Optimal. Leaf size=22 \[ \frac{2 x^{3/2}}{3 a (a-b x)^{3/2}} \]
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Rubi [A] time = 0.0016898, 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} \[ \frac{2 x^{3/2}}{3 a (a-b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{(a-b x)^{5/2}} \, dx &=\frac{2 x^{3/2}}{3 a (a-b x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0055816, size = 22, normalized size = 1. \[ \frac{2 x^{3/2}}{3 a (a-b x)^{3/2}} \]
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}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05743, size = 22, normalized size = 1. \begin{align*} \frac{2 \, x^{\frac{3}{2}}}{3 \,{\left (-b x + a\right )}^{\frac{3}{2}} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77506, size = 78, normalized size = 3.55 \begin{align*} \frac{2 \, \sqrt{-b x + a} x^{\frac{3}{2}}}{3 \,{\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.25948, size = 97, normalized size = 4.41 \begin{align*} \begin{cases} \frac{2 i x^{\frac{3}{2}}}{- 3 a^{\frac{5}{2}} \sqrt{-1 + \frac{b x}{a}} + 3 a^{\frac{3}{2}} b x \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{2 x^{\frac{3}{2}}}{- 3 a^{\frac{5}{2}} \sqrt{1 - \frac{b x}{a}} + 3 a^{\frac{3}{2}} b x \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1171, size = 138, normalized size = 6.27 \begin{align*} \frac{4 \,{\left (3 \,{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt{-b} + a^{2} \sqrt{-b} b^{2}\right )}{\left | b \right |}}{3 \,{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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