Integrand size = 15, antiderivative size = 22 \[ \int \frac {1}{a^3+\sqrt {-a} x} \, dx=\frac {\log \left (a^3+\sqrt {-a} x\right )}{\sqrt {-a}} \]
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Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {31} \[ \int \frac {1}{a^3+\sqrt {-a} x} \, dx=\frac {\log \left (a^3+\sqrt {-a} x\right )}{\sqrt {-a}} \]
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Rule 31
Rubi steps \begin{align*} \text {integral}& = \frac {\log \left (a^3+\sqrt {-a} x\right )}{\sqrt {-a}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^3+\sqrt {-a} x} \, dx=\frac {\log \left (a^3+\sqrt {-a} x\right )}{\sqrt {-a}} \]
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Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
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none
Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{a^3+\sqrt {-a} x} \, dx=-\frac {\sqrt {-a} \log \left (-\sqrt {-a} a^{2} + x\right )}{a} \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a^3+\sqrt {-a} x} \, dx=\frac {\log {\left (a^{3} + x \sqrt {- a} \right )}}{\sqrt {- a}} \]
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none
Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{a^3+\sqrt {-a} x} \, dx=\frac {\log \left (a^{3} + \sqrt {-a} x\right )}{\sqrt {-a}} \]
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none
Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a^3+\sqrt {-a} x} \, dx=\frac {\log \left ({\left | a^{3} + \sqrt {-a} x \right |}\right )}{\sqrt {-a}} \]
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Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {1}{a^3+\sqrt {-a} x} \, dx=\frac {\ln \left (x-{\left (-a\right )}^{5/2}\right )}{\sqrt {-a}} \]
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