Optimal. Leaf size=30 \[ -\frac{x^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{2 a (p+1)} \]
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Rubi [A] time = 0.0060034, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ -\frac{x^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{2 a (p+1)} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int x^{-3-2 p} \left (a+b x^2\right )^p \, dx &=-\frac{x^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 a (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0116172, size = 29, normalized size = 0.97 \[ \frac{x^{-2 p-2} \left (a+b x^2\right )^{p+1}}{a (-2 p-2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 29, normalized size = 1. \begin{align*} -{\frac{{x}^{-2-2\,p} \left ( b{x}^{2}+a \right ) ^{1+p}}{2\,a \left ( 1+p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.83406, size = 50, normalized size = 1.67 \begin{align*} -\frac{{\left (b x^{2} + a\right )} e^{\left (p \log \left (b x^{2} + a\right ) - 2 \, p \log \left (x\right )\right )}}{2 \, a{\left (p + 1\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61178, size = 77, normalized size = 2.57 \begin{align*} -\frac{{\left (b x^{3} + a x\right )}{\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 3}}{2 \,{\left (a p + a\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 - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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