Optimal. Leaf size=28 \[ -\frac{\sqrt{a+b x}}{2 x^2 \sqrt{-a-b x}} \]
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Rubi [A] time = 0.0033396, antiderivative size = 28, 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, 30} \[ -\frac{\sqrt{a+b x}}{2 x^2 \sqrt{-a-b x}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x}}{x^3 \sqrt{-a-b x}} \, dx &=\frac{\sqrt{a+b x} \int \frac{1}{x^3} \, dx}{\sqrt{-a-b x}}\\ &=-\frac{\sqrt{a+b x}}{2 x^2 \sqrt{-a-b x}}\\ \end{align*}
Mathematica [A] time = 0.0042085, size = 28, normalized size = 1. \[ -\frac{\sqrt{a+b x}}{2 x^2 \sqrt{-a-b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 23, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{x}^{2}}\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 = 1.86342, size = 4, normalized size = 0.14 \begin{align*} 0 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.08265, size = 88, normalized size = 3.14 \begin{align*} \frac{2 i a b^{3} \left (\frac{a}{b} + x\right )}{2 a^{4} - 4 a^{3} b \left (\frac{a}{b} + x\right ) + 2 a^{2} b^{2} \left (\frac{a}{b} + x\right )^{2}} - \frac{i b^{4} \left (\frac{a}{b} + x\right )^{2}}{2 a^{4} - 4 a^{3} b \left (\frac{a}{b} + x\right ) + 2 a^{2} b^{2} \left (\frac{a}{b} + x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.41103, size = 26, normalized size = 0.93 \begin{align*} -\frac{i \,{\left (\frac{b^{3}}{a^{2}} - \frac{b}{x^{2}}\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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