Optimal. Leaf size=69 \[ -\frac{3 \sqrt{x} \sqrt{2-b x}}{2 b^2}+\frac{3 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}-\frac{x^{3/2} \sqrt{2-b x}}{2 b} \]
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Rubi [A] time = 0.01492, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {50, 54, 216} \[ -\frac{3 \sqrt{x} \sqrt{2-b x}}{2 b^2}+\frac{3 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}-\frac{x^{3/2} \sqrt{2-b x}}{2 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\sqrt{2-b x}} \, dx &=-\frac{x^{3/2} \sqrt{2-b x}}{2 b}+\frac{3 \int \frac{\sqrt{x}}{\sqrt{2-b x}} \, dx}{2 b}\\ &=-\frac{3 \sqrt{x} \sqrt{2-b x}}{2 b^2}-\frac{x^{3/2} \sqrt{2-b x}}{2 b}+\frac{3 \int \frac{1}{\sqrt{x} \sqrt{2-b x}} \, dx}{2 b^2}\\ &=-\frac{3 \sqrt{x} \sqrt{2-b x}}{2 b^2}-\frac{x^{3/2} \sqrt{2-b x}}{2 b}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-b x^2}} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=-\frac{3 \sqrt{x} \sqrt{2-b x}}{2 b^2}-\frac{x^{3/2} \sqrt{2-b x}}{2 b}+\frac{3 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0295956, size = 52, normalized size = 0.75 \[ \frac{3 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}-\frac{\sqrt{x} \sqrt{2-b x} (b x+3)}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 84, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,b}{x}^{{\frac{3}{2}}}\sqrt{-bx+2}}-{\frac{3}{2\,{b}^{2}}\sqrt{x}\sqrt{-bx+2}}+{\frac{3}{2}\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{x}}}} \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 = 1.56412, size = 292, normalized size = 4.23 \begin{align*} \left [-\frac{{\left (b^{2} x + 3 \, b\right )} \sqrt{-b x + 2} \sqrt{x} + 3 \, \sqrt{-b} \log \left (-b x + \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right )}{2 \, b^{3}}, -\frac{{\left (b^{2} x + 3 \, b\right )} \sqrt{-b x + 2} \sqrt{x} + 6 \, \sqrt{b} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{2 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.37159, size = 163, normalized size = 2.36 \begin{align*} \begin{cases} - \frac{i x^{\frac{5}{2}}}{2 \sqrt{b x - 2}} - \frac{i x^{\frac{3}{2}}}{2 b \sqrt{b x - 2}} + \frac{3 i \sqrt{x}}{b^{2} \sqrt{b x - 2}} - \frac{3 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\\frac{x^{\frac{5}{2}}}{2 \sqrt{- b x + 2}} + \frac{x^{\frac{3}{2}}}{2 b \sqrt{- b x + 2}} - \frac{3 \sqrt{x}}{b^{2} \sqrt{- b x + 2}} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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