\(\int x^2 (a+b (c x^q)^n)^p \, dx\) [3047]

Optimal result

Integrand size = 17, antiderivative size = 73 \[ \int x^2 \left (a+b \left (c x^q\right )^n\right )^p \, dx=\frac {1}{3} x^3 \left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {3}{n q},1+\frac {3}{n q},-\frac {b \left (c x^q\right )^n}{a}\right ) \]

[Out]

1/3*x^3*(a+b*(c*x^q)^n)^p*hypergeom([-p, 3/n/q],[1+3/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 = 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 = {377, 372, 371} \[ \int x^2 \left (a+b \left (c x^q\right )^n\right )^p \, dx=\frac {1}{3} x^3 \left (a+b \left (c x^q\right )^n\right )^p \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {3}{n q},1+\frac {3}{n q},-\frac {b \left (c x^q\right )^n}{a}\right ) \]

[In]

Int[x^2*(a + b*(c*x^q)^n)^p,x]

[Out]

(x^3*(a + b*(c*x^q)^n)^p*Hypergeometric2F1[-p, 3/(n*q), 1 + 3/(n*q), -((b*(c*x^q)^n)/a)])/(3*(1 + (b*(c*x^q)^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 377

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

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^2 \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 x^2 \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 ) \\ & = \frac {1}{3} x^3 \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 {3}{n q};1+\frac {3}{n q};-\frac {b \left (c x^q\right )^n}{a}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \left (c x^q\right )^n\right )^p \, dx=\frac {1}{3} x^3 \left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {3}{n q},1+\frac {3}{n q},-\frac {b \left (c x^q\right )^n}{a}\right ) \]

[In]

Integrate[x^2*(a + b*(c*x^q)^n)^p,x]

[Out]

(x^3*(a + b*(c*x^q)^n)^p*Hypergeometric2F1[-p, 3/(n*q), 1 + 3/(n*q), -((b*(c*x^q)^n)/a)])/(3*(1 + (b*(c*x^q)^n
)/a)^p)

Maple [F]

\[\int x^{2} {\left (a +b \left (c \,x^{q}\right )^{n}\right )}^{p}d x\]

[In]

int(x^2*(a+b*(c*x^q)^n)^p,x)

[Out]

int(x^2*(a+b*(c*x^q)^n)^p,x)

Fricas [F]

\[ \int x^2 \left (a+b \left (c x^q\right )^n\right )^p \, dx=\int { {\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*(c*x^q)^n)^p,x, algorithm="fricas")

[Out]

integral(((c*x^q)^n*b + a)^p*x^2, x)

Sympy [F]

\[ \int x^2 \left (a+b \left (c x^q\right )^n\right )^p \, dx=\int x^{2} \left (a + b \left (c x^{q}\right )^{n}\right )^{p}\, dx \]

[In]

integrate(x**2*(a+b*(c*x**q)**n)**p,x)

[Out]

Integral(x**2*(a + b*(c*x**q)**n)**p, x)

Maxima [F]

\[ \int x^2 \left (a+b \left (c x^q\right )^n\right )^p \, dx=\int { {\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*(c*x^q)^n)^p,x, algorithm="maxima")

[Out]

integrate(((c*x^q)^n*b + a)^p*x^2, x)

Giac [F]

\[ \int x^2 \left (a+b \left (c x^q\right )^n\right )^p \, dx=\int { {\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*(c*x^q)^n)^p,x, algorithm="giac")

[Out]

integrate(((c*x^q)^n*b + a)^p*x^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \left (c x^q\right )^n\right )^p \, dx=\int x^2\,{\left (a+b\,{\left (c\,x^q\right )}^n\right )}^p \,d x \]

[In]

int(x^2*(a + b*(c*x^q)^n)^p,x)

[Out]

int(x^2*(a + b*(c*x^q)^n)^p, x)