\(\int (e x)^m (a+\frac {b}{c+d x})^p \, dx\) [62]

Optimal result

Integrand size = 19, antiderivative size = 86 \[ \int (e x)^m \left (a+\frac {b}{c+d x}\right )^p \, dx=\frac {(e x)^{1+m} \left (1+\frac {d x}{c}\right )^p \left (1+\frac {a d x}{b+a c}\right )^{-p} \left (a+\frac {b}{c+d x}\right )^p \operatorname {AppellF1}\left (1+m,p,-p,2+m,-\frac {d x}{c},-\frac {a d x}{b+a c}\right )}{e (1+m)} \] Output:

(e*x)^(1+m)*(1+d*x/c)^p*(a+b/(d*x+c))^p*AppellF1(1+m,p,-p,2+m,-d*x/c,-a*d* 
x/(a*c+b))/e/(1+m)/((1+a*d*x/(a*c+b))^p)
 

Mathematica [F]

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

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

Output:

Integrate[(e*x)^m*(a + b/(c + d*x))^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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((e*x)^m*((a*d*x + a*c + b)/(d*x + c))^p, x)
 

Sympy [F]

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

integrate((e*x)**m*(a+b/(d*x+c))**p,x)
 

Output:

Integral((e*x)**m*((a*c + a*d*x + b)/(c + d*x))**p, x)
 

Maxima [F]

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

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

Output:

integrate((e*x)^m*(a + b/(d*x + c))^p, x)
 

Giac [F]

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

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

Output:

integrate((e*x)^m*(a + b/(d*x + c))^p, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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