Integrand size = 13, antiderivative size = 66 \[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=x \left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{n q},1+\frac {1}{n q},-\frac {b \left (c x^q\right )^n}{a}\right ) \] Output:
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)
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=x \left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{n q},1+\frac {1}{n q},-\frac {b \left (c x^q\right )^n}{a}\right ) \] Input:
Integrate[(a + b*(c*x^q)^n)^p,x]
Output:
(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
Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {788, 779, 778}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a+b \left (c x^q\right )^n\right )^p \, dx\) |
\(\Big \downarrow \) 788 |
\(\displaystyle \int \left (a+b \left (c x^q\right )^n\right )^pdx\) |
\(\Big \downarrow \) 779 |
\(\displaystyle \left (a+b \left (c x^q\right )^n\right )^p \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^{-p} \int \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^pdx\) |
\(\Big \downarrow \) 778 |
\(\displaystyle x \left (a+b \left (c x^q\right )^n\right )^p \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{n q},1+\frac {1}{n q},-\frac {b \left (c x^q\right )^n}{a}\right )\) |
Input:
Int[(a + b*(c*x^q)^n)^p,x]
Output:
(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
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-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])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x ^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p, x], x ] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p, 0] && !IntegerQ[1/n] && !ILtQ[Si mplify[1/n + p], 0] && !(IntegerQ[p] || GtQ[a, 0])
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]
\[\int {\left (a +b \left (c \,x^{q}\right )^{n}\right )}^{p}d x\]
Input:
int((a+b*(c*x^q)^n)^p,x)
Output:
int((a+b*(c*x^q)^n)^p,x)
\[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=\int { {\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} \,d x } \] Input:
integrate((a+b*(c*x^q)^n)^p,x, algorithm="fricas")
Output:
integral(((c*x^q)^n*b + a)^p, x)
\[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=\int \left (a + b \left (c x^{q}\right )^{n}\right )^{p}\, dx \] Input:
integrate((a+b*(c*x**q)**n)**p,x)
Output:
Integral((a + b*(c*x**q)**n)**p, x)
\[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=\int { {\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} \,d x } \] Input:
integrate((a+b*(c*x^q)^n)^p,x, algorithm="maxima")
Output:
integrate(((c*x^q)^n*b + a)^p, x)
\[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=\int { {\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} \,d x } \] Input:
integrate((a+b*(c*x^q)^n)^p,x, algorithm="giac")
Output:
integrate(((c*x^q)^n*b + a)^p, x)
Timed out. \[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=\int {\left (a+b\,{\left (c\,x^q\right )}^n\right )}^p \,d x \] Input:
int((a + b*(c*x^q)^n)^p,x)
Output:
int((a + b*(c*x^q)^n)^p, x)
\[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=\frac {\left (x^{n q} c^{n} b +a \right )^{p} x +\left (\int \frac {\left (x^{n q} c^{n} b +a \right )^{p}}{x^{n q} c^{n} b n p q +x^{n q} c^{n} b +a n p q +a}d x \right ) a \,n^{2} p^{2} q^{2}+\left (\int \frac {\left (x^{n q} c^{n} b +a \right )^{p}}{x^{n q} c^{n} b n p q +x^{n q} c^{n} b +a n p q +a}d x \right ) a n p q}{n p q +1} \] Input:
int((a+b*(c*x^q)^n)^p,x)
Output:
((x**(n*q)*c**n*b + a)**p*x + int((x**(n*q)*c**n*b + a)**p/(x**(n*q)*c**n* b*n*p*q + x**(n*q)*c**n*b + a*n*p*q + a),x)*a*n**2*p**2*q**2 + int((x**(n* q)*c**n*b + a)**p/(x**(n*q)*c**n*b*n*p*q + x**(n*q)*c**n*b + a*n*p*q + a), x)*a*n*p*q)/(n*p*q + 1)