Optimal. Leaf size=22 \[ -\frac{2 (a-b x)^{3/2}}{3 a x^{3/2}} \]
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Rubi [A] time = 0.0017392, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {37} \[ -\frac{2 (a-b x)^{3/2}}{3 a x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{a-b x}}{x^{5/2}} \, dx &=-\frac{2 (a-b x)^{3/2}}{3 a x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0063745, size = 22, normalized size = 1. \[ -\frac{2 (a-b x)^{3/2}}{3 a x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 17, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,a} \left ( -bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02236, size = 22, normalized size = 1. \begin{align*} -\frac{2 \,{\left (-b x + a\right )}^{\frac{3}{2}}}{3 \, a x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59402, size = 57, normalized size = 2.59 \begin{align*} \frac{2 \,{\left (b x - a\right )} \sqrt{-b x + a}}{3 \, a x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.13464, size = 92, normalized size = 4.18 \begin{align*} \begin{cases} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} - 1}}{3 x} + \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}}{3 a} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{2 i \sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{3 x} + \frac{2 i b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{3 a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24095, size = 57, normalized size = 2.59 \begin{align*} \frac{2 \,{\left (b x - a\right )} \sqrt{-b x + a} b^{4}}{3 \,{\left ({\left (b x - a\right )} b + a b\right )}^{\frac{3}{2}} a{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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