3.383 \(\int (a-b x^n)^p (a+b x^n)^p (a^2+b^2 x^{2 n})^p \, dx\)

Optimal. Leaf size=87 \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac{b^4 x^{4 n}}{a^4}\right )^{-p} \, _2F_1\left (\frac{1}{4 n},-p;\frac{1}{4} \left (4+\frac{1}{n}\right );\frac{b^4 x^{4 n}}{a^4}\right ) \]

[Out]

(x*(a - b*x^n)^p*(a + b*x^n)^p*(a^2 + b^2*x^(2*n))^p*Hypergeometric2F1[1/(4*n), -p, (4 + n^(-1))/4, (b^4*x^(4*
n))/a^4])/(1 - (b^4*x^(4*n))/a^4)^p

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Rubi [A]  time = 0.0587613, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {519, 253, 246, 245} \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac{b^4 x^{4 n}}{a^4}\right )^{-p} \, _2F_1\left (\frac{1}{4 n},-p;\frac{1}{4} \left (4+\frac{1}{n}\right );\frac{b^4 x^{4 n}}{a^4}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a - b*x^n)^p*(a + b*x^n)^p*(a^2 + b^2*x^(2*n))^p*Hypergeometric2F1[1/(4*n), -p, (4 + n^(-1))/4, (b^4*x^(4*
n))/a^4])/(1 - (b^4*x^(4*n))/a^4)^p

Rule 519

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracP
art[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

Rule 253

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[((a1 + b1*x^n)^FracPa
rt[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]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[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 245

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])

Rubi steps

\begin{align*} \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \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 (a^2-b^2 x^{2 n}\right )^{-p}\right ) \int \left (a^2-b^2 x^{2 n}\right )^p \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 (a^2+b^2 x^{2 n}\right )^p \left (a^4-b^4 x^{4 n}\right )^{-p}\right ) \int \left (a^4-b^4 x^{4 n}\right )^p \, dx\\ &=\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 \left (1-\frac{b^4 x^{4 n}}{a^4}\right )^{-p}\right ) \int \left (1-\frac{b^4 x^{4 n}}{a^4}\right )^p \, dx\\ &=x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac{b^4 x^{4 n}}{a^4}\right )^{-p} \, _2F_1\left (\frac{1}{4 n},-p;\frac{1}{4} \left (4+\frac{1}{n}\right );\frac{b^4 x^{4 n}}{a^4}\right )\\ \end{align*}

Mathematica [A]  time = 0.0419371, size = 87, normalized size = 1. \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac{b^4 x^{4 n}}{a^4}\right )^{-p} \, _2F_1\left (\frac{1}{4 n},-p;1+\frac{1}{4 n};\frac{b^4 x^{4 n}}{a^4}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a - b*x^n)^p*(a + b*x^n)^p*(a^2 + b^2*x^(2*n))^p*Hypergeometric2F1[1/(4*n), -p, 1 + 1/(4*n), (b^4*x^(4*n))
/a^4])/(1 - (b^4*x^(4*n))/a^4)^p

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Maple [F]  time = 1.464, size = 0, normalized size = 0. \begin{align*} \int \left ( a-b{x}^{n} \right ) ^{p} \left ( a+b{x}^{n} \right ) ^{p} \left ({a}^{2}+{b}^{2}{x}^{2\,n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b^{2} x^{2 \, n} + a^{2}\right )}^{p}{\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-b*x^n)^p*(a+b*x^n)^p*(a^2+b^2*x^(2*n))^p,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} x^{2 \, n} + a^{2}\right )}^{p}{\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-b*x^n)^p*(a+b*x^n)^p*(a^2+b^2*x^(2*n))^p,x, algorithm="fricas")

[Out]

integral((b^2*x^(2*n) + a^2)^p*(b*x^n + a)^p*(-b*x^n + a)^p, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b^{2} x^{2 \, n} + a^{2}\right )}^{p}{\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-b*x^n)^p*(a+b*x^n)^p*(a^2+b^2*x^(2*n))^p,x, algorithm="giac")

[Out]

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