\(\int x^{-1-n (1+2 p)} (a+b x^n)^p (c+d x^n)^p \, dx\) [567]

Optimal result

Integrand size = 31, antiderivative size = 106 \[ \int x^{-1-n (1+2 p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=-\frac {x^{-n (1+2 p)} \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^p \left (\frac {a \left (c+d x^n\right )}{c \left (a+b x^n\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{a n (1+2 p)} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.17 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.01 \[ \int x^{-1-n (1+2 p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \, dx=-\frac {x^{-n (1+2 p)} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^{1+p} \left (1+\frac {d x^n}{c}\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )}\right )}{c n (1+2 p)} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.48 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, 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.

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x^(-1-n*(1+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 - n - 1), x)
 

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*x**n*(x**n*d + c)**p*(x**n*b + a)**p*b*d - (x**n*d + c)**p*(x**n*b + 
 a)**p*a*d - (x**n*d + c)**p*(x**n*b + a)**p*b*c + 2*x**(2*n*p + 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 + 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 + 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...