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

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 = 0.150388, 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 \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac{b^4 x^{4 n}}{a^4}\right )^{-p} \, _2F_1\left (\frac{1}{4 n},-p;\frac{1}{4} \left (4+\frac{1}{n}\right );\frac{b^4 x^{4 n}}{a^4}\right ) \]

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

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Rubi in Sympy [A]  time = 32.5251, size = 70, normalized size = 0.8 \[ x \left (1 - \frac{b^{4} x^{4 n}}{a^{4}}\right )^{- p} \left (a - b x^{n}\right )^{p} \left (a + b x^{n}\right )^{p} \left (a^{2} + b^{2} x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{4 n} \\ \frac{n + \frac{1}{4}}{n} \end{matrix}\middle |{\frac{b^{4} x^{4 n}}{a^{4}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 - b**4*x**(4*n)/a**4)**(-p)*(a - b*x**n)**p*(a + b*x**n)**p*(a**2 + b**2*x*
*(2*n))**p*hyper((-p, 1/(4*n)), ((n + 1/4)/n,), b**4*x**(4*n)/a**4)

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Mathematica [A]  time = 0.278254, size = 0, normalized size = 0. \[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.806, size = 0, normalized size = 0. \[ \int \left ( a-b{x}^{n} \right ) ^{p} \left ( a+b{x}^{n} \right ) ^{p} \left ({a}^{2}+{b}^{2}{x}^{2\,n} \right ) ^{p}\, dx \]

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. \[ \int{\left (b^{2} x^{2 \, n} + a^{2}\right )}^{p}{\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + a^2)^p*(b*x^n + a)^p*(-b*x^n + a)^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)