\(\int \frac {(a+\frac {b}{c+d x^n})^p}{x^3} \, dx\) [338]

Optimal result

Integrand size = 19, antiderivative size = 96 \[ \int \frac {\left (a+\frac {b}{c+d x^n}\right )^p}{x^3} \, dx=-\frac {\left (1+\frac {d x^n}{c}\right )^p \left (1+\frac {a d x^n}{b+a c}\right )^{-p} \left (a+\frac {b}{c+d x^n}\right )^p \operatorname {AppellF1}\left (-\frac {2}{n},p,-p,-\frac {2-n}{n},-\frac {d x^n}{c},-\frac {a d x^n}{b+a c}\right )}{2 x^2} \] Output:

-1/2*(1+d*x^n/c)^p*(a+b/(c+d*x^n))^p*AppellF1(-2/n,p,-p,-(2-n)/n,-d*x^n/c, 
-a*d*x^n/(a*c+b))/x^2/((1+a*d*x^n/(a*c+b))^p)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2057, 2058, 1013, 1013, 1012}

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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {\left (\frac {d x^n}{c}+1\right )^p \left (\frac {a c+a d x^n+b}{c+d x^n}\right )^p \left (\frac {a d x^n}{a c+b}+1\right )^{-p} \operatorname {AppellF1}\left (-\frac {2}{n},p,-p,-\frac {2-n}{n},-\frac {d x^n}{c},-\frac {a d x^n}{b+a c}\right )}{2 x^2}\)

Input:

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

Output:

-1/2*(((b + a*c + a*d*x^n)/(c + d*x^n))^p*(1 + (d*x^n)/c)^p*AppellF1[-2/n, 
 p, -p, -((2 - n)/n), -((d*x^n)/c), -((a*d*x^n)/(b + a*c))])/(x^2*(1 + (a* 
d*x^n)/(b + a*c))^p)
 

Defintions of rubi rules used

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

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

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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