\(\int x^{-1+n} (d x^n)^p (a+b x^n)^q \, dx\) [487]

Optimal result

Integrand size = 22, antiderivative size = 63 \[ \int x^{-1+n} \left (d x^n\right )^p \left (a+b x^n\right )^q \, dx=\frac {\left (d x^n\right )^{1+p} \left (a+b x^n\right )^q \left (1+\frac {b x^n}{a}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,-\frac {b x^n}{a}\right )}{d n (1+p)} \] Output:

(d*x^n)^(p+1)*(a+b*x^n)^q*hypergeom([-q, p+1],[2+p],-b*x^n/a)/d/n/(p+1)/(( 
1+b*x^n/a)^q)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int x^{-1+n} \left (d x^n\right )^p \left (a+b x^n\right )^q \, dx=\frac {x^n \left (d x^n\right )^p \left (a+b x^n\right )^q \left (1+\frac {b x^n}{a}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,-\frac {b x^n}{a}\right )}{n+n p} \] Input:

Integrate[x^(-1 + n)*(d*x^n)^p*(a + b*x^n)^q,x]
 

Output:

(x^n*(d*x^n)^p*(a + b*x^n)^q*Hypergeometric2F1[1 + p, -q, 2 + p, -((b*x^n) 
/a)])/((n + n*p)*(1 + (b*x^n)/a)^q)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {31, 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.

\(\displaystyle \int x^{n-1} \left (d x^n\right )^p \left (a+b x^n\right )^q \, dx\)

\(\Big \downarrow \) 31

\(\displaystyle x^{-n p} \left (d x^n\right )^p \int x^{p n+n-1} \left (b x^n+a\right )^qdx\)

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

\(\displaystyle \frac {x^{n (p+1)-n p} \left (d x^n\right )^p \left (a+b x^n\right )^q \left (\frac {b x^n}{a}+1\right )^{-q} \operatorname {Hypergeometric2F1}\left (p+1,-q,p+2,-\frac {b x^n}{a}\right )}{n (p+1)}\)

Input:

Int[x^(-1 + n)*(d*x^n)^p*(a + b*x^n)^q,x]
 

Output:

(x^(-(n*p) + n*(1 + p))*(d*x^n)^p*(a + b*x^n)^q*Hypergeometric2F1[1 + p, - 
q, 2 + p, -((b*x^n)/a)])/(n*(1 + p)*(1 + (b*x^n)/a)^q)
 

Defintions of rubi rules used

rule 31

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[(b* 
x^i)^p/(a*x)^(i*p)   Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p} 
, x] &&  !IntegerQ[p]
 

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])
 

Maple [F]

\[\int x^{-1+n} \left (d \,x^{n}\right )^{p} \left (a +b \,x^{n}\right )^{q}d x\]

Input:

int(x^(-1+n)*(d*x^n)^p*(a+b*x^n)^q,x)
 

Output:

int(x^(-1+n)*(d*x^n)^p*(a+b*x^n)^q,x)
 

Fricas [F]

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

integrate(x^(-1+n)*(d*x^n)^p*(a+b*x^n)^q,x, algorithm="fricas")
 

Output:

integral((b*x^n + a)^q*(d*x^n)^p*x^(n - 1), x)
 

Sympy [F]

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

integrate(x**(-1+n)*(d*x**n)**p*(a+b*x**n)**q,x)
 

Output:

Integral(x**(n - 1)*(d*x**n)**p*(a + b*x**n)**q, x)
 

Maxima [F]

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

integrate(x^(-1+n)*(d*x^n)^p*(a+b*x^n)^q,x, algorithm="maxima")
 

Output:

integrate((b*x^n + a)^q*(d*x^n)^p*x^(n - 1), x)
 

Giac [F]

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

integrate(x^(-1+n)*(d*x^n)^p*(a+b*x^n)^q,x, algorithm="giac")
 

Output:

integrate((b*x^n + a)^q*(d*x^n)^p*x^(n - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{-1+n} \left (d x^n\right )^p \left (a+b x^n\right )^q \, dx=\int x^{n-1}\,{\left (d\,x^n\right )}^p\,{\left (a+b\,x^n\right )}^q \,d x \] Input:

int(x^(n - 1)*(d*x^n)^p*(a + b*x^n)^q,x)
 

Output:

int(x^(n - 1)*(d*x^n)^p*(a + b*x^n)^q, x)
 

