\(\int x^{-1-n (2+2 p)} (a+b x^n)^p (c+d x^n)^p \, dx\) [566]

Optimal result

Integrand size = 31, antiderivative size = 167 \[ \int x^{-1-n (2+2 p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=-\frac {x^{-2 n (1+p)} \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{1+p}}{2 a c n (1+p)}+\frac {(b c+a d) x^{-n (1+2 p)} \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^p \left (\frac {a \left (c+d x^n\right )}{c \left (a+b x^n\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{2 a^2 c n (1+2 p)} \] Output:

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

Mathematica [F]

\[ \int x^{-1-n (2+2 p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=\int x^{-1-n (2+2 p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx \] Input:

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

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 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int x^{-1-n \left (2+2 p \right )} \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{p}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int x^{-1-n (2+2 p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} x^{-2 \, n {\left (p + 1\right )} - 1} \,d x } \] Input:

integrate(x^(-1-n*(2+2*p))*(a+b*x^n)^p*(c+d*x^n)^p,x, algorithm="fricas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int x^{-1-n (2+2 p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(x**(-1-n*(2+2*p))*(a+b*x**n)**p*(c+d*x**n)**p,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int x^{-1-n (2+2 p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} x^{-2 \, n {\left (p + 1\right )} - 1} \,d x } \] Input:

integrate(x^(-1-n*(2+2*p))*(a+b*x^n)^p*(c+d*x^n)^p,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int x^{-1-n (2+2 p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} x^{-2 \, n {\left (p + 1\right )} - 1} \,d x } \] Input:

integrate(x^(-1-n*(2+2*p))*(a+b*x^n)^p*(c+d*x^n)^p,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int x^{-1-n (2+2 p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=\int \frac {{\left (a+b\,x^n\right )}^p\,{\left (c+d\,x^n\right )}^p}{x^{n\,\left (2\,p+2\right )+1}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x^{-1-n (2+2 p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=\text {too large to display} \] Input:

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

Output:

(x**(2*n)*(x**n*d + c)**p*(x**n*b + a)**p*b*d - x**n*(x**n*d + c)**p*(x**n 
*b + a)**p*a*d*p - x**n*(x**n*d + c)**p*(x**n*b + a)**p*b*c*p - 2*(x**n*d 
+ c)**p*(x**n*b + a)**p*a*c*p - (x**n*d + c)**p*(x**n*b + a)**p*a*c - 2*x* 
*(2*n*p + 2*n)*int(((x**n*d + c)**p*(x**n*b + a)**p)/(2*x**(2*n*p + 2*n)*a 
*b*d**2*p*x + x**(2*n*p + 2*n)*a*b*d**2*x + 2*x**(2*n*p + 2*n)*b**2*c*d*p* 
x + x**(2*n*p + 2*n)*b**2*c*d*x + 2*x**(2*n*p + n)*a**2*d**2*p*x + x**(2*n 
*p + n)*a**2*d**2*x + 4*x**(2*n*p + n)*a*b*c*d*p*x + 2*x**(2*n*p + n)*a*b* 
c*d*x + 2*x**(2*n*p + n)*b**2*c**2*p*x + x**(2*n*p + n)*b**2*c**2*x + 2*x* 
*(2*n*p)*a**2*c*d*p*x + x**(2*n*p)*a**2*c*d*x + 2*x**(2*n*p)*a*b*c**2*p*x 
+ x**(2*n*p)*a*b*c**2*x),x)*a**3*d**3*n*p**3 - 3*x**(2*n*p + 2*n)*int(((x* 
*n*d + c)**p*(x**n*b + a)**p)/(2*x**(2*n*p + 2*n)*a*b*d**2*p*x + x**(2*n*p 
 + 2*n)*a*b*d**2*x + 2*x**(2*n*p + 2*n)*b**2*c*d*p*x + x**(2*n*p + 2*n)*b* 
*2*c*d*x + 2*x**(2*n*p + n)*a**2*d**2*p*x + x**(2*n*p + n)*a**2*d**2*x + 4 
*x**(2*n*p + n)*a*b*c*d*p*x + 2*x**(2*n*p + n)*a*b*c*d*x + 2*x**(2*n*p + n 
)*b**2*c**2*p*x + x**(2*n*p + n)*b**2*c**2*x + 2*x**(2*n*p)*a**2*c*d*p*x + 
 x**(2*n*p)*a**2*c*d*x + 2*x**(2*n*p)*a*b*c**2*p*x + x**(2*n*p)*a*b*c**2*x 
),x)*a**3*d**3*n*p**2 - x**(2*n*p + 2*n)*int(((x**n*d + c)**p*(x**n*b + a) 
**p)/(2*x**(2*n*p + 2*n)*a*b*d**2*p*x + x**(2*n*p + 2*n)*a*b*d**2*x + 2*x* 
*(2*n*p + 2*n)*b**2*c*d*p*x + x**(2*n*p + 2*n)*b**2*c*d*x + 2*x**(2*n*p + 
n)*a**2*d**2*p*x + x**(2*n*p + n)*a**2*d**2*x + 4*x**(2*n*p + n)*a*b*c*...