3.382 \(\int (a-b x^n)^p (a+b x^n)^p (c+d x^{2 n})^q \, dx\)

Optimal. Leaf size=113 \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \left (c+d x^{2 n}\right )^q \left (\frac{d x^{2 n}}{c}+1\right )^{-q} F_1\left (\frac{1}{2 n};-p,-q;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2},-\frac{d x^{2 n}}{c}\right ) \]

[Out]

(x*(a - b*x^n)^p*(a + b*x^n)^p*(c + d*x^(2*n))^q*AppellF1[1/(2*n), -p, -q, (2 + n^(-1))/2, (b^2*x^(2*n))/a^2,
-((d*x^(2*n))/c)])/((1 - (b^2*x^(2*n))/a^2)^p*(1 + (d*x^(2*n))/c)^q)

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Rubi [A]  time = 0.0904555, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {519, 430, 429} \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \left (c+d x^{2 n}\right )^q \left (\frac{d x^{2 n}}{c}+1\right )^{-q} F_1\left (\frac{1}{2 n};-p,-q;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2},-\frac{d x^{2 n}}{c}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a - b*x^n)^p*(a + b*x^n)^p*(c + d*x^(2*n))^q*AppellF1[1/(2*n), -p, -q, (2 + n^(-1))/2, (b^2*x^(2*n))/a^2,
-((d*x^(2*n))/c)])/((1 - (b^2*x^(2*n))/a^2)^p*(1 + (d*x^(2*n))/c)^q)

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 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (c+d x^{2 n}\right )^q \, 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 (c+d x^{2 n}\right )^q \, 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 \left (c+d x^{2 n}\right )^q \, 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} \left (c+d x^{2 n}\right )^q \left (1+\frac{d x^{2 n}}{c}\right )^{-q}\right ) \int \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^p \left (1+\frac{d x^{2 n}}{c}\right )^q \, 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} \left (c+d x^{2 n}\right )^q \left (1+\frac{d x^{2 n}}{c}\right )^{-q} F_1\left (\frac{1}{2 n};-p,-q;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2},-\frac{d x^{2 n}}{c}\right )\\ \end{align*}

Mathematica [F]  time = 0.238882, size = 0, normalized size = 0. \[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (c+d x^{2 n}\right )^q \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 1.492, size = 0, normalized size = 0. \begin{align*} \int \left ( a-b{x}^{n} \right ) ^{p} \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{2\,n} \right ) ^{q}\, 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*(c+d*x^(2*n))^q,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{2 \, n} + c\right )}^{q}{\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*(c+d*x^(2*n))^q,x, algorithm="maxima")

[Out]

integrate((d*x^(2*n) + c)^q*(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 (d x^{2 \, n} + c\right )}^{q}{\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*(c+d*x^(2*n))^q,x, algorithm="fricas")

[Out]

integral((d*x^(2*n) + c)^q*(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*(c+d*x**(2*n))**q,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{2 \, n} + c\right )}^{q}{\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*(c+d*x^(2*n))^q,x, algorithm="giac")

[Out]

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