\(\int (a-b x^n)^p (a+b x^n)^p \, dx\) [66]

Optimal result

Integrand size = 20, antiderivative size = 72 \[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \, dx=x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,\frac {1}{2} \left (2+\frac {1}{n}\right ),\frac {b^2 x^{2 n}}{a^2}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \, dx=x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,1+\frac {1}{2 n},\frac {b^2 x^{2 n}}{a^2}\right ) \] Input:

Integrate[(a - b*x^n)^p*(a + b*x^n)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {785, 779, 778}

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.

\(\displaystyle \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \, dx\)

\(\Big \downarrow \) 785

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

\(\Big \downarrow \) 779

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

\(\Big \downarrow \) 778

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

Input:

Int[(a - b*x^n)^p*(a + b*x^n)^p,x]
 

Output:

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

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x 
^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(1 + b*(x^n/a))^p, x], x 
] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Si 
mplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 785

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Sy 
mbol] :> Simp[(a1 + b1*x^n)^FracPart[p]*((a2 + b2*x^n)^FracPart[p]/(a1*a2 + 
 b1*b2*x^(2*n))^FracPart[p])   Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /; Fre 
eQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]
 

Maple [F]

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

Input:

int((a-b*x^n)^p*(a+b*x^n)^p,x)
 

Output:

int((a-b*x^n)^p*(a+b*x^n)^p,x)
 

Fricas [F]

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

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

Output:

integral((b*x^n + a)^p*(-b*x^n + a)^p, x)
 

Sympy [F]

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

integrate((a-b*x**n)**p*(a+b*x**n)**p,x)
 

Output:

Integral((a - b*x**n)**p*(a + b*x**n)**p, x)
 

Maxima [F]

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

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

Output:

integrate((b*x^n + a)^p*(-b*x^n + a)^p, x)
 

Giac [F]

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

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

Output:

integrate((b*x^n + a)^p*(-b*x^n + a)^p, x)
 

Mupad [F(-1)]

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

int((a + b*x^n)^p*(a - b*x^n)^p,x)
 

Output:

int((a + b*x^n)^p*(a - b*x^n)^p, x)
 

Reduce [F]

\[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \, dx=\frac {\left (x^{n} b +a \right )^{p} \left (-x^{n} b +a \right )^{p} x -4 \left (\int \frac {\left (x^{n} b +a \right )^{p} \left (-x^{n} b +a \right )^{p}}{2 x^{2 n} b^{2} n p +x^{2 n} b^{2}-2 a^{2} n p -a^{2}}d x \right ) a^{2} n^{2} p^{2}-2 \left (\int \frac {\left (x^{n} b +a \right )^{p} \left (-x^{n} b +a \right )^{p}}{2 x^{2 n} b^{2} n p +x^{2 n} b^{2}-2 a^{2} n p -a^{2}}d x \right ) a^{2} n p}{2 n p +1} \] Input:

int((a-b*x^n)^p*(a+b*x^n)^p,x)
 

Output:

((x**n*b + a)**p*( - x**n*b + a)**p*x - 4*int(((x**n*b + a)**p*( - x**n*b 
+ a)**p)/(2*x**(2*n)*b**2*n*p + x**(2*n)*b**2 - 2*a**2*n*p - a**2),x)*a**2 
*n**2*p**2 - 2*int(((x**n*b + a)**p*( - x**n*b + a)**p)/(2*x**(2*n)*b**2*n 
*p + x**(2*n)*b**2 - 2*a**2*n*p - a**2),x)*a**2*n*p)/(2*n*p + 1)