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

Optimal. Leaf size=76 \[ -\frac{\left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2 n},-p;1-\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right )}{x} \]

[Out]

-(((-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])/(x*(1 - (b^2*x^(2*n))/a^2)^p))

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Rubi [A]  time = 0.117493, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2 n},-p;1-\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right )}{x} \]

Antiderivative was successfully verified.

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

[Out]

-(((-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])/(x*(1 - (b^2*x^(2*n))/a^2)^p))

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Rubi in Sympy [A]  time = 21.2007, size = 60, normalized size = 0.79 \[ - \frac{\left (1 - \frac{b^{2} x^{2 n}}{a^{2}}\right )^{- p} \left (- a + b x^{n}\right )^{p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{1}{2 n} \\ \frac{n - \frac{1}{2}}{n} \end{matrix}\middle |{\frac{b^{2} x^{2 n}}{a^{2}}} \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(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)/x

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Mathematica [A]  time = 0.105017, size = 76, normalized size = 1. \[ -\frac{\left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2 n},-p;1-\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right )}{x} \]

Antiderivative was successfully verified.

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

[Out]

-(((-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])/(x*(1 - (b^2*x^(2*n))/a^2)^p))

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Maple [F]  time = 0.222, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{n}-a \right ) ^{p} \left ( a+b{x}^{n} \right ) ^{p}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((b*x^n-a)^p*(a+b*x^n)^p/x^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^p*(b*x^n - a)^p/x^2,x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^p*(b*x^n - a)^p/x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{n} + a\right )}^{p}{\left (b x^{n} - a\right )}^{p}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^p*(b*x^n - a)^p/x^2,x, algorithm="fricas")

[Out]

integral((b*x^n + a)^p*(b*x^n - a)^p/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^p*(b*x^n - a)^p/x^2, x)