Optimal. Leaf size=57 \[ -\frac{(b x-a)^{3/2}}{x}+3 b \sqrt{b x-a}-3 \sqrt{a} b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0143466, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {47, 50, 63, 205} \[ -\frac{(b x-a)^{3/2}}{x}+3 b \sqrt{b x-a}-3 \sqrt{a} b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{(-a+b x)^{3/2}}{x^2} \, dx &=-\frac{(-a+b x)^{3/2}}{x}+\frac{1}{2} (3 b) \int \frac{\sqrt{-a+b x}}{x} \, dx\\ &=3 b \sqrt{-a+b x}-\frac{(-a+b x)^{3/2}}{x}-\frac{1}{2} (3 a b) \int \frac{1}{x \sqrt{-a+b x}} \, dx\\ &=3 b \sqrt{-a+b x}-\frac{(-a+b x)^{3/2}}{x}-(3 a) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )\\ &=3 b \sqrt{-a+b x}-\frac{(-a+b x)^{3/2}}{x}-3 \sqrt{a} b \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0112193, size = 36, normalized size = 0.63 \[ \frac{2 b (b x-a)^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};1-\frac{b x}{a}\right )}{5 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 48, normalized size = 0.8 \begin{align*} 2\,b\sqrt{bx-a}+{\frac{a}{x}\sqrt{bx-a}}-3\,b\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) \sqrt{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 = 1.60363, size = 246, normalized size = 4.32 \begin{align*} \left [\frac{3 \, \sqrt{-a} b x \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + 2 \,{\left (2 \, b x + a\right )} \sqrt{b x - a}}{2 \, x}, -\frac{3 \, \sqrt{a} b x \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) -{\left (2 \, b x + a\right )} \sqrt{b x - a}}{x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.56156, size = 201, normalized size = 3.53 \begin{align*} \begin{cases} - 3 i \sqrt{a} b \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + \frac{i a^{2}}{\sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{i a \sqrt{b}}{\sqrt{x} \sqrt{\frac{a}{b x} - 1}} - \frac{2 i b^{\frac{3}{2}} \sqrt{x}}{\sqrt{\frac{a}{b x} - 1}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\3 \sqrt{a} b \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} - \frac{a^{2}}{\sqrt{b} x^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}} - \frac{a \sqrt{b}}{\sqrt{x} \sqrt{- \frac{a}{b x} + 1}} + \frac{2 b^{\frac{3}{2}} \sqrt{x}}{\sqrt{- \frac{a}{b x} + 1}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23978, size = 78, normalized size = 1.37 \begin{align*} -\frac{3 \, \sqrt{a} b^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) - 2 \, \sqrt{b x - a} b^{2} - \frac{\sqrt{b x - a} a b}{x}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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