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

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 = 0.23465, 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 \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \left (c+d x^{2 n}\right )^q \left (\frac{d x^{2 n}}{c}+1\right )^{-q} F_1\left (\frac{1}{2 n};-p,-q;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2},-\frac{d x^{2 n}}{c}\right ) \]

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)

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Rubi in Sympy [A]  time = 37.8593, size = 88, normalized size = 0.78 \[ x \left (1 - \frac{b^{2} x^{2 n}}{a^{2}}\right )^{- p} \left (1 + \frac{d x^{2 n}}{c}\right )^{- q} \left (a - b x^{n}\right )^{p} \left (a + b x^{n}\right )^{p} \left (c + d x^{2 n}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{1}{2 n},- p,- q,\frac{n + \frac{1}{2}}{n},\frac{b^{2} x^{2 n}}{a^{2}},- \frac{d x^{2 n}}{c} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 - b**2*x**(2*n)/a**2)**(-p)*(1 + d*x**(2*n)/c)**(-q)*(a - b*x**n)**p*(a + b
*x**n)**p*(c + d*x**(2*n))**q*appellf1(1/(2*n), -p, -q, (n + 1/2)/n, b**2*x**(2*
n)/a**2, -d*x**(2*n)/c)

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Mathematica [A]  time = 0.370362, 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 = 0.813, 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 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. \[ \int{\left (d x^{2 \, n} + c\right )}^{q}{\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^(2*n) + c)^q*(b*x^n + a)^p*(-b*x^n + a)^p,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. \[{\rm integral}\left ({\left (d x^{2 \, n} + c\right )}^{q}{\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^(2*n) + c)^q*(b*x^n + a)^p*(-b*x^n + a)^p,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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^(2*n) + c)^q*(b*x^n + a)^p*(-b*x^n + a)^p,x, algorithm="giac")

[Out]

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