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

Optimal. Leaf size=78 \[ \frac{1}{3} x^3 \left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac{3}{2 n},-p;1+\frac{3}{2 n};\frac{b^2 x^{2 n}}{a^2}\right ) \]

[Out]

(x^3*(-a + b*x^n)^p*(a + b*x^n)^p*Hypergeometric2F1[3/(2*n), -p, 1 + 3/(2*n), (b
^2*x^(2*n))/a^2])/(3*(1 - (b^2*x^(2*n))/a^2)^p)

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Rubi [A]  time = 0.120972, antiderivative size = 78, 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 \[ \frac{1}{3} x^3 \left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac{3}{2 n},-p;1+\frac{3}{2 n};\frac{b^2 x^{2 n}}{a^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(-a + b*x^n)^p*(a + b*x^n)^p*Hypergeometric2F1[3/(2*n), -p, 1 + 3/(2*n), (b
^2*x^(2*n))/a^2])/(3*(1 - (b^2*x^(2*n))/a^2)^p)

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Rubi in Sympy [A]  time = 21.3161, size = 60, normalized size = 0.77 \[ \frac{x^{3} \left (1 - \frac{b^{2} x^{2 n}}{a^{2}}\right )^{- p} \left (- a + b x^{n}\right )^{p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2 n} \\ \frac{n + \frac{3}{2}}{n} \end{matrix}\middle |{\frac{b^{2} x^{2 n}}{a^{2}}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*(1 - b**2*x**(2*n)/a**2)**(-p)*(-a + b*x**n)**p*(a + b*x**n)**p*hyper((-p,
3/(2*n)), ((n + 3/2)/n,), b**2*x**(2*n)/a**2)/3

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Mathematica [A]  time = 0.153832, size = 78, normalized size = 1. \[ \frac{1}{3} x^3 \left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac{3}{2 n},-p;1+\frac{3}{2 n};\frac{b^2 x^{2 n}}{a^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(-a + b*x^n)^p*(a + b*x^n)^p*Hypergeometric2F1[3/(2*n), -p, 1 + 3/(2*n), (b
^2*x^(2*n))/a^2])/(3*(1 - (b^2*x^(2*n))/a^2)^p)

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Maple [F]  time = 0.217, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( b{x}^{n}-a \right ) ^{p} \left ( a+b{x}^{n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

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)