\(\int \frac {(a+\frac {b}{x})^n}{(c+d x)^2} \, dx\) [293]

Optimal result

Integrand size = 17, antiderivative size = 56 \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{(c+d x)^2} \, dx=-\frac {b \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{(a c-b d)^2 (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{(c+d x)^2} \, dx=-\frac {b \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,-\frac {c \left (a+\frac {b}{x}\right )}{-a c+b d}\right )}{(-a c+b d)^2 (1+n)} \] Input:

Integrate[(a + b/x)^n/(c + d*x)^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {941, 946, 78}

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.

\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^n}{(c+d x)^2} \, dx\)

\(\Big \downarrow \) 941

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

\(\Big \downarrow \) 946

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

\(\Big \downarrow \) 78

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

Input:

Int[(a + b/x)^n/(c + d*x)^2,x]
 

Output:

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

Defintions of rubi rules used

rule 78

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

rule 941

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

rule 946

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^n], 
x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n 
+ 1, 0]
 

Maple [F]

\[\int \frac {\left (a +\frac {b}{x}\right )^{n}}{\left (d x +c \right )^{2}}d x\]

Input:

int((a+b/x)^n/(d*x+c)^2,x)
 

Output:

int((a+b/x)^n/(d*x+c)^2,x)
 

Fricas [F]

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

integrate((a+b/x)^n/(d*x+c)^2,x, algorithm="fricas")
 

Output:

integral(((a*x + b)/x)^n/(d^2*x^2 + 2*c*d*x + c^2), x)
 

Sympy [F]

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

integrate((a+b/x)**n/(d*x+c)**2,x)
 

Output:

Integral((a + b/x)**n/(c + d*x)**2, x)
 

Maxima [F]

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

integrate((a+b/x)^n/(d*x+c)^2,x, algorithm="maxima")
 

Output:

integrate((a + b/x)^n/(d*x + c)^2, x)
 

Giac [F]

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

integrate((a+b/x)^n/(d*x+c)^2,x, algorithm="giac")
 

Output:

integrate((a + b/x)^n/(d*x + c)^2, x)
 

Mupad [F(-1)]

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

int((a + b/x)^n/(c + d*x)^2,x)
 

Output:

int((a + b/x)^n/(c + d*x)^2, x)
 

Reduce [F]

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

int((a+b/x)^n/(d*x+c)^2,x)
 

Output:

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