3.1067 \(\int x^{-1-2 p} \left (a+b x^2\right )^p \, dx\)

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]

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

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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]

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

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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]

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

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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]

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

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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]

int(x^(-1-2*p)*(b*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 1}\,{d x} \]

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]

integrate((b*x^2 + a)^p*x^(-2*p - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 1}, x\right ) \]

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]

integral((b*x^2 + a)^p*x^(-2*p - 1), 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**(-1-2*p)*(b*x**2+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 1}\,{d x} \]

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]

integrate((b*x^2 + a)^p*x^(-2*p - 1), x)