\(\int \frac {(a+\frac {b}{x})^n}{x^5 (c+d x)} \, dx\) [290]

Optimal result

Integrand size = 20, antiderivative size = 207 \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^5 (c+d x)} \, dx=\frac {(a c+b d) \left (a^2 c^2+b^2 d^2\right ) \left (a+\frac {b}{x}\right )^{1+n}}{b^4 c^4 (1+n)}-\frac {\left (3 a^2 c^2+2 a b c d+b^2 d^2\right ) \left (a+\frac {b}{x}\right )^{2+n}}{b^4 c^3 (2+n)}+\frac {(3 a c+b d) \left (a+\frac {b}{x}\right )^{3+n}}{b^4 c^2 (3+n)}-\frac {\left (a+\frac {b}{x}\right )^{4+n}}{b^4 c (4+n)}+\frac {d^4 \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c^4 (a c-b d) (1+n)} \] Output:

(a*c+b*d)*(a^2*c^2+b^2*d^2)*(a+b/x)^(1+n)/b^4/c^4/(1+n)-(3*a^2*c^2+2*a*b*c 
*d+b^2*d^2)*(a+b/x)^(2+n)/b^4/c^3/(2+n)+(3*a*c+b*d)*(a+b/x)^(3+n)/b^4/c^2/ 
(3+n)-(a+b/x)^(4+n)/b^4/c/(4+n)+d^4*(a+b/x)^(1+n)*hypergeom([1, 1+n],[2+n] 
,c*(a+b/x)/(a*c-b*d))/c^4/(a*c-b*d)/(1+n)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^5 (c+d x)} \, dx=\frac {\left (a+\frac {b}{x}\right )^{1+n} \left (\frac {(a c+b d) \left (a^2 c^2+b^2 d^2\right )}{b^4 (1+n)}-\frac {c \left (3 a^2 c^2+2 a b c d+b^2 d^2\right ) \left (a+\frac {b}{x}\right )}{b^4 (2+n)}+\frac {c^2 (3 a c+b d) \left (a+\frac {b}{x}\right )^2}{b^4 (3+n)}-\frac {c^3 \left (a+\frac {b}{x}\right )^3}{b^4 (4+n)}+\frac {d^4 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{(a c-b d) (1+n)}\right )}{c^4} \] Input:

Integrate[(a + b/x)^n/(x^5*(c + d*x)),x]
 

Output:

((a + b/x)^(1 + n)*(((a*c + b*d)*(a^2*c^2 + b^2*d^2))/(b^4*(1 + n)) - (c*( 
3*a^2*c^2 + 2*a*b*c*d + b^2*d^2)*(a + b/x))/(b^4*(2 + n)) + (c^2*(3*a*c + 
b*d)*(a + b/x)^2)/(b^4*(3 + n)) - (c^3*(a + b/x)^3)/(b^4*(4 + n)) + (d^4*H 
ypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/((a*c - b*d) 
*(1 + n))))/c^4
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 207, 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.

\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^n}{x^5 (c+d x)} \, dx\)

\(\Big \downarrow \) 1016

\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^n}{x^6 \left (\frac {c}{x}+d\right )}dx\)

\(\Big \downarrow \) 948

\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^n}{\left (\frac {c}{x}+d\right ) x^4}d\frac {1}{x}\)

\(\Big \downarrow \) 99

\(\displaystyle -\int \left (\frac {(a c+b d) \left (-a^2 c^2-b^2 d^2\right ) \left (a+\frac {b}{x}\right )^n}{b^3 c^4}+\frac {d^4 \left (a+\frac {b}{x}\right )^n}{c^4 \left (\frac {c}{x}+d\right )}+\frac {\left (3 a^2 c^2+2 a b d c+b^2 d^2\right ) \left (a+\frac {b}{x}\right )^{n+1}}{b^3 c^3}+\frac {(-3 a c-b d) \left (a+\frac {b}{x}\right )^{n+2}}{b^3 c^2}+\frac {\left (a+\frac {b}{x}\right )^{n+3}}{b^3 c}\right )d\frac {1}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(a c+b d) \left (a^2 c^2+b^2 d^2\right ) \left (a+\frac {b}{x}\right )^{n+1}}{b^4 c^4 (n+1)}-\frac {\left (3 a^2 c^2+2 a b c d+b^2 d^2\right ) \left (a+\frac {b}{x}\right )^{n+2}}{b^4 c^3 (n+2)}+\frac {(3 a c+b d) \left (a+\frac {b}{x}\right )^{n+3}}{b^4 c^2 (n+3)}-\frac {\left (a+\frac {b}{x}\right )^{n+4}}{b^4 c (n+4)}+\frac {d^4 \left (a+\frac {b}{x}\right )^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c^4 (n+1) (a c-b d)}\)

Input:

Int[(a + b/x)^n/(x^5*(c + d*x)),x]
 

Output:

