\(\int x^{-1-n} (a+b x^n)^p (c+d x^n)^q \, dx\) [574]

Optimal result

Integrand size = 26, antiderivative size = 97 \[ \int x^{-1-n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\frac {b \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^q \left (\frac {b \left (c+d x^n\right )}{b c-a d}\right )^{-q} \operatorname {AppellF1}\left (1+p,2,-q,2+p,1+\frac {b x^n}{a},-\frac {d \left (a+b x^n\right )}{b c-a d}\right )}{a^2 n (1+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.13 \[ \int x^{-1-n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\frac {x^{-n} \left (1+\frac {a x^{-n}}{b}\right )^{-p} \left (1+\frac {c x^{-n}}{d}\right )^{-q} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \operatorname {AppellF1}\left (1-p-q,-p,-q,2-p-q,-\frac {a x^{-n}}{b},-\frac {c x^{-n}}{d}\right )}{n (-1+p+q)} \] Input:

Integrate[x^(-1 - n)*(a + b*x^n)^p*(c + d*x^n)^q,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {948, 154, 153}

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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 154

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

\(\Big \downarrow \) 153

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

Input:

Int[x^(-1 - n)*(a + b*x^n)^p*(c + d*x^n)^q,x]
 

Output:

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

Defintions of rubi rules used

rule 153

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(b*e - a*f)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*Simp 
lify[b/(b*c - a*d)]^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c 
 - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, 
n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[p] && GtQ[Simplify[b/( 
b*c - a*d)], 0] &&  !(GtQ[Simplify[d/(d*a - c*b)], 0] && SimplerQ[c + d*x, 
a + b*x])
 

rule 154

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(c + d*x)^FracPart[n]/(Simplify[b/(b*c - a*d)]^IntPart[n 
]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c 
 - a*d)) + b*d*(x/(b*c - a*d)), x]^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[p] &&  !G 
tQ[Simplify[b/(b*c - a*d)], 0] &&  !SimplerQ[c + d*x, a + b*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]]
 

Maple [F]

\[\int x^{-1-n} \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{q}d x\]

Input:

int(x^(-1-n)*(a+b*x^n)^p*(c+d*x^n)^q,x)
 

Output:

int(x^(-1-n)*(a+b*x^n)^p*(c+d*x^n)^q,x)
 

Fricas [F]

\[ \int x^{-1-n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} x^{-n - 1} \,d x } \] Input:

integrate(x^(-1-n)*(a+b*x^n)^p*(c+d*x^n)^q,x, algorithm="fricas")
 

Output:

integral((b*x^n + a)^p*(d*x^n + c)^q*x^(-n - 1), x)
 

Sympy [F(-2)]

Exception generated. \[ \int x^{-1-n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(x**(-1-n)*(a+b*x**n)**p*(c+d*x**n)**q,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int x^{-1-n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} x^{-n - 1} \,d x } \] Input:

integrate(x^(-1-n)*(a+b*x^n)^p*(c+d*x^n)^q,x, algorithm="maxima")
 

Output:

integrate((b*x^n + a)^p*(d*x^n + c)^q*x^(-n - 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int x^{-1-n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^(-1-n)*(a+b*x^n)^p*(c+d*x^n)^q,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{-1,[0,0,4,3,8,2,3,3,9,3]%%%}+%%%{6,[0,0,4,3,7,3,3,3,8,4]%% 
%}+%%%{-1
 

Mupad [F(-1)]

Timed out. \[ \int x^{-1-n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\int \frac {{\left (a+b\,x^n\right )}^p\,{\left (c+d\,x^n\right )}^q}{x^{n+1}} \,d x \] Input:

int(((a + b*x^n)^p*(c + d*x^n)^q)/x^(n + 1),x)
 

Output:

int(((a + b*x^n)^p*(c + d*x^n)^q)/x^(n + 1), x)
 

Reduce [F]

\[ \int x^{-1-n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\text {too large to display} \] Input:

int(x^(-1-n)*(a+b*x^n)^p*(c+d*x^n)^q,x)
 

Output:

( - x**n*(x**n*d + c)**q*(x**n*b + a)**p*a*b*d**2*p - x**n*(x**n*d + c)**q 
*(x**n*b + a)**p*b**2*c*d*q - (x**n*d + c)**q*(x**n*b + a)**p*a**2*d**2*q* 
*2 + (x**n*d + c)**q*(x**n*b + a)**p*a**2*d**2*q - 2*(x**n*d + c)**q*(x**n 
*b + a)**p*a*b*c*d*p*q + (x**n*d + c)**q*(x**n*b + a)**p*a*b*c*d*p + (x**n 
*d + c)**q*(x**n*b + a)**p*a*b*c*d*q - (x**n*d + c)**q*(x**n*b + a)**p*b** 
2*c**2*p**2 + (x**n*d + c)**q*(x**n*b + a)**p*b**2*c**2*p + x**n*int(((x** 
n*d + c)**q*(x**n*b + a)**p)/(x**(2*n)*a**2*b*d**3*q**2*x - x**(2*n)*a**2* 
b*d**3*q*x + 2*x**(2*n)*a*b**2*c*d**2*p*q*x - x**(2*n)*a*b**2*c*d**2*p*x - 
 x**(2*n)*a*b**2*c*d**2*q*x + x**(2*n)*b**3*c**2*d*p**2*x - x**(2*n)*b**3* 
c**2*d*p*x + x**n*a**3*d**3*q**2*x - x**n*a**3*d**3*q*x + 2*x**n*a**2*b*c* 
d**2*p*q*x - x**n*a**2*b*c*d**2*p*x + x**n*a**2*b*c*d**2*q**2*x - 2*x**n*a 
**2*b*c*d**2*q*x + x**n*a*b**2*c**2*d*p**2*x + 2*x**n*a*b**2*c**2*d*p*q*x 
- 2*x**n*a*b**2*c**2*d*p*x - x**n*a*b**2*c**2*d*q*x + x**n*b**3*c**3*p**2* 
x - x**n*b**3*c**3*p*x + a**3*c*d**2*q**2*x - a**3*c*d**2*q*x + 2*a**2*b*c 
**2*d*p*q*x - a**2*b*c**2*d*p*x - a**2*b*c**2*d*q*x + a*b**2*c**3*p**2*x - 
 a*b**2*c**3*p*x),x)*a**5*d**5*n*q**5 - 2*x**n*int(((x**n*d + c)**q*(x**n* 
b + a)**p)/(x**(2*n)*a**2*b*d**3*q**2*x - x**(2*n)*a**2*b*d**3*q*x + 2*x** 
(2*n)*a*b**2*c*d**2*p*q*x - x**(2*n)*a*b**2*c*d**2*p*x - x**(2*n)*a*b**2*c 
*d**2*q*x + x**(2*n)*b**3*c**2*d*p**2*x - x**(2*n)*b**3*c**2*d*p*x + x**n* 
a**3*d**3*q**2*x - x**n*a**3*d**3*q*x + 2*x**n*a**2*b*c*d**2*p*q*x - x*...