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

Optimal. Leaf size=69 \[ \frac{b x^{-p q} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \, _2F_1\left (2,p+1;p+2;\frac{b x^{n-q}}{a}+1\right )}{a^2 (p+1) (n-q)} \]

[Out]

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

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Rubi [A]  time = 0.077541, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2032, 266, 65} \[ \frac{b x^{-p q} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \, _2F_1\left (2,p+1;p+2;\frac{b x^{n-q}}{a}+1\right )}{a^2 (p+1) (n-q)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 266

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

Rule 65

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

Rubi steps

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

Mathematica [A]  time = 0.159438, size = 82, normalized size = 1.19 \[ \frac{x^{-n-p q+q} \left (a x^q+b x^n\right )^p \left (\frac{a x^{q-n}}{b}+1\right )^{-p} \, _2F_1\left (1-p,-p;2-p;-\frac{a x^{q-n}}{b}\right )}{(p-1) (n-q)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.51, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-n- \left ( p-1 \right ) q} \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^(-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)

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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^{-{\left (p - 1\right )} q - n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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^(-(p - 1)*q - n - 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^{-{\left (p - 1\right )} q - n - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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^(-(p - 1)*q - n - 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**(-1-n-(-1+p)*q)*(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^{-{\left (p - 1\right )} q - n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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^(-(p - 1)*q - n - 1), x)