3.2797 \(\int (c x)^{-1-n p} (a+b x^n)^p \, dx\)

Optimal. Leaf size=63 \[ -\frac{(c x)^{-n p} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{b x^n}{a}\right )}{c n p} \]

[Out]

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

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Rubi [A]  time = 0.0241856, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {365, 364} \[ -\frac{(c x)^{-n p} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{b x^n}{a}\right )}{c n p} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 - n*p)*(a + b*x^n)^p,x]

[Out]

-(((a + b*x^n)^p*Hypergeometric2F1[-p, -p, 1 - p, -((b*x^n)/a)])/(c*n*p*(c*x)^(n*p)*(1 + (b*x^n)/a)^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 (c x)^{-1-n p} \left (a+b x^n\right )^p \, dx &=\left (\left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int (c x)^{-1-n p} \left (1+\frac{b x^n}{a}\right )^p \, dx\\ &=-\frac{(c x)^{-n p} \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{b x^n}{a}\right )}{c n p}\\ \end{align*}

Mathematica [A]  time = 0.0134042, size = 63, normalized size = 1. \[ -\frac{x (c x)^{-n p-1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{b x^n}{a}\right )}{n p} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^(-1 - n*p)*(a + b*x^n)^p,x]

[Out]

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

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{-np-1} \left ( a+b{x}^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(-n*p-1)*(a+b*x^n)^p,x)

[Out]

int((c*x)^(-n*p-1)*(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 (c x\right )^{-n p - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-n*p-1)*(a+b*x^n)^p,x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^p*(c*x)^(-n*p - 1), 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 (c x\right )^{-n p - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-n*p-1)*(a+b*x^n)^p,x, algorithm="fricas")

[Out]

integral((b*x^n + a)^p*(c*x)^(-n*p - 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**(-n*p-1)*(a+b*x**n)**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{p} \left (c x\right )^{-n p - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-n*p-1)*(a+b*x^n)^p,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^p*(c*x)^(-n*p - 1), x)