b+a c}\right )}{(b+a c)^2 e n (1+p)} \] Output:

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

Mathematica [F]

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

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

Output:

Integrate[(e*x)^(-1 - n)*(a + b/(c + d*x^n))^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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2054

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

\(\Big \downarrow \) 2053

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2053

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) 
))^(p_), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( 
(a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, 
 x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2054

Int[((f_)*(x_))^(m_)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_) 
^(n_.)))^(p_), x_Symbol] :> Simp[Simp[(c*x)^m/x^m]   Int[x^m*(e*((a + b*x^n 
)/(c + d*x^n)))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Integ 
erQ[Simplify[(m + 1)/n]]
 

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((x**n*a*d + a*c + b)**p/(x**n*(x**n*d + c)**p*x),x)/(e**n*e)