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

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 = 0.0530113, 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 \[ -\frac{x^{-2 p-7} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} (-2 p-7),-p;\frac{1}{2} (-2 p-5);-\frac{b x^2}{a}\right )}{2 p+7} \]

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))

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Rubi in Sympy [A]  time = 8.67323, size = 54, normalized size = 1.02 \[ - \frac{x^{- 2 p - 7} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - p - \frac{7}{2} \\ - p - \frac{5}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{2 p + 7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-8-2*p)*(b*x**2+a)**p,x)

[Out]

-x**(-2*p - 7)*(1 + b*x**2/a)**(-p)*(a + b*x**2)**p*hyper((-p, -p - 7/2), (-p -
5/2,), -b*x**2/a)/(2*p + 7)

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Mathematica [A]  time = 0.0539069, 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 = 0.083, size = 0, normalized size = 0. \[ \int{x}^{-8-2\,p} \left ( b{x}^{2}+a \right ) ^{p}\, dx \]

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. \[ \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 8}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*x^(-2*p - 8),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. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 8}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*x^(-2*p - 8),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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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