3.1066 \(\int x^{-2-2 p} \left (a+b x^2\right )^p \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0533584, 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 \[ -\frac{x^{-2 p-1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} (-2 p-1),-p;\frac{1}{2} (1-2 p);-\frac{b x^2}{a}\right )}{2 p+1} \]

Antiderivative was successfully verified.

[In]  Int[x^(-2 - 2*p)*(a + b*x^2)^p,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.20424, size = 53, normalized size = 1. \[ - \frac{x^{- 2 p - 1} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - p - \frac{1}{2} \\ - p + \frac{1}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{2 p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0542976, size = 66, normalized size = 1.25 \[ -\frac{x^{-2 p-1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (-p-\frac{1}{2},-p;\frac{1}{2}-p;-\frac{b x^2}{a}\right )}{2 p+1} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-2-2*p)*(b*x^2+a)^p,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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**(-2-2*p)*(b*x**2+a)**p,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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