Optimal. Leaf size=20 \[ \frac {\log \left ((-a)^{5/2} x+1\right )}{\sqrt {-a}} \]
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Rubi [A] time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\log \left ((-a)^{5/2} x+1\right )}{\sqrt {-a}} \end {gather*}
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*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.10 \begin {gather*} \frac {\log \left (\frac {1}{a^2}+\sqrt {-a} x\right )}{\sqrt {-a}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\frac {1}{a^2}+\sqrt {-a} x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.01, size = 24, normalized size = 1.20 \begin {gather*} -\frac {\sqrt {-a} \log \left (a^{3} x - \sqrt {-a}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 19, normalized size = 0.95 \begin {gather*} \frac {\log \left ({\left | \sqrt {-a} x + \frac {1}{a^{2}} \right |}\right )}{\sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.95 \begin {gather*} \frac {\ln \left (\sqrt {-a}\, x +\frac {1}{a^{2}}\right )}{\sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 18, normalized size = 0.90 \begin {gather*} \frac {\log \left (\sqrt {-a} x + \frac {1}{a^{2}}\right )}{\sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 14, normalized size = 0.70 \begin {gather*} \frac {\ln \left (x+\frac {1}{{\left (-a\right )}^{5/2}}\right )}{\sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 20, normalized size = 1.00 \begin {gather*} \frac {\log {\left (a^{2} x \sqrt {- a} + 1 \right )}}{\sqrt {- a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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