Optimal. Leaf size=43 \[ -\frac{x^{-2 p} \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,1;1-p;-\frac{b x^2}{a}\right )}{2 a p} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0454165, antiderivative size = 56, normalized size of antiderivative = 1.3, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{x^{-2 p} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{b x^2}{a}\right )}{2 p} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - 2*p)*(a + b*x^2)^p,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.49927, size = 42, normalized size = 0.98 \[ - \frac{x^{- 2 p} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ - p + 1 \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{2 p} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-2*p)*(b*x**2+a)**p,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0356083, size = 56, normalized size = 1.3 \[ -\frac{x^{-2 p} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{b x^2}{a}\right )}{2 p} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - 2*p)*(a + b*x^2)^p,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.074, size = 0, normalized size = 0. \[ \int{x}^{-1-2\,p} \left ( b{x}^{2}+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-2*p)*(b*x^2+a)^p,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*x^(-2*p - 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*x^(-2*p - 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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**(-1-2*p)*(b*x**2+a)**p,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*x^(-2*p - 1),x, algorithm="giac")
[Out]