Optimal. Leaf size=43 \[ -\frac{x^{-2 p} \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,1;1-p;-\frac{b x^2}{a}\right )}{2 a p} \]
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Rubi [A] time = 0.0157644, 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, Rules used = {365, 364} \[ -\frac{x^{-2 p} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{b x^2}{a}\right )}{2 p} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int x^{-1-2 p} \left (a+b x^2\right )^p \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int x^{-1-2 p} \left (1+\frac{b x^2}{a}\right )^p \, dx\\ &=-\frac{x^{-2 p} \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{b x^2}{a}\right )}{2 p}\\ \end{align*}
Mathematica [A] time = 0.0071147, 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.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-2\,p} \left ( b{x}^{2}+a \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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