\(\int (d x)^m (a+b (c x)^n)^p \, dx\) [2923]

Optimal result

Integrand size = 17, antiderivative size = 73 \[ \int (d x)^m \left (a+b (c x)^n\right )^p \, dx=\frac {(d x)^{1+m} \left (a+b (c x)^n\right )^p \left (1+\frac {b (c x)^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b (c x)^n}{a}\right )}{d (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {374, 372, 371} \[ \int (d x)^m \left (a+b (c x)^n\right )^p \, dx=\frac {(d x)^{m+1} \left (a+b (c x)^n\right )^p \left (\frac {b (c x)^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{n},-p,\frac {m+n+1}{n},-\frac {b (c x)^n}{a}\right )}{d (m+1)} \]

[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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 374

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 ) \text {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] (verified)

Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int (d x)^m \left (a+b (c x)^n\right )^p \, dx=\frac {x (d x)^m \left (a+b (c x)^n\right )^p \left (1+\frac {b (c x)^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,1+\frac {1+m}{n},-\frac {b (c x)^n}{a}\right )}{1+m} \]

[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]

\[\int \left (d x \right )^{m} \left (a +b \left (c x \right )^{n}\right )^{p}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (d x)^m \left (a+b (c x)^n\right )^p \, dx=\int { {\left (\left (c x\right )^{n} b + a\right )}^{p} \left (d x\right )^{m} \,d x } \]

[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]

\[ \int (d x)^m \left (a+b (c x)^n\right )^p \, dx=\int \left (d x\right )^{m} \left (a + b \left (c x\right )^{n}\right )^{p}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (d x)^m \left (a+b (c x)^n\right )^p \, dx=\int { {\left (\left (c x\right )^{n} b + a\right )}^{p} \left (d x\right )^{m} \,d x } \]

[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)

Giac [F]

\[ \int (d x)^m \left (a+b (c x)^n\right )^p \, dx=\int { {\left (\left (c x\right )^{n} b + a\right )}^{p} \left (d x\right )^{m} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \left (a+b (c x)^n\right )^p \, dx=\int {\left (d\,x\right )}^m\,{\left (a+b\,{\left (c\,x\right )}^n\right )}^p \,d x \]

[In]

int((d*x)^m*(a + b*(c*x)^n)^p,x)

[Out]

int((d*x)^m*(a + b*(c*x)^n)^p, x)