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,1+1/2/n,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 = 0.09, antiderivative size = 113, normalized size of antiderivative = 1.00, 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 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])

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 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])

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*}

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Mathematica [F]  time = 0.44, size = 0, normalized size = 0.00 \[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (c+d x^{2 n}\right )^q \, dx \]

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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fricas [F]  time = 1.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x^{2 \, n} + c\right )}^{q} {\left (b x^{n} + a\right )}^{p} {\left (-b x^{n} + a\right )}^{p}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{2 \, n} + c\right )}^{q} {\left (b x^{n} + a\right )}^{p} {\left (-b x^{n} + a\right )}^{p}\,{d x} \]

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)

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maple [F]  time = 1.60, size = 0, normalized size = 0.00 \[ \int \left (-b \,x^{n}+a \right )^{p} \left (b \,x^{n}+a \right )^{p} \left (d \,x^{2 n}+c \right )^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{2 \, n} + c\right )}^{q} {\left (b x^{n} + a\right )}^{p} {\left (-b x^{n} + a\right )}^{p}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c+d\,x^{2\,n}\right )}^q\,{\left (a+b\,x^n\right )}^p\,{\left (a-b\,x^n\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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