\(\int (d x)^m (a+b (c x^q)^n)^p \, dx\) [144]

Optimal result

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

(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)
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int (d x)^m \left (a+b \left (c x^q\right )^n\right )^p \, dx=\frac {x (d x)^m \left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1+m}{n q},1+\frac {1+m}{n q},-\frac {b \left (c x^q\right )^n}{a}\right )}{1+m} \] Input:

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

Output:

(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)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {894, 889, 888}

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.

\(\displaystyle \int (d x)^m \left (a+b \left (c x^q\right )^n\right )^p \, dx\)

\(\Big \downarrow \) 894

\(\displaystyle \int (d x)^m \left (a+b \left (c x^q\right )^n\right )^pdx\)

\(\Big \downarrow \) 889

\(\displaystyle \left (a+b \left (c x^q\right )^n\right )^p \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^{-p} \int (d x)^m \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^pdx\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {(d x)^{m+1} \left (a+b \left (c x^q\right )^n\right )^p \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {m+1}{n q},\frac {m+1}{n q}+1,-\frac {b \left (c x^q\right )^n}{a}\right )}{d (m+1)}\)

Input:

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

Output:

((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)
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[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 894

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo 
l] :> 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(d**m*(x**m*(x**(n*q)*c**n*b + a)**p*x + int((x**m*(x**(n*q)*c**n*b + a)** 
p)/(x**(n*q)*c**n*b*m + x**(n*q)*c**n*b*n*p*q + x**(n*q)*c**n*b + a*m + a* 
n*p*q + a),x)*a*m*n*p*q + int((x**m*(x**(n*q)*c**n*b + a)**p)/(x**(n*q)*c* 
*n*b*m + x**(n*q)*c**n*b*n*p*q + x**(n*q)*c**n*b + a*m + a*n*p*q + a),x)*a 
*n**2*p**2*q**2 + int((x**m*(x**(n*q)*c**n*b + a)**p)/(x**(n*q)*c**n*b*m + 
 x**(n*q)*c**n*b*n*p*q + x**(n*q)*c**n*b + a*m + a*n*p*q + a),x)*a*n*p*q)) 
/(m + n*p*q + 1)