Reduce [F]

\[ \int x^{-1+n} \left (d x^n\right )^p \left (a+b x^n\right )^q \, dx=\frac {d^{p} \left (x^{n p +n} \left (x^{n} b +a \right )^{q} b p +x^{n p +n} \left (x^{n} b +a \right )^{q} b q +x^{n p} \left (x^{n} b +a \right )^{q} a q -\left (\int \frac {x^{n p} \left (x^{n} b +a \right )^{q}}{x^{n} b \,p^{2} x +2 x^{n} b p q x +x^{n} b p x +x^{n} b \,q^{2} x +x^{n} b q x +a \,p^{2} x +2 a p q x +a p x +a \,q^{2} x +a q x}d x \right ) a^{2} n \,p^{3} q -2 \left (\int \frac {x^{n p} \left (x^{n} b +a \right )^{q}}{x^{n} b \,p^{2} x +2 x^{n} b p q x +x^{n} b p x +x^{n} b \,q^{2} x +x^{n} b q x +a \,p^{2} x +2 a p q x +a p x +a \,q^{2} x +a q x}d x \right ) a^{2} n \,p^{2} q^{2}-\left (\int \frac {x^{n p} \left (x^{n} b +a \right )^{q}}{x^{n} b \,p^{2} x +2 x^{n} b p q x +x^{n} b p x +x^{n} b \,q^{2} x +x^{n} b q x +a \,p^{2} x +2 a p q x +a p x +a \,q^{2} x +a q x}d x \right ) a^{2} n \,p^{2} q -\left (\int \frac {x^{n p} \left (x^{n} b +a \right )^{q}}{x^{n} b \,p^{2} x +2 x^{n} b p q x +x^{n} b p x +x^{n} b \,q^{2} x +x^{n} b q x +a \,p^{2} x +2 a p q x +a p x +a \,q^{2} x +a q x}d x \right ) a^{2} n p \,q^{3}-\left (\int \frac {x^{n p} \left (x^{n} b +a \right )^{q}}{x^{n} b \,p^{2} x +2 x^{n} b p q x +x^{n} b p x +x^{n} b \,q^{2} x +x^{n} b q x +a \,p^{2} x +2 a p q x +a p x +a \,q^{2} x +a q x}d x \right ) a^{2} n p \,q^{2}\right )}{b n \left (p^{2}+2 p q +q^{2}+p +q \right )} \] Input:

int(x^(-1+n)*(d*x^n)^p*(a+b*x^n)^q,x)
 

Output:

(d**p*(x**(n*p + n)*(x**n*b + a)**q*b*p + x**(n*p + n)*(x**n*b + a)**q*b*q 
 + x**(n*p)*(x**n*b + a)**q*a*q - int((x**(n*p)*(x**n*b + a)**q)/(x**n*b*p 
**2*x + 2*x**n*b*p*q*x + x**n*b*p*x + x**n*b*q**2*x + x**n*b*q*x + a*p**2* 
x + 2*a*p*q*x + a*p*x + a*q**2*x + a*q*x),x)*a**2*n*p**3*q - 2*int((x**(n* 
p)*(x**n*b + a)**q)/(x**n*b*p**2*x + 2*x**n*b*p*q*x + x**n*b*p*x + x**n*b* 
q**2*x + x**n*b*q*x + a*p**2*x + 2*a*p*q*x + a*p*x + a*q**2*x + a*q*x),x)* 
a**2*n*p**2*q**2 - int((x**(n*p)*(x**n*b + a)**q)/(x**n*b*p**2*x + 2*x**n* 
b*p*q*x + x**n*b*p*x + x**n*b*q**2*x + x**n*b*q*x + a*p**2*x + 2*a*p*q*x + 
 a*p*x + a*q**2*x + a*q*x),x)*a**2*n*p**2*q - int((x**(n*p)*(x**n*b + a)** 
q)/(x**n*b*p**2*x + 2*x**n*b*p*q*x + x**n*b*p*x + x**n*b*q**2*x + x**n*b*q 
*x + a*p**2*x + 2*a*p*q*x + a*p*x + a*q**2*x + a*q*x),x)*a**2*n*p*q**3 - i 
nt((x**(n*p)*(x**n*b + a)**q)/(x**n*b*p**2*x + 2*x**n*b*p*q*x + x**n*b*p*x 
 + x**n*b*q**2*x + x**n*b*q*x + a*p**2*x + 2*a*p*q*x + a*p*x + a*q**2*x + 
a*q*x),x)*a**2*n*p*q**2))/(b*n*(p**2 + 2*p*q + p + q**2 + q))