\(\int \frac {(a+\frac {b}{x})^n}{x^3 (c+d x)} \, dx\) [289]

Optimal result

Integrand size = 20, antiderivative size = 115 \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx=\frac {(a c+b d) \left (a+\frac {b}{x}\right )^{1+n}}{b^2 c^2 (1+n)}-\frac {\left (a+\frac {b}{x}\right )^{2+n}}{b^2 c (2+n)}+\frac {d^2 \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c^2 (a c-b d) (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx=-\frac {\left (a+\frac {b}{x}\right )^n (b+a x) \left ((a c-b d) (-b c (1+n)+a c x+b d (2+n) x)+b^2 d^2 (2+n) x \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )\right )}{b^2 c^2 (-a c+b d) (1+n) (2+n) x^2} \] Input:

Integrate[(a + b/x)^n/(x^3*(c + d*x)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1016, 948, 99, 2009}

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.

\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx\)

\(\Big \downarrow \) 1016

\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^n}{x^4 \left (\frac {c}{x}+d\right )}dx\)

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b/x)^n/(x^3*(c + d*x)),x]
 

Output:

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

Defintions of rubi rules used

rule 99

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

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])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {\left (a +\frac {b}{x}\right )^{n}}{x^{3} \left (d x +c \right )}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b/x)^n/x^3/(d*x+c),x, algorithm="fricas")
 

Output:

integral(((a*x + b)/x)^n/(d*x^4 + c*x^3), x)
 

Sympy [F]

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

integrate((a+b/x)**n/x**3/(d*x+c),x)
 

Output:

Integral((a + b/x)**n/(x**3*(c + d*x)), x)
 

Maxima [F]

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

integrate((a+b/x)^n/x^3/(d*x+c),x, algorithm="maxima")
 

Output:

integrate((a + b/x)^n/((d*x + c)*x^3), x)
 

Giac [F]

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

integrate((a+b/x)^n/x^3/(d*x+c),x, algorithm="giac")
 

Output:

integrate((a + b/x)^n/((d*x + c)*x^3), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((a*x + b)**n*a**2*c*n*x**2 - (a*x + b)**n*a*b*c*n**2*x - (a*x + b)**n*a*b 
*d*n*x**2 - 2*(a*x + b)**n*a*b*d*x**2 - (a*x + b)**n*b**2*c*n**2 - (a*x + 
b)**n*b**2*c*n + (a*x + b)**n*b**2*d*n**2*x + 2*(a*x + b)**n*b**2*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)*a*b**2*c*d*n**3*x**2 - 3*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)*a*b**2*c*d*n**2*x**2 - 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)*a*b**2*c*d*n*x**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**3*d**2*n**3*x**2 + 3*x**n*i 
nt((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**3*d**2*n**2*x**2 + 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**3*d**2*n*x**2)/(x**n*b**2*c** 
2*n*x**2*(n**2 + 3*n + 2))