\(\int (a+b (c x^q)^n)^p \, dx\) [3049]

Optimal result

Integrand size = 13, antiderivative size = 66 \[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=x \left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{n q},1+\frac {1}{n q},-\frac {b \left (c x^q\right )^n}{a}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {262, 252, 251} \[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=x \left (a+b \left (c x^q\right )^n\right )^p \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{n q},1+\frac {1}{n q},-\frac {b \left (c x^q\right )^n}{a}\right ) \]

[In]

Int[(a + b*(c*x^q)^n)^p,x]

[Out]

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

Rule 251

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])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[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 262

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

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (a+b c^n x^{n q}\right )^p \, dx,x^{n q},c^{-n} \left (c x^q\right )^n\right ) \\ & = \text {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 \left (1+\frac {b c^n x^{n q}}{a}\right )^p \, dx,x^{n q},c^{-n} \left (c x^q\right )^n\right ) \\ & = x \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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=x \left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{n q},1+\frac {1}{n q},-\frac {b \left (c x^q\right )^n}{a}\right ) \]

[In]

Integrate[(a + b*(c*x^q)^n)^p,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(((c*x^q)^n*b + a)^p, x)

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*(c*x**q)**n)**p, x)

Maxima [F]

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

[In]

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

[Out]

integrate(((c*x^q)^n*b + a)^p, x)

Giac [F]

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

[In]

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

[Out]

integrate(((c*x^q)^n*b + a)^p, x)

Mupad [F(-1)]

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

[In]

int((a + b*(c*x^q)^n)^p,x)

[Out]

int((a + b*(c*x^q)^n)^p, x)