Optimal. Leaf size=55 \[ 2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )-2 a \sqrt{b x-a}+\frac{2}{3} (b x-a)^{3/2} \]
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Rubi [A] time = 0.0145422, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {50, 63, 205} \[ 2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )-2 a \sqrt{b x-a}+\frac{2}{3} (b x-a)^{3/2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{(-a+b x)^{3/2}}{x} \, dx &=\frac{2}{3} (-a+b x)^{3/2}-a \int \frac{\sqrt{-a+b x}}{x} \, dx\\ &=-2 a \sqrt{-a+b x}+\frac{2}{3} (-a+b x)^{3/2}+a^2 \int \frac{1}{x \sqrt{-a+b x}} \, dx\\ &=-2 a \sqrt{-a+b x}+\frac{2}{3} (-a+b x)^{3/2}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )}{b}\\ &=-2 a \sqrt{-a+b x}+\frac{2}{3} (-a+b x)^{3/2}+2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0742335, size = 48, normalized size = 0.87 \[ 2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )+\frac{2}{3} (b x-4 a) \sqrt{b x-a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 44, normalized size = 0.8 \begin{align*}{\frac{2}{3} \left ( bx-a \right ) ^{{\frac{3}{2}}}}+2\,{a}^{3/2}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) -2\,a\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 = 1.57924, size = 225, normalized size = 4.09 \begin{align*} \left [\sqrt{-a} a \log \left (\frac{b x + 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + \frac{2}{3} \, \sqrt{b x - a}{\left (b x - 4 \, a\right )}, 2 \, a^{\frac{3}{2}} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + \frac{2}{3} \, \sqrt{b x - a}{\left (b x - 4 \, a\right )}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.19287, size = 189, normalized size = 3.44 \begin{align*} \begin{cases} - \frac{8 a^{\frac{3}{2}} \sqrt{-1 + \frac{b x}{a}}}{3} - i a^{\frac{3}{2}} \log{\left (\frac{b x}{a} \right )} + 2 i a^{\frac{3}{2}} \log{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} - 2 a^{\frac{3}{2}} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 \sqrt{a} b x \sqrt{-1 + \frac{b x}{a}}}{3} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{8 i a^{\frac{3}{2}} \sqrt{1 - \frac{b x}{a}}}{3} - i a^{\frac{3}{2}} \log{\left (\frac{b x}{a} \right )} + 2 i a^{\frac{3}{2}} \log{\left (\sqrt{1 - \frac{b x}{a}} + 1 \right )} + \frac{2 i \sqrt{a} b x \sqrt{1 - \frac{b x}{a}}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16273, size = 58, normalized size = 1.05 \begin{align*} 2 \, a^{\frac{3}{2}} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + \frac{2}{3} \,{\left (b x - a\right )}^{\frac{3}{2}} - 2 \, \sqrt{b x - a} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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