\(\int x^{-1-2 n p} (a+b x^n)^p (c+d x^n)^p \, dx\) [568]

Optimal result

Integrand size = 27, antiderivative size = 95 \[ \int x^{-1-2 n p} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=-\frac {x^{-2 n p} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^p \left (1+\frac {d x^n}{c}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{2 n p} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.02 \[ \int x^{-1-2 n p} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=-\frac {x^{-2 n p} \left (a+b x^n\right )^p \left (\frac {a+b x^n}{a}\right )^{-p} \left (c+d x^n\right )^p \left (\frac {c+d x^n}{c}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{2 n p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {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.

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

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

Input:

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

Output:

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x^(-2*n*p-1)*(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 - 1), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^(-2*n*p-1)*(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), x)
 

Giac [F]

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

integrate(x^(-2*n*p-1)*(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), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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