\(\int \frac {(a+\frac {b}{x})^n x}{c+d x} \, dx\) [285]

Optimal result

Integrand size = 18, antiderivative size = 131 \[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx=\frac {\left (a+\frac {b}{x}\right )^{1+n} x}{a d}+\frac {c^2 \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d^2 (a c-b d) (1+n)}-\frac {(a c-b d n) \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b}{a x}\right )}{a^2 d^2 (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx=\frac {\left (a+\frac {b}{x}\right )^n (b+a x) \left (a^2 c^2 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )+(a c-b d) \left (a d (1+n) x+(-a c+b d n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b}{a x}\right )\right )\right )}{a^2 d^2 (a c-b d) (1+n) x} \] Input:

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

Output:

((a + b/x)^n*(b + a*x)*(a^2*c^2*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + 
 b/x))/(a*c - b*d)] + (a*c - b*d)*(a*d*(1 + n)*x + (-(a*c) + b*d*n)*Hyperg 
eometric2F1[1, 1 + n, 2 + n, 1 + b/(a*x)])))/(a^2*d^2*(a*c - b*d)*(1 + n)* 
x)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1016, 899, 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.

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

\(\Big \downarrow \) 1016

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 75

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

((a + b/x)^(1 + n)*x)/(a*d) + ((a*c^2*(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*n)*(a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + b/(a* 
x)])/(a*d*(1 + n)))/(a*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 899

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol 
] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, 
 b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx=\frac {-\left (a x +b \right )^{n} c n +\left (a x +b \right )^{n} c +\left (a x +b \right )^{n} d n x -x^{n} \left (\int \frac {\left (a x +b \right )^{n}}{x^{n} a c \,x^{2}+x^{n} a d \,x^{3}+x^{n} b c x +x^{n} b d \,x^{2}}d x \right ) b \,c^{2} n^{2}+x^{n} \left (\int \frac {\left (a x +b \right )^{n}}{x^{n} a c \,x^{2}+x^{n} a d \,x^{3}+x^{n} b c x +x^{n} b d \,x^{2}}d x \right ) b \,c^{2} n -x^{n} \left (\int \frac {\left (a x +b \right )^{n} x}{x^{n} a c x +x^{n} a d \,x^{2}+x^{n} b c +x^{n} b d x}d x \right ) a c d n +x^{n} \left (\int \frac {\left (a x +b \right )^{n} x}{x^{n} a c x +x^{n} a d \,x^{2}+x^{n} b c +x^{n} b d x}d x \right ) b \,d^{2} n^{2}}{x^{n} d^{2} n} \] Input:

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

Output:

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