\(\int (a+b (c x^q)^n)^p \, dx\) [147]

Optimal result

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

Mathematica [A] (verified)

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
 

Rubi [A] (verified)

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
 

Defintions of rubi rules used

rule 778

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])
 

rule 779

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])
 

rule 788

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a+b \left (c x^q\right )^n\right )^p \, dx=\frac {\left (x^{n q} c^{n} b +a \right )^{p} x +\left (\int \frac {\left (x^{n q} c^{n} b +a \right )^{p}}{x^{n q} c^{n} b n p q +x^{n q} c^{n} b +a n p q +a}d x \right ) a \,n^{2} p^{2} q^{2}+\left (\int \frac {\left (x^{n q} c^{n} b +a \right )^{p}}{x^{n q} c^{n} b n p q +x^{n q} c^{n} b +a n p q +a}d x \right ) a n p q}{n p q +1} \] 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)