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