3.184 \(\int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \, dx\)

Optimal. Leaf size=72 \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2}\right ) \]

[Out]

(x*(a - b*x^n)^p*(a + b*x^n)^p*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, (b
^2*x^(2*n))/a^2])/(1 - (b^2*x^(2*n))/a^2)^p

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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]

(x*(a - b*x^n)^p*(a + b*x^n)^p*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, (b
^2*x^(2*n))/a^2])/(1 - (b^2*x^(2*n))/a^2)^p

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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)

[Out]

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

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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]

[Out]

(x*(a - b*x^n)^p*(a + b*x^n)^p*Hypergeometric2F1[1/(2*n), -p, 1 + 1/(2*n), (b^2*
x^(2*n))/a^2])/(1 - (b^2*x^(2*n))/a^2)^p

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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]

int((a-b*x^n)^p*(a+b*x^n)^p,x)

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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]

integrate((b*x^n + a)^p*(-b*x^n + a)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}, x\right ) \]

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]

integral((b*x^n + a)^p*(-b*x^n + a)^p, x)

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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]

Integral((a - b*x**n)**p*(a + b*x**n)**p, x)

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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]

integrate((b*x^n + a)^p*(-b*x^n + a)^p, x)