Optimal. Leaf size=24 \[ \frac {\sqrt {a+b x} \log (x)}{\sqrt {-a-b x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {23, 29}
\begin {gather*} \frac {\log (x) \sqrt {a+b x}}{\sqrt {-a-b x}} \end {gather*}
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*}
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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+b x} \log (x)}{\sqrt {-a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 22, normalized size = 0.92
method | result | size |
default | \(-\frac {\sqrt {-b x -a}\, \ln \left (x \right )}{\sqrt {b x +a}}\) | \(22\) |
risch | \(-\frac {i \sqrt {\frac {-b x -a}{b x +a}}\, \sqrt {b x +a}\, \ln \left (x \right )}{\sqrt {-b x -a}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.32, size = 53, normalized size = 2.21 \begin {gather*} b \sqrt {-\frac {1}{b^{2}}} \log \left (x + \frac {a}{b}\right ) - i \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.83, size = 37, normalized size = 1.54 \begin {gather*} \begin {cases} - i \log {\left (-1 + \frac {b \left (\frac {a}{b} + x\right )}{a} \right )} & \text {for}\: \left |{\frac {b \left (\frac {a}{b} + x\right )}{a}}\right | > 1 \\- i \log {\left (1 - \frac {b \left (\frac {a}{b} + x\right )}{a} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 2.37, size = 7, normalized size = 0.29 \begin {gather*} -i \, \log \left ({\left | b x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {a+b\,x}}{x\,\sqrt {-a-b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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