Optimal. Leaf size=84 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{3} x^{3/2} (2-b x)^{3/2}+\frac{1}{2} x^{3/2} \sqrt{2-b x}-\frac{\sqrt{x} \sqrt{2-b x}}{2 b} \]
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Rubi [A] time = 0.0157307, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {50, 54, 216} \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{3} x^{3/2} (2-b x)^{3/2}+\frac{1}{2} x^{3/2} \sqrt{2-b x}-\frac{\sqrt{x} \sqrt{2-b x}}{2 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \sqrt{x} (2-b x)^{3/2} \, dx &=\frac{1}{3} x^{3/2} (2-b x)^{3/2}+\int \sqrt{x} \sqrt{2-b x} \, dx\\ &=\frac{1}{2} x^{3/2} \sqrt{2-b x}+\frac{1}{3} x^{3/2} (2-b x)^{3/2}+\frac{1}{2} \int \frac{\sqrt{x}}{\sqrt{2-b x}} \, dx\\ &=-\frac{\sqrt{x} \sqrt{2-b x}}{2 b}+\frac{1}{2} x^{3/2} \sqrt{2-b x}+\frac{1}{3} x^{3/2} (2-b x)^{3/2}+\frac{\int \frac{1}{\sqrt{x} \sqrt{2-b x}} \, dx}{2 b}\\ &=-\frac{\sqrt{x} \sqrt{2-b x}}{2 b}+\frac{1}{2} x^{3/2} \sqrt{2-b x}+\frac{1}{3} x^{3/2} (2-b x)^{3/2}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{2-b x^2}} \, dx,x,\sqrt{x}\right )}{b}\\ &=-\frac{\sqrt{x} \sqrt{2-b x}}{2 b}+\frac{1}{2} x^{3/2} \sqrt{2-b x}+\frac{1}{3} x^{3/2} (2-b x)^{3/2}+\frac{\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.040948, size = 60, normalized size = 0.71 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}-\frac{\sqrt{x} \sqrt{2-b x} \left (2 b^2 x^2-7 b x+3\right )}{6 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 94, normalized size = 1.1 \begin{align*}{\frac{1}{3}{x}^{{\frac{3}{2}}} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}}+{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{-bx+2}}-{\frac{1}{2\,b}\sqrt{x}\sqrt{-bx+2}}+{\frac{1}{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{3}{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 = 1.86114, size = 329, normalized size = 3.92 \begin{align*} \left [-\frac{{\left (2 \, b^{3} x^{2} - 7 \, 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 )}{6 \, b^{2}}, -\frac{{\left (2 \, b^{3} x^{2} - 7 \, 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 )}{6 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.46052, size = 199, normalized size = 2.37 \begin{align*} \begin{cases} - \frac{i b^{2} x^{\frac{7}{2}}}{3 \sqrt{b x - 2}} + \frac{11 i b x^{\frac{5}{2}}}{6 \sqrt{b x - 2}} - \frac{17 i x^{\frac{3}{2}}}{6 \sqrt{b x - 2}} + \frac{i \sqrt{x}}{b \sqrt{b x - 2}} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\\frac{b^{2} x^{\frac{7}{2}}}{3 \sqrt{- b x + 2}} - \frac{11 b x^{\frac{5}{2}}}{6 \sqrt{- b x + 2}} + \frac{17 x^{\frac{3}{2}}}{6 \sqrt{- b x + 2}} - \frac{\sqrt{x}}{b \sqrt{- b x + 2}} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \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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