3.1060 \(\int x^{-8-2 p} (a+b x^2)^p \, dx\)

Optimal. Leaf size=53 \[ -\frac{x^{-2 p-7} \left (a+b x^2\right )^{p+1} \, _2F_1\left (-\frac{5}{2},1;\frac{1}{2} (-2 p-5);-\frac{b x^2}{a}\right )}{a (2 p+7)} \]

[Out]

-((x^(-7 - 2*p)*(a + b*x^2)^(1 + p)*Hypergeometric2F1[-5/2, 1, (-5 - 2*p)/2, -((b*x^2)/a)])/(a*(7 + 2*p)))

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Rubi [A]  time = 0.0208351, antiderivative size = 70, normalized size of antiderivative = 1.32, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {365, 364} \[ -\frac{x^{-2 p-7} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} (-2 p-7),-p;\frac{1}{2} (-2 p-5);-\frac{b x^2}{a}\right )}{2 p+7} \]

Antiderivative was successfully verified.

[In]

Int[x^(-8 - 2*p)*(a + b*x^2)^p,x]

[Out]

-((x^(-7 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[(-7 - 2*p)/2, -p, (-5 - 2*p)/2, -((b*x^2)/a)])/((7 + 2*p)*(1 +
(b*x^2)/a)^p))

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^{-8-2 p} \left (a+b x^2\right )^p \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int x^{-8-2 p} \left (1+\frac{b x^2}{a}\right )^p \, dx\\ &=-\frac{x^{-7-2 p} \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \, _2F_1\left (\frac{1}{2} (-7-2 p),-p;\frac{1}{2} (-5-2 p);-\frac{b x^2}{a}\right )}{7+2 p}\\ \end{align*}

Mathematica [A]  time = 0.0153562, size = 66, normalized size = 1.25 \[ -\frac{x^{-2 p-7} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (-p-\frac{7}{2},-p;-p-\frac{5}{2};-\frac{b x^2}{a}\right )}{2 p+7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-8 - 2*p)*(a + b*x^2)^p,x]

[Out]

-((x^(-7 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[-7/2 - p, -p, -5/2 - p, -((b*x^2)/a)])/((7 + 2*p)*(1 + (b*x^2)
/a)^p))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{x}^{-8-2\,p} \left ( b{x}^{2}+a \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-8-2*p)*(b*x^2+a)^p,x)

[Out]

int(x^(-8-2*p)*(b*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 8}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-8-2*p)*(b*x^2+a)^p,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^p*x^(-2*p - 8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 8}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-8-2*p)*(b*x^2+a)^p,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^p*x^(-2*p - 8), 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**(-8-2*p)*(b*x**2+a)**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 8}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-8-2*p)*(b*x^2+a)^p,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p*x^(-2*p - 8), x)