((a*c + b*d)*(a^2*c^2 + b^2*d^2)*(a + b/x)^(1 + n))/(b^4*c^4*(1 + n)) - (( 
3*a^2*c^2 + 2*a*b*c*d + b^2*d^2)*(a + b/x)^(2 + n))/(b^4*c^3*(2 + n)) + (( 
3*a*c + b*d)*(a + b/x)^(3 + n))/(b^4*c^2*(3 + n)) - (a + b/x)^(4 + n)/(b^4 
*c*(4 + n)) + (d^4*(a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (c 
*(a + b/x))/(a*c - b*d)])/(c^4*(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]

\[\int \frac {\left (a +\frac {b}{x}\right )^{n}}{x^{5} \left (d x +c \right )}d x\]

Input:

int((a+b/x)^n/x^5/(d*x+c),x)
 

Output:

int((a+b/x)^n/x^5/(d*x+c),x)
 

Fricas [F]

\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^5 (c+d x)} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )} x^{5}} \,d x } \] Input:

integrate((a+b/x)^n/x^5/(d*x+c),x, algorithm="fricas")
 

Output:

integral(((a*x + b)/x)^n/(d*x^6 + c*x^5), x)
 

Sympy [F]

\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^5 (c+d x)} \, dx=\int \frac {\left (a + \frac {b}{x}\right )^{n}}{x^{5} \left (c + d x\right )}\, dx \] Input:

integrate((a+b/x)**n/x**5/(d*x+c),x)
 

Output:

Integral((a + b/x)**n/(x**5*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^5 (c+d x)} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )} x^{5}} \,d x } \] Input:

integrate((a+b/x)^n/x^5/(d*x+c),x, algorithm="maxima")
 

Output:

integrate((a + b/x)^n/((d*x + c)*x^5), x)
 

Giac [F]

\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^5 (c+d x)} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )} x^{5}} \,d x } \] Input:

integrate((a+b/x)^n/x^5/(d*x+c),x, algorithm="giac")
 

Output:

integrate((a + b/x)^n/((d*x + c)*x^5), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^5 (c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{x}\right )}^n}{x^5\,\left (c+d\,x\right )} \,d x \] Input:

int((a + b/x)^n/(x^5*(c + d*x)),x)
 

Output:

int((a + b/x)^n/(x^5*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {\left (a+\frac {b}{x}\right )^n}{x^5 (c+d x)} \, dx=\text {too large to display} \] Input:

int((a+b/x)^n/x^5/(d*x+c),x)
 

Output:

(6*(a*x + b)**n*a**5*c**3*n*x**4 - 6*(a*x + b)**n*a**4*b*c**3*n**2*x**3 + 
6*(a*x + b)**n*a**4*b*c**2*d*n**2*x**4 - 24*(a*x + b)**n*a**4*b*c**2*d*x** 
4 + 3*(a*x + b)**n*a**3*b**2*c**3*n**3*x**2 + 3*(a*x + b)**n*a**3*b**2*c** 
3*n**2*x**2 - 6*(a*x + b)**n*a**3*b**2*c**2*d*n**3*x**3 + 24*(a*x + b)**n* 
a**3*b**2*c**2*d*n*x**3 - 6*(a*x + b)**n*a**3*b**2*c*d**2*n**2*x**4 - 30*( 
a*x + b)**n*a**3*b**2*c*d**2*n*x**4 - 24*(a*x + b)**n*a**3*b**2*c*d**2*x** 
4 - (a*x + b)**n*a**2*b**3*c**3*n**4*x - 3*(a*x + b)**n*a**2*b**3*c**3*n** 
3*x - 2*(a*x + b)**n*a**2*b**3*c**3*n**2*x + 4*(a*x + b)**n*a**2*b**3*c**2 
*d*n**4*x**2 + 11*(a*x + b)**n*a**2*b**3*c**2*d*n**3*x**2 + 7*(a*x + b)**n 
*a**2*b**3*c**2*d*n**2*x**2 - 2*(a*x + b)**n*a**2*b**3*c*d**2*n**4*x**3 - 
10*(a*x + b)**n*a**2*b**3*c*d**2*n**3*x**3 - 8*(a*x + b)**n*a**2*b**3*c*d* 
*2*n**2*x**3 - (a*x + b)**n*a*b**4*c**3*n**4 - 6*(a*x + b)**n*a*b**4*c**3* 
n**3 - 11*(a*x + b)**n*a*b**4*c**3*n**2 - 6*(a*x + b)**n*a*b**4*c**3*n - ( 
a*x + b)**n*a*b**4*c**2*d*n**5*x - 3*(a*x + b)**n*a*b**4*c**2*d*n**4*x + 2 
*(a*x + b)**n*a*b**4*c**2*d*n**3*x + 12*(a*x + b)**n*a*b**4*c**2*d*n**2*x 
+ 8*(a*x + b)**n*a*b**4*c**2*d*n*x + (a*x + b)**n*a*b**4*c*d**2*n**5*x**2 
+ 5*(a*x + b)**n*a*b**4*c*d**2*n**4*x**2 + (a*x + b)**n*a*b**4*c*d**2*n**3 
*x**2 - 15*(a*x + b)**n*a*b**4*c*d**2*n**2*x**2 - 12*(a*x + b)**n*a*b**4*c 
*d**2*n*x**2 + 2*(a*x + b)**n*a*b**4*d**3*n**4*x**3 + 16*(a*x + b)**n*a*b* 
*4*d**3*n**3*x**3 + 38*(a*x + b)**n*a*b**4*d**3*n**2*x**3 + 24*(a*x + b...