Optimal. Leaf size=66 \[ -\frac{2 (a-b x)^{3/2}}{\sqrt{x}}-3 b \sqrt{x} \sqrt{a-b x}-3 a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right ) \]
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Rubi [A] time = 0.0203585, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {47, 50, 63, 217, 203} \[ -\frac{2 (a-b x)^{3/2}}{\sqrt{x}}-3 b \sqrt{x} \sqrt{a-b x}-3 a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{(a-b x)^{3/2}}{x^{3/2}} \, dx &=-\frac{2 (a-b x)^{3/2}}{\sqrt{x}}-(3 b) \int \frac{\sqrt{a-b x}}{\sqrt{x}} \, dx\\ &=-3 b \sqrt{x} \sqrt{a-b x}-\frac{2 (a-b x)^{3/2}}{\sqrt{x}}-\frac{1}{2} (3 a b) \int \frac{1}{\sqrt{x} \sqrt{a-b x}} \, dx\\ &=-3 b \sqrt{x} \sqrt{a-b x}-\frac{2 (a-b x)^{3/2}}{\sqrt{x}}-(3 a b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x^2}} \, dx,x,\sqrt{x}\right )\\ &=-3 b \sqrt{x} \sqrt{a-b x}-\frac{2 (a-b x)^{3/2}}{\sqrt{x}}-(3 a b) \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a-b x}}\right )\\ &=-3 b \sqrt{x} \sqrt{a-b x}-\frac{2 (a-b x)^{3/2}}{\sqrt{x}}-3 a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0114121, size = 47, normalized size = 0.71 \[ -\frac{2 a \sqrt{a-b x} \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{b x}{a}\right )}{\sqrt{x} \sqrt{1-\frac{b x}{a}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 74, normalized size = 1.1 \begin{align*} -{(bx+2\,a)\sqrt{-bx+a}{\frac{1}{\sqrt{x}}}}-{\frac{3\,a}{2}\sqrt{b}\arctan \left ({\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ) \sqrt{x \left ( -bx+a \right ) }{\frac{1}{\sqrt{x}}}{\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 = 1.90025, size = 292, normalized size = 4.42 \begin{align*} \left [\frac{3 \, a \sqrt{-b} x \log \left (-2 \, b x + 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) - 2 \,{\left (b x + 2 \, a\right )} \sqrt{-b x + a} \sqrt{x}}{2 \, x}, \frac{3 \, a \sqrt{b} x \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (b x + 2 \, a\right )} \sqrt{-b x + a} \sqrt{x}}{x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.21243, size = 199, normalized size = 3.02 \begin{align*} \begin{cases} \frac{2 i a^{\frac{3}{2}}}{\sqrt{x} \sqrt{-1 + \frac{b x}{a}}} - \frac{i \sqrt{a} b \sqrt{x}}{\sqrt{-1 + \frac{b x}{a}}} + 3 i a \sqrt{b} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} - \frac{i b^{2} x^{\frac{3}{2}}}{\sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{2 a^{\frac{3}{2}}}{\sqrt{x} \sqrt{1 - \frac{b x}{a}}} + \frac{\sqrt{a} b \sqrt{x}}{\sqrt{1 - \frac{b x}{a}}} - 3 a \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} + \frac{b^{2} x^{\frac{3}{2}}}{\sqrt{a} \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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