Optimal. Leaf size=26 \[ -\frac {\sqrt {a+b x}}{x \sqrt {-a-b x}} \]
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Rubi [A] time = 0.00, antiderivative size = 26, 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, 30} \[ -\frac {\sqrt {a+b x}}{x \sqrt {-a-b x}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{x^2 \sqrt {-a-b x}} \, dx &=\frac {\sqrt {a+b x} \int \frac {1}{x^2} \, dx}{\sqrt {-a-b x}}\\ &=-\frac {\sqrt {a+b x}}{x \sqrt {-a-b x}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 26, normalized size = 1.00 \[ -\frac {\sqrt {a+b x}}{x \sqrt {-a-b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 1, normalized size = 0.04 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.20, size = 5, normalized size = 0.19 \[ \frac {i}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.88 \[ -\frac {\sqrt {b x +a}}{\sqrt {-b x -a}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 28, normalized size = 1.08 \[ \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2}}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 21, normalized size = 0.81 \[ \frac {\sqrt {-a-b\,x}}{x\,\sqrt {a+b\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.51, size = 20, normalized size = 0.77 \[ \frac {i b^{2} \left (\frac {a}{b} + x\right )}{- a^{2} + a b \left (\frac {a}{b} + x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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