\(\int (d x)^m (a+b (c x^q)^n)^p \, dx\) [3046]

Optimal result

Integrand size = 19, antiderivative size = 86 \[ \int (d x)^m \left (a+b \left (c x^q\right )^n\right )^p \, dx=\frac {(d x)^{1+m} \left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1+m}{n q},1+\frac {1+m}{n q},-\frac {b \left (c x^q\right )^n}{a}\right )}{d (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {377, 372, 371} \[ \int (d x)^m \left (a+b \left (c x^q\right )^n\right )^p \, dx=\frac {(d x)^{m+1} \left (a+b \left (c x^q\right )^n\right )^p \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {m+1}{n q},\frac {m+1}{n q}+1,-\frac {b \left (c x^q\right )^n}{a}\right )}{d (m+1)} \]

[In]

Int[(d*x)^m*(a + b*(c*x^q)^n)^p,x]

[Out]

((d*x)^(1 + m)*(a + b*(c*x^q)^n)^p*Hypergeometric2F1[-p, (1 + m)/(n*q), 1 + (1 + m)/(n*q), -((b*(c*x^q)^n)/a)]
)/(d*(1 + m)*(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 (d x)^m \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 (d x)^m \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 {(d x)^{1+m} \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+m}{n q};1+\frac {1+m}{n q};-\frac {b \left (c x^q\right )^n}{a}\right )}{d (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int (d x)^m \left (a+b \left (c x^q\right )^n\right )^p \, dx=\frac {x (d x)^m \left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1+m}{n q},1+\frac {1+m}{n q},-\frac {b \left (c x^q\right )^n}{a}\right )}{1+m} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((d*x)^m*(a + b*(c*x^q)^n)^p,x)

[Out]

int((d*x)^m*(a + b*(c*x^q)^n)^p, x)