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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]