Optimal. Leaf size=112 \[ -\frac{5 x^{3/2} \sqrt{2-b x}}{24 b^2}-\frac{5 \sqrt{x} \sqrt{2-b x}}{8 b^3}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}+\frac{1}{4} x^{7/2} \sqrt{2-b x}-\frac{x^{5/2} \sqrt{2-b x}}{12 b} \]
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Rubi [A] time = 0.0286051, antiderivative size = 112, 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} \[ -\frac{5 x^{3/2} \sqrt{2-b x}}{24 b^2}-\frac{5 \sqrt{x} \sqrt{2-b x}}{8 b^3}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}+\frac{1}{4} x^{7/2} \sqrt{2-b x}-\frac{x^{5/2} \sqrt{2-b x}}{12 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int x^{5/2} \sqrt{2-b x} \, dx &=\frac{1}{4} x^{7/2} \sqrt{2-b x}+\frac{1}{4} \int \frac{x^{5/2}}{\sqrt{2-b x}} \, dx\\ &=-\frac{x^{5/2} \sqrt{2-b x}}{12 b}+\frac{1}{4} x^{7/2} \sqrt{2-b x}+\frac{5 \int \frac{x^{3/2}}{\sqrt{2-b x}} \, dx}{12 b}\\ &=-\frac{5 x^{3/2} \sqrt{2-b x}}{24 b^2}-\frac{x^{5/2} \sqrt{2-b x}}{12 b}+\frac{1}{4} x^{7/2} \sqrt{2-b x}+\frac{5 \int \frac{\sqrt{x}}{\sqrt{2-b x}} \, dx}{8 b^2}\\ &=-\frac{5 \sqrt{x} \sqrt{2-b x}}{8 b^3}-\frac{5 x^{3/2} \sqrt{2-b x}}{24 b^2}-\frac{x^{5/2} \sqrt{2-b x}}{12 b}+\frac{1}{4} x^{7/2} \sqrt{2-b x}+\frac{5 \int \frac{1}{\sqrt{x} \sqrt{2-b x}} \, dx}{8 b^3}\\ &=-\frac{5 \sqrt{x} \sqrt{2-b x}}{8 b^3}-\frac{5 x^{3/2} \sqrt{2-b x}}{24 b^2}-\frac{x^{5/2} \sqrt{2-b x}}{12 b}+\frac{1}{4} x^{7/2} \sqrt{2-b x}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-b x^2}} \, dx,x,\sqrt{x}\right )}{4 b^3}\\ &=-\frac{5 \sqrt{x} \sqrt{2-b x}}{8 b^3}-\frac{5 x^{3/2} \sqrt{2-b x}}{24 b^2}-\frac{x^{5/2} \sqrt{2-b x}}{12 b}+\frac{1}{4} x^{7/2} \sqrt{2-b x}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0450378, size = 71, normalized size = 0.63 \[ \frac{\sqrt{x} \sqrt{2-b x} \left (6 b^3 x^3-2 b^2 x^2-5 b x-15\right )}{24 b^3}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 116, normalized size = 1. \begin{align*} -{\frac{1}{4\,b}{x}^{{\frac{5}{2}}} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}}-{\frac{5}{12\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}}-{\frac{5}{8\,{b}^{3}} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5}{8\,{b}^{3}}\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{7}{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 = 1.51176, size = 367, normalized size = 3.28 \begin{align*} \left [\frac{{\left (6 \, b^{4} x^{3} - 2 \, b^{3} x^{2} - 5 \, 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^{4}}, \frac{{\left (6 \, b^{4} x^{3} - 2 \, b^{3} x^{2} - 5 \, 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^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.6091, size = 252, normalized size = 2.25 \begin{align*} \begin{cases} \frac{i b x^{\frac{9}{2}}}{4 \sqrt{b x - 2}} - \frac{7 i x^{\frac{7}{2}}}{12 \sqrt{b x - 2}} - \frac{i x^{\frac{5}{2}}}{24 b \sqrt{b x - 2}} - \frac{5 i x^{\frac{3}{2}}}{24 b^{2} \sqrt{b x - 2}} + \frac{5 i \sqrt{x}}{4 b^{3} \sqrt{b x - 2}} - \frac{5 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 b^{\frac{7}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{b x^{\frac{9}{2}}}{4 \sqrt{- b x + 2}} + \frac{7 x^{\frac{7}{2}}}{12 \sqrt{- b x + 2}} + \frac{x^{\frac{5}{2}}}{24 b \sqrt{- b x + 2}} + \frac{5 x^{\frac{3}{2}}}{24 b^{2} \sqrt{- b x + 2}} - \frac{5 \sqrt{x}}{4 b^{3} \sqrt{- b x + 2}} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 b^{\frac{7}{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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