Optimal. Leaf size=20 \[ \frac{\log \left ((-a)^{5/2} x+1\right )}{\sqrt{-a}} \]
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Rubi [A] time = 0.0042515, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {31} \[ \frac{\log \left ((-a)^{5/2} x+1\right )}{\sqrt{-a}} \]
Antiderivative was successfully verified.
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Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\frac{1}{a^2}+\sqrt{-a} x} \, dx &=\frac{\log \left (1+(-a)^{5/2} x\right )}{\sqrt{-a}}\\ \end{align*}
Mathematica [A] time = 0.0183021, size = 22, normalized size = 1.1 \[ \frac{\log \left (\frac{1}{a^2}+\sqrt{-a} x\right )}{\sqrt{-a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 19, normalized size = 1. \begin{align*}{\ln \left ({a}^{-2}+x\sqrt{-a} \right ){\frac{1}{\sqrt{-a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03575, size = 24, normalized size = 1.2 \begin{align*} \frac{\log \left (\sqrt{-a} x + \frac{1}{a^{2}}\right )}{\sqrt{-a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45429, size = 47, normalized size = 2.35 \begin{align*} -\frac{\sqrt{-a} \log \left (a^{3} x - \sqrt{-a}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.11012, size = 20, normalized size = 1. \begin{align*} \frac{\log{\left (a^{2} x \sqrt{- a} + 1 \right )}}{\sqrt{- a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2728, size = 26, normalized size = 1.3 \begin{align*} \frac{\log \left ({\left | \sqrt{-a} x + \frac{1}{a^{2}} \right |}\right )}{\sqrt{-a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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