\(\int (e x)^{-1-n} (a+b (c+d x^n))^p \, dx\) [350]

Optimal result

Integrand size = 23, antiderivative size = 69 \[ \int (e x)^{-1-n} \left (a+b \left (c+d x^n\right )\right )^p \, dx=\frac {b d x^n (e x)^{-n} \left (a+b c+b d x^n\right )^{1+p} \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1+\frac {b d x^n}{a+b c}\right )}{(a+b c)^2 e n (1+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.29 \[ \int (e x)^{-1-n} \left (a+b \left (c+d x^n\right )\right )^p \, dx=\frac {x (e x)^{-1-n} \left (1+\frac {(a+b c) x^{-n}}{b d}\right )^{-p} \left (a+b c+b d x^n\right )^p \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,\frac {(-a-b c) x^{-n}}{b d}\right )}{n (-1+p)} \] Input:

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

Output:

(x*(e*x)^(-1 - n)*(a + b*c + b*d*x^n)^p*Hypergeometric2F1[1 - p, -p, 2 - p 
, (-a - b*c)/(b*d*x^n)])/(n*(-1 + p)*(1 + (a + b*c)/(b*d*x^n))^p)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2073, 800, 798, 75}

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.

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

\(\Big \downarrow \) 2073

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

\(\Big \downarrow \) 800

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

\(\Big \downarrow \) 798

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

\(\Big \downarrow \) 75

\(\displaystyle \frac {b d x^n (e x)^{-n} \left (a+b c+b d x^n\right )^{p+1} \operatorname {Hypergeometric2F1}\left (2,p+1,p+2,\frac {b d x^n}{a+b c}+1\right )}{e n (p+1) (a+b c)^2}\)

Input:

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

Output:

(b*d*x^n*(a + b*c + b*d*x^n)^(1 + p)*Hypergeometric2F1[2, 1 + p, 2 + p, 1 
+ (b*d*x^n)/(a + b*c)])/((a + b*c)^2*e*n*(1 + p)*(e*x)^n)
 

Defintions of rubi rules used

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 798

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

rule 800

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

rule 2073

Int[(u_)^(p_.)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x)^m*ExpandToSum[u, 
x]^p, x] /; FreeQ[{c, m, p}, x] && BinomialQ[u, x] &&  !BinomialMatchQ[u, x 
]
 

Maple [F]

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

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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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