Optimal. Leaf size=106 \[ \frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{3/2}}+\frac{1}{4} x^{3/2} (2-b x)^{5/2}+\frac{5}{12} x^{3/2} (2-b x)^{3/2}+\frac{5}{8} x^{3/2} \sqrt{2-b x}-\frac{5 \sqrt{x} \sqrt{2-b x}}{8 b} \]
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Rubi [A] time = 0.022558, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {50, 54, 216} \[ \frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{3/2}}+\frac{1}{4} x^{3/2} (2-b x)^{5/2}+\frac{5}{12} x^{3/2} (2-b x)^{3/2}+\frac{5}{8} x^{3/2} \sqrt{2-b x}-\frac{5 \sqrt{x} \sqrt{2-b x}}{8 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \sqrt{x} (2-b x)^{5/2} \, dx &=\frac{1}{4} x^{3/2} (2-b x)^{5/2}+\frac{5}{4} \int \sqrt{x} (2-b x)^{3/2} \, dx\\ &=\frac{5}{12} x^{3/2} (2-b x)^{3/2}+\frac{1}{4} x^{3/2} (2-b x)^{5/2}+\frac{5}{4} \int \sqrt{x} \sqrt{2-b x} \, dx\\ &=\frac{5}{8} x^{3/2} \sqrt{2-b x}+\frac{5}{12} x^{3/2} (2-b x)^{3/2}+\frac{1}{4} x^{3/2} (2-b x)^{5/2}+\frac{5}{8} \int \frac{\sqrt{x}}{\sqrt{2-b x}} \, dx\\ &=-\frac{5 \sqrt{x} \sqrt{2-b x}}{8 b}+\frac{5}{8} x^{3/2} \sqrt{2-b x}+\frac{5}{12} x^{3/2} (2-b x)^{3/2}+\frac{1}{4} x^{3/2} (2-b x)^{5/2}+\frac{5 \int \frac{1}{\sqrt{x} \sqrt{2-b x}} \, dx}{8 b}\\ &=-\frac{5 \sqrt{x} \sqrt{2-b x}}{8 b}+\frac{5}{8} x^{3/2} \sqrt{2-b x}+\frac{5}{12} x^{3/2} (2-b x)^{3/2}+\frac{1}{4} x^{3/2} (2-b x)^{5/2}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-b x^2}} \, dx,x,\sqrt{x}\right )}{4 b}\\ &=-\frac{5 \sqrt{x} \sqrt{2-b x}}{8 b}+\frac{5}{8} x^{3/2} \sqrt{2-b x}+\frac{5}{12} x^{3/2} (2-b x)^{3/2}+\frac{1}{4} x^{3/2} (2-b x)^{5/2}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0437954, size = 71, normalized size = 0.67 \[ \frac{\sqrt{x} \sqrt{2-b x} \left (6 b^3 x^3-34 b^2 x^2+59 b x-15\right )}{24 b}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 107, normalized size = 1. \begin{align*}{\frac{1}{4}{x}^{{\frac{3}{2}}} \left ( -bx+2 \right ) ^{{\frac{5}{2}}}}+{\frac{5}{12}{x}^{{\frac{3}{2}}} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}}+{\frac{5}{8}{x}^{{\frac{3}{2}}}\sqrt{-bx+2}}-{\frac{5}{8\,b}\sqrt{x}\sqrt{-bx+2}}+{\frac{5}{8}\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 = 1.88979, size = 373, normalized size = 3.52 \begin{align*} \left [\frac{{\left (6 \, b^{4} x^{3} - 34 \, b^{3} x^{2} + 59 \, b^{2} x - 15 \, b\right )} \sqrt{-b x + 2} \sqrt{x} - 15 \, \sqrt{-b} \log \left (-b x + \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right )}{24 \, b^{2}}, \frac{{\left (6 \, b^{4} x^{3} - 34 \, b^{3} x^{2} + 59 \, b^{2} x - 15 \, b\right )} \sqrt{-b x + 2} \sqrt{x} - 30 \, \sqrt{b} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{24 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.8152, size = 255, normalized size = 2.41 \begin{align*} \begin{cases} \frac{i b^{3} x^{\frac{9}{2}}}{4 \sqrt{b x - 2}} - \frac{23 i b^{2} x^{\frac{7}{2}}}{12 \sqrt{b x - 2}} + \frac{127 i b x^{\frac{5}{2}}}{24 \sqrt{b x - 2}} - \frac{133 i x^{\frac{3}{2}}}{24 \sqrt{b x - 2}} + \frac{5 i \sqrt{x}}{4 b \sqrt{b x - 2}} - \frac{5 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 b^{\frac{3}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{b^{3} x^{\frac{9}{2}}}{4 \sqrt{- b x + 2}} + \frac{23 b^{2} x^{\frac{7}{2}}}{12 \sqrt{- b x + 2}} - \frac{127 b x^{\frac{5}{2}}}{24 \sqrt{- b x + 2}} + \frac{133 x^{\frac{3}{2}}}{24 \sqrt{- b x + 2}} - \frac{5 \sqrt{x}}{4 b \sqrt{- b x + 2}} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 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(-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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