Optimal. Leaf size=24 \[ \frac{\log (x) \sqrt{a+b x}}{\sqrt{-a-b x}} \]
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Rubi [A] time = 0.0063245, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {23, 29} \[ \frac{\log (x) \sqrt{a+b x}}{\sqrt{-a-b x}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 29
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x}}{x \sqrt{-a-b x}} \, dx &=\frac{\sqrt{a+b x} \int \frac{1}{x} \, dx}{\sqrt{-a-b x}}\\ &=\frac{\sqrt{a+b x} \log (x)}{\sqrt{-a-b x}}\\ \end{align*}
Mathematica [A] time = 0.0043989, size = 24, normalized size = 1. \[ \frac{\log (x) \sqrt{a+b x}}{\sqrt{-a-b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 22, normalized size = 0.9 \begin{align*} -{\ln \left ( x \right ) \sqrt{-bx-a}{\frac{1}{\sqrt{bx+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 = 1.78665, size = 4, normalized size = 0.17 \begin{align*} 0 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.20083, size = 39, normalized size = 1.62 \begin{align*} \begin{cases} - i \log{\left (-1 + \frac{b \left (\frac{a}{b} + x\right )}{a} \right )} & \text{for}\: \frac{\left |{b \left (\frac{a}{b} + x\right )}\right |}{\left |{a}\right |} > 1 \\- i \log{\left (1 - \frac{b \left (\frac{a}{b} + x\right )}{a} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.34537, size = 18, normalized size = 0.75 \begin{align*} -i \, \log \left ({\left | b x \right |}\right ) + i \, \log \left ({\left | a \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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