\(\int (a+\frac {b}{x})^m (c+d x)^n \, dx\) [595]

Optimal result

Integrand size = 17, antiderivative size = 80 \[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^n \, dx=\frac {\left (a+\frac {b}{x}\right )^m x \left (1+\frac {a x}{b}\right )^{-m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1-m,-m,-n,2-m,-\frac {a x}{b},-\frac {d x}{c}\right )}{1-m} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {942, 152, 152, 150}

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.

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

\(\Big \downarrow \) 942

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {x \left (a+\frac {b}{x}\right )^m \left (\frac {a x}{b}+1\right )^{-m} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (1-m,-m,-n,2-m,-\frac {a x}{b},-\frac {d x}{c}\right )}{1-m}\)

Input:

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

Output:

((a + b/x)^m*x*(c + d*x)^n*AppellF1[1 - m, -m, -n, 2 - m, -((a*x)/b), -((d 
*x)/c)])/((1 - m)*(1 + (a*x)/b)^m*(1 + (d*x)/c)^n)
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 152

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) 
Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]
 

rule 942

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symb 
ol] :> Simp[x^(n*FracPart[q])*((c + d/x^n)^FracPart[q]/(d + c*x^n)^FracPart 
[q])   Int[(a + b*x^n)^p*((d + c*x^n)^q/x^(n*q)), x], x] /; FreeQ[{a, b, c, 
 d, n, p, q}, x] && EqQ[mn, -n] &&  !IntegerQ[q] &&  !IntegerQ[p]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^n \, dx=\text {too large to display} \] Input:

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

Output:

( - (c + d*x)**n*(a*x + b)**m*c*m - (c + d*x)**n*(a*x + b)**m*c*n + (c + d 
*x)**n*(a*x + b)**m*d*m*x - (c + d*x)**n*(a*x + b)**m*d*n*x + 2*x**m*int(( 
(c + d*x)**n*(a*x + b)**m*x)/(x**m*a*c*m*n*x + x**m*a*c*m*x - x**m*a*c*n** 
2*x - x**m*a*c*n*x + x**m*a*d*m*n*x**2 + x**m*a*d*m*x**2 - x**m*a*d*n**2*x 
**2 - x**m*a*d*n*x**2 + x**m*b*c*m*n + x**m*b*c*m - x**m*b*c*n**2 - x**m*b 
*c*n + x**m*b*d*m*n*x + x**m*b*d*m*x - x**m*b*d*n**2*x - x**m*b*d*n*x),x)* 
a*c*d*m**2*n**2 + 2*x**m*int(((c + d*x)**n*(a*x + b)**m*x)/(x**m*a*c*m*n*x 
 + x**m*a*c*m*x - x**m*a*c*n**2*x - x**m*a*c*n*x + x**m*a*d*m*n*x**2 + x** 
m*a*d*m*x**2 - x**m*a*d*n**2*x**2 - x**m*a*d*n*x**2 + x**m*b*c*m*n + x**m* 
b*c*m - x**m*b*c*n**2 - x**m*b*c*n + x**m*b*d*m*n*x + x**m*b*d*m*x - x**m* 
b*d*n**2*x - x**m*b*d*n*x),x)*a*c*d*m**2*n - 2*x**m*int(((c + d*x)**n*(a*x 
 + b)**m*x)/(x**m*a*c*m*n*x + x**m*a*c*m*x - x**m*a*c*n**2*x - x**m*a*c*n* 
x + x**m*a*d*m*n*x**2 + x**m*a*d*m*x**2 - x**m*a*d*n**2*x**2 - x**m*a*d*n* 
x**2 + x**m*b*c*m*n + x**m*b*c*m - x**m*b*c*n**2 - x**m*b*c*n + x**m*b*d*m 
*n*x + x**m*b*d*m*x - x**m*b*d*n**2*x - x**m*b*d*n*x),x)*a*c*d*m*n**3 - 2* 
x**m*int(((c + d*x)**n*(a*x + b)**m*x)/(x**m*a*c*m*n*x + x**m*a*c*m*x - x* 
*m*a*c*n**2*x - x**m*a*c*n*x + x**m*a*d*m*n*x**2 + x**m*a*d*m*x**2 - x**m* 
a*d*n**2*x**2 - x**m*a*d*n*x**2 + x**m*b*c*m*n + x**m*b*c*m - x**m*b*c*n** 
2 - x**m*b*c*n + x**m*b*d*m*n*x + x**m*b*d*m*x - x**m*b*d*n**2*x - x**m*b* 
d*n*x),x)*a*c*d*m*n**2 + x**m*int(((c + d*x)**n*(a*x + b)**m*x)/(x**m*a...