\(\int x^{-1-n (-1+p)-q} (b x^n+a x^q)^p \, dx\) [443]

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2057, 372, 371} \[ \int x^{-1-n (-1+p)-q} \left (b x^n+a x^q\right )^p \, dx=\frac {x^{n (-p)+n-q} \left (\frac {b x^{n-q}}{a}+1\right )^{-p} \left (a x^q+b x^n\right )^p \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^{n-q}}{a}\right )}{(1-p) (n-q)} \]

[In]

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

[Out]

(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)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[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 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(FracPart[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] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rubi steps \begin{align*} \text {integral}& = \left (x^{-p q} \left (a+b x^{n-q}\right )^{-p} \left (b x^n+a x^q\right )^p\right ) \int x^{-1-n (-1+p)-q+p q} \left (a+b x^{n-q}\right )^p \, dx \\ & = \left (x^{-p q} \left (1+\frac {b x^{n-q}}{a}\right )^{-p} \left (b x^n+a x^q\right )^p\right ) \int x^{-1-n (-1+p)-q+p q} \left (1+\frac {b x^{n-q}}{a}\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 \, _2F_1\left (1-p,-p;2-p;-\frac {b x^{n-q}}{a}\right )}{(1-p) (n-q)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99 \[ \int x^{-1-n (-1+p)-q} \left (b x^n+a x^q\right )^p \, dx=-\frac {x^{n-n p-q} \left (1+\frac {b x^{n-q}}{a}\right )^{-p} \left (b x^n+a x^q\right )^p \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^{n-q}}{a}\right )}{(-1+p) (n-q)} \]

[In]

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

[Out]

-((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))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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