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

Optimal. Leaf size=66 \[ -\frac{x^{-n p} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \, _2F_1\left (1,1;1-p;-\frac{b x^{n-q}}{a}\right )}{a p (n-q)} \]

[Out]

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

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Rubi [A]  time = 0.0675878, antiderivative size = 74, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2032, 365, 364} \[ -\frac{x^{-n p} \left (\frac{b x^{n-q}}{a}+1\right )^{-p} \left (a x^q+b x^n\right )^p \, _2F_1\left (-p,-p;1-p;-\frac{b x^{n-q}}{a}\right )}{p (n-q)} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n*p)*(b*x^n + a*x^q)^p,x]

[Out]

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

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[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]

Rule 365

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 364

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

Rubi steps

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

Mathematica [A]  time = 0.0955013, size = 74, normalized size = 1.12 \[ -\frac{x^{-n p} \left (\frac{b x^{n-q}}{a}+1\right )^{-p} \left (a x^q+b x^n\right )^p \, _2F_1\left (-p,-p;1-p;-\frac{b x^{n-q}}{a}\right )}{p (n-q)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.511, size = 0, normalized size = 0. \begin{align*} \int{x}^{-np-1} \left ( b{x}^{n}+a{x}^{q} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a x^{q}\right )}^{p} x^{-n p - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{n} + a x^{q}\right )}^{p} x^{-n p - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a x^{q}\right )}^{p} x^{-n p - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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