3.2923 \(\int (d x)^m (a+b (c x)^n)^p \, dx\)

Optimal. Leaf size=73 \[ \frac{(d x)^{m+1} \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b (c x)^n}{a}\right )}{d (m+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0443885, antiderivative size = 73, 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 = {367, 365, 364} \[ \frac{(d x)^{m+1} \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b (c x)^n}{a}\right )}{d (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*(c*x)^n)^p,x]

[Out]

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

Mathematica [A]  time = 0.104617, size = 70, normalized size = 0.96 \[ \frac{x (d x)^m \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+1}{n}+1;-\frac{b (c x)^n}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*(c*x)^n)^p,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.306, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b \left ( cx \right ) ^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((c*x)^n*b + a)^p*(d*x)^m, x)

________________________________________________________________________________________

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

[Out]

Integral((d*x)**m*(a + b*(c*x)**n)**p, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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