3.1019 \(\int x (-a+b x^n)^p (a+b x^n)^p \, dx\)

Optimal. Leaf size=70 \[ \frac{1}{2} x^2 \left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};\frac{b^2 x^{2 n}}{a^2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0307147, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {366, 365, 364} \[ \frac{1}{2} x^2 \left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};\frac{b^2 x^{2 n}}{a^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*(-a + b*x^n)^p*(a + b*x^n)^p,x]

[Out]

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

Rule 366

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

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

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

Rubi steps

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

Mathematica [A]  time = 0.0144613, size = 72, normalized size = 1.03 \[ \frac{1}{2} x^2 \left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \text{HypergeometricPFQ}\left (\left \{\frac{1}{n},-p\right \},\left \{\frac{1}{n}+1\right \},\frac{b^2 x^{2 n}}{a^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.142, size = 0, normalized size = 0. \begin{align*} \int x \left ( -a+b{x}^{n} \right ) ^{p} \left ( a+b{x}^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-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, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-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, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{p}{\left (b x^{n} - a\right )}^{p} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-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, x)