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