\(\int (a-b x^n)^p (a+b x^n)^p \, dx\) [283]

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 0.02 (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 = {259, 252, 251} \[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \, dx=x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,\frac {1}{2} \left (2+\frac {1}{n}\right ),\frac {b^2 x^{2 n}}{a^2}\right ) \]

[In]

Int[(a - b*x^n)^p*(a + b*x^n)^p,x]

[Out]

(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

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-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 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 259

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a1 + b1*x^n)^FracPar
t[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] /;
FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \left (\left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2-b^2 x^{2 n}\right )^{-p}\right ) \int \left (a^2-b^2 x^{2 n}\right )^p \, dx \\ & = \left (\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}\right ) \int \left (1-\frac {b^2 x^{2 n}}{a^2}\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} \, _2F_1\left (\frac {1}{2 n},-p;\frac {1}{2} \left (2+\frac {1}{n}\right );\frac {b^2 x^{2 n}}{a^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \, dx=x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,1+\frac {1}{2 n},\frac {b^2 x^{2 n}}{a^2}\right ) \]

[In]

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

[Out]

(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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*x^n)^p*(a - b*x^n)^p,x)

[Out]

int((a + b*x^n)^p*(a - b*x^n)^p, x)