\(\int \frac {(a+\frac {b}{x})^n x}{(c+d x)^2} \, dx\) [292]

Optimal result

Integrand size = 18, antiderivative size = 150 \[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{(c+d x)^2} \, dx=-\frac {c \left (a+\frac {b}{x}\right )^{1+n}}{d (a c-b d) \left (d+\frac {c}{x}\right )}-\frac {c (a c-b d (1-n)) \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d^2 (a c-b d)^2 (1+n)}+\frac {\left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b}{a x}\right )}{a d^2 (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{(c+d x)^2} \, dx=\frac {\left (a+\frac {b}{x}\right )^{1+n} \left (-\frac {c d x}{(a c-b d) (c+d x)}-\frac {c (a c+b d (-1+n)) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{(a c-b d)^2 (1+n)}+\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b}{a x}\right )}{a (1+n)}\right )}{d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1016, 948, 114, 174, 75, 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.

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

\(\Big \downarrow \) 1016

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 75

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

-((c*(a + b/x)^(1 + n))/(d*(a*c - b*d)*(d + c/x))) - ((c*(a*c - b*d*(1 - n 
))*(a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c 
 - b*d)])/(d*(a*c - b*d)*(1 + n)) - ((a*c - b*d)*(a + b/x)^(1 + n)*Hyperge 
ometric2F1[1, 1 + n, 2 + n, 1 + b/(a*x)])/(a*d*(1 + n)))/(d*(a*c - b*d))
 

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 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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1016

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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