3.2926 \(\int (a+b (c x)^n)^p \, dx\)

Optimal. Leaf size=52 \[ x \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c x)^n}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0256385, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 246, 245} \[ x \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c x)^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x)^n)^p,x]

[Out]

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

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 245

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

Rubi steps

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

Mathematica [A]  time = 0.0073823, size = 52, normalized size = 1. \[ x \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c x)^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x)^n)^p,x]

[Out]

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

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( cx \right ) ^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((c*x)^n*b + a)^p, x)

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

[Out]

Integral((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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b + a)^p, x)