3.2794 \(\int (c x)^{-n} (a+b x^n)^p \, dx\)

Optimal. Leaf size=53 \[ \frac{(c x)^{1-n} \left (a+b x^n\right )^{p+1} \, _2F_1\left (1,p+\frac{1}{n};\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (1-n)} \]

[Out]

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

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Rubi [A]  time = 0.0229762, antiderivative size = 64, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {365, 364} \[ \frac{(c x)^{1-n} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n}-1,-p;\frac{1}{n};-\frac{b x^n}{a}\right )}{c (1-n)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^p/(c*x)^n,x]

[Out]

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

Mathematica [A]  time = 0.0140734, size = 59, normalized size = 1.11 \[ -\frac{x (c x)^{-n} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n}-1,-p;\frac{1}{n};-\frac{b x^n}{a}\right )}{n-1} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^p/(c*x)^n,x]

[Out]

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

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{x}^{n} \right ) ^{p}}{ \left ( cx \right ) ^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^p/(c*x)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^n + a)^p/(c*x)^n, x)

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Sympy [C]  time = 9.48317, size = 44, normalized size = 0.83 \begin{align*} \frac{a^{p} c^{- n} x x^{- n} \Gamma \left (-1 + \frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, -1 + \frac{1}{n} \\ \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**p*c**(-n)*x*x**(-n)*gamma(-1 + 1/n)*hyper((-p, -1 + 1/n), (1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(1/n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^p/(c*x)^n, x)