3.11.20 \(\int (-a+b x^n)^p (a+b x^n)^p \, dx\) [1020]

Optimal. Leaf size=73 \[ x \left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac {1}{2 n},-p;\frac {1}{2} \left (2+\frac {1}{n}\right );\frac {b^2 x^{2 n}}{a^2}\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {259, 252, 251} \begin {gather*} x \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;\frac {1}{2} \left (2+\frac {1}{n}\right );\frac {b^2 x^{2 n}}{a^2}\right ) \end {gather*}

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

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 259

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a1 + b1*x^n)^FracPar
t[p]*((a2 + b2*x^n)^FracPart[p]/(a1*a2 + b1*b2*x^(2*n))^FracPart[p]), Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /;
FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \left (-a+b x^n\right )^p \left (a+b x^n\right )^p \, dx &=\left (\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (-a^2+b^2 x^{2 n}\right )^{-p}\right ) \int \left (-a^2+b^2 x^{2 n}\right )^p \, dx\\ &=\left (\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}\right ) \int \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^p \, dx\\ &=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 )\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 73, normalized size = 1.00 \begin {gather*} 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 ) \end {gather*}

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 = 0.13, size = 0, normalized size = 0.00 \[\int \left (b \,x^{n}-a \right )^{p} \left (a +b \,x^{n}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a+b*x^n)^p*(a+b*x^n)^p,x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a+b*x^n)^p*(a+b*x^n)^p,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- a + b x^{n}\right )^{p} \left (a + b x^{n}\right )^{p}\, dx \end {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a+b*x^n)^p*(a+b*x^n)^p,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,x^n\right )}^p\,{\left (b\,x^n-a\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^p*(b*x^n - a)^p,x)

[Out]

int((a + b*x^n)^p*(b*x^n - a)^p, x)

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