Optimal. Leaf size=72 \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2}\right ) \]
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Rubi [A] time = 0.0733766, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a - b*x^n)^p*(a + b*x^n)^p,x]
[Out]
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Rubi in Sympy [A] time = 21.4, size = 56, normalized size = 0.78 \[ x \left (1 - \frac{b^{2} x^{2 n}}{a^{2}}\right )^{- p} \left (a - b x^{n}\right )^{p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{\frac{b^{2} x^{2 n}}{a^{2}}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a-b*x**n)**p*(a+b*x**n)**p,x)
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Mathematica [A] time = 0.126143, size = 72, normalized size = 1. \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{2 n},-p;1+\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a - b*x^n)^p*(a + b*x^n)^p,x]
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Maple [F] time = 0.242, size = 0, normalized size = 0. \[ \int \left ( a-b{x}^{n} \right ) ^{p} \left ( a+b{x}^{n} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a-b*x^n)^p*(a+b*x^n)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*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. \[{\rm integral}\left ({\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^p*(-b*x^n + a)^p,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a - b x^{n}\right )^{p} \left (a + b x^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a-b*x**n)**p*(a+b*x**n)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^p*(-b*x^n + a)^p,x, algorithm="giac")
[Out]