3.2925 \(\int x (a+b (c x)^n)^p \, dx\)

Optimal. Leaf size=61 \[ \frac{1}{2} x^2 \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{2}{n},-p;\frac{n+2}{n};-\frac{b (c x)^n}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0326114, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {367, 12, 365, 364} \[ \frac{1}{2} x^2 \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{2}{n},-p;\frac{n+2}{n};-\frac{b (c x)^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*(c*x)^n)^p,x]

[Out]

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

Rule 367

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Dist[1/c, Subst[Int[((d*x)/c)^m*(a
+ b*x^n)^p, x], x, c*x], x] /; FreeQ[{a, b, c, d, m, n, p}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [A]  time = 0.0339293, size = 61, normalized size = 1. \[ \frac{1}{2} x^2 \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{2}{n},-p;1+\frac{2}{n};-\frac{b (c x)^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*(c*x)^n)^p,x]

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b \left ( cx \right ) ^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b + a)^p*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((c*x)^n*b + a)^p*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a + b*(c*x)**n)**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b + a)^p*x, x)