\(\int x^{-1-n (-1+p)-q} (b x^n+a x^q)^p \, dx\) [426]

Optimal result

Integrand size = 27, antiderivative size = 84 \[ \int x^{-1-n (-1+p)-q} \left (b x^n+a x^q\right )^p \, dx=\frac {x^{n-n p-q} \left (1+\frac {b x^{n-q}}{a}\right )^{-p} \left (b x^n+a x^q\right )^p \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^{n-q}}{a}\right )}{(1-p) (n-q)} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99 \[ \int x^{-1-n (-1+p)-q} \left (b x^n+a x^q\right )^p \, dx=-\frac {x^{n-n p-q} \left (1+\frac {b x^{n-q}}{a}\right )^{-p} \left (b x^n+a x^q\right )^p \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^{n-q}}{a}\right )}{(-1+p) (n-q)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1938, 882, 74}

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.

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

\(\Big \downarrow \) 1938

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

\(\Big \downarrow \) 882

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

\(\Big \downarrow \) 74

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 882

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^Simplify[ 
(m + 1)/n + p]*x^m*(a + b*x^n)^p*((x^n/(a + b*x^n))^p/(n*x^Simplify[m + n*p 
]))   Subst[Int[x^((m + 1)/n - 1)/(1 - b*x)^(Simplify[(m + 1)/n + p] + 1), 
x], x, x^n/(a + b*x^n)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simpli 
fy[(m + 1)/n + p]]
 

rule 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(x**n*(x**n*b + x**q*a)**p + x**(n*p + q)*int((x**n*(x**n*b + x**q*a)**p)/ 
(x**(n*p + n)*b*x + x**(n*p + q)*a*x),x)*a*n*p - x**(n*p + q)*int((x**n*(x 
**n*b + x**q*a)**p)/(x**(n*p + n)*b*x + x**(n*p + q)*a*x),x)*a*p*q)/(x**(n 
*p + q)*(n - q))