3.5.41 \(\int x^{-1-n p} (b x^n+a x^q)^p \, dx\) [441]

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

[Out]

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

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Rubi [A]
time = 0.05, 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 = {2057, 372, 371} \begin {gather*} -\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)} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.09, size = 74, normalized size = 1.12 \begin {gather*} -\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 {gather*}

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 = 0.21, size = 0, normalized size = 0.00 \[\int x^{-n p -1} \left (b \,x^{n}+a \,x^{q}\right )^{p}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{- n p - 1} \left (a x^{q} + b x^{n}\right )^{p}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (b\,x^n+a\,x^q\right )}^p}{x^{n\,p+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n + a*x^q)^p/x^(n*p + 1),x)

[Out]

int((b*x^n + a*x^q)^p/x^(n*p + 1), x)

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