Optimal. Leaf size=19 \[ -\frac{\sqrt{a+b x^2}}{a x} \]
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Rubi [A] time = 0.0044502, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {5, 264} \[ -\frac{\sqrt{a+b x^2}}{a x} \]
Antiderivative was successfully verified.
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Rule 5
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac{1}{x^2 \sqrt{a+b x^2}} \, dx\\ &=-\frac{\sqrt{a+b x^2}}{a x}\\ \end{align*}
Mathematica [A] time = 0.003643, size = 19, normalized size = 1. \[ -\frac{\sqrt{a+b x^2}}{a x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 18, normalized size = 1. \begin{align*} -{\frac{1}{ax}\sqrt{b{x}^{2}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52835, size = 32, normalized size = 1.68 \begin{align*} -\frac{\sqrt{b x^{2} + a}}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.57357, size = 19, normalized size = 1. \begin{align*} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18817, size = 41, normalized size = 2.16 \begin{align*} \frac{2 \, \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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