3.3051 \(\int \frac{(a+b (c x^q)^n)^p}{x^2} \, dx\)

Optimal. Leaf size=71 \[ -\frac{\left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-\frac{1}{n q};1-\frac{1}{n q};-\frac{b \left (c x^q\right )^n}{a}\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.0290384, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {370, 365, 364} \[ -\frac{\left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-\frac{1}{n q};1-\frac{1}{n q};-\frac{b \left (c x^q\right )^n}{a}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^q)^n)^p/x^2,x]

[Out]

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

Rule 370

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Subst[Int[(d*x)^m*(a + b*c^n*
x^(n*q))^p, x], x^(n*q), (c*x^q)^n/c^n] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&  !RationalQ[n]

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 \frac{\left (a+b \left (c x^q\right )^n\right )^p}{x^2} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b c^n x^{n q}\right )^p}{x^2} \, dx,x^{n q},c^{-n} \left (c x^q\right )^n\right )\\ &=\operatorname{Subst}\left (\left (\left (a+b c^n x^{n q}\right )^p \left (1+\frac{b c^n x^{n q}}{a}\right )^{-p}\right ) \int \frac{\left (1+\frac{b c^n x^{n q}}{a}\right )^p}{x^2} \, dx,x^{n q},c^{-n} \left (c x^q\right )^n\right )\\ &=-\frac{\left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac{b \left (c x^q\right )^n}{a}\right )^{-p} \, _2F_1\left (-p,-\frac{1}{n q};1-\frac{1}{n q};-\frac{b \left (c x^q\right )^n}{a}\right )}{x}\\ \end{align*}

Mathematica [A]  time = 0.0800314, size = 71, normalized size = 1. \[ -\frac{\left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-\frac{1}{n q};1-\frac{1}{n q};-\frac{b \left (c x^q\right )^n}{a}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x^q)^n)^p/x^2,x]

[Out]

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

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Maple [F]  time = 0.931, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b \left ( c{x}^{q} \right ) ^{n} \right ) ^{p}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*(c*x^q)^n)^p/x^2,x)

[Out]

int((a+b*(c*x^q)^n)^p/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^q)^n)^p/x^2,x, algorithm="maxima")

[Out]

integrate(((c*x^q)^n*b + a)^p/x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^q)^n)^p/x^2,x, algorithm="fricas")

[Out]

integral(((c*x^q)^n*b + a)^p/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \left (c x^{q}\right )^{n}\right )^{p}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x**q)**n)**p/x**2,x)

[Out]

Integral((a + b*(c*x**q)**n)**p/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^q)^n)^p/x^2,x, algorithm="giac")

[Out]

integrate(((c*x^q)^n*b + a)^p/x^2, x)