Optimal. Leaf size=89 \[ \frac{5 x^{3/2} \sqrt{2-b x}}{2 b^2}+\frac{15 \sqrt{x} \sqrt{2-b x}}{2 b^3}-\frac{15 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}+\frac{2 x^{5/2}}{b \sqrt{2-b x}} \]
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Rubi [A] time = 0.0207852, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {47, 50, 54, 216} \[ \frac{5 x^{3/2} \sqrt{2-b x}}{2 b^2}+\frac{15 \sqrt{x} \sqrt{2-b x}}{2 b^3}-\frac{15 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}+\frac{2 x^{5/2}}{b \sqrt{2-b x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{(2-b x)^{3/2}} \, dx &=\frac{2 x^{5/2}}{b \sqrt{2-b x}}-\frac{5 \int \frac{x^{3/2}}{\sqrt{2-b x}} \, dx}{b}\\ &=\frac{2 x^{5/2}}{b \sqrt{2-b x}}+\frac{5 x^{3/2} \sqrt{2-b x}}{2 b^2}-\frac{15 \int \frac{\sqrt{x}}{\sqrt{2-b x}} \, dx}{2 b^2}\\ &=\frac{2 x^{5/2}}{b \sqrt{2-b x}}+\frac{15 \sqrt{x} \sqrt{2-b x}}{2 b^3}+\frac{5 x^{3/2} \sqrt{2-b x}}{2 b^2}-\frac{15 \int \frac{1}{\sqrt{x} \sqrt{2-b x}} \, dx}{2 b^3}\\ &=\frac{2 x^{5/2}}{b \sqrt{2-b x}}+\frac{15 \sqrt{x} \sqrt{2-b x}}{2 b^3}+\frac{5 x^{3/2} \sqrt{2-b x}}{2 b^2}-\frac{15 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-b x^2}} \, dx,x,\sqrt{x}\right )}{b^3}\\ &=\frac{2 x^{5/2}}{b \sqrt{2-b x}}+\frac{15 \sqrt{x} \sqrt{2-b x}}{2 b^3}+\frac{5 x^{3/2} \sqrt{2-b x}}{2 b^2}-\frac{15 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0061238, size = 30, normalized size = 0.34 \[ \frac{x^{7/2} \, _2F_1\left (\frac{3}{2},\frac{7}{2};\frac{9}{2};\frac{b x}{2}\right )}{7 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 138, normalized size = 1.6 \begin{align*} -{\frac{ \left ( bx+7 \right ) \left ( bx-2 \right ) }{2\,{b}^{3}}\sqrt{x}\sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{-x \left ( bx-2 \right ) }}}{\frac{1}{\sqrt{-bx+2}}}}-{ \left ({\frac{15}{2}\arctan \left ({\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){b}^{-{\frac{7}{2}}}}+8\,{\frac{1}{{b}^{4}}\sqrt{-b \left ( x-2\,{b}^{-1} \right ) ^{2}-2\,x+4\,{b}^{-1}} \left ( x-2\,{b}^{-1} \right ) ^{-1}} \right ) \sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}} \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 = 1.53743, size = 387, normalized size = 4.35 \begin{align*} \left [-\frac{15 \,{\left (b x - 2\right )} \sqrt{-b} \log \left (-b x - \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right ) -{\left (b^{3} x^{2} + 5 \, b^{2} x - 30 \, b\right )} \sqrt{-b x + 2} \sqrt{x}}{2 \,{\left (b^{5} x - 2 \, b^{4}\right )}}, \frac{30 \,{\left (b x - 2\right )} \sqrt{b} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) +{\left (b^{3} x^{2} + 5 \, b^{2} x - 30 \, b\right )} \sqrt{-b x + 2} \sqrt{x}}{2 \,{\left (b^{5} x - 2 \, b^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.8081, size = 173, normalized size = 1.94 \begin{align*} \begin{cases} \frac{i x^{\frac{5}{2}}}{2 b \sqrt{b x - 2}} + \frac{5 i x^{\frac{3}{2}}}{2 b^{2} \sqrt{b x - 2}} - \frac{15 i \sqrt{x}}{b^{3} \sqrt{b x - 2}} + \frac{15 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{x^{\frac{5}{2}}}{2 b \sqrt{- b x + 2}} - \frac{5 x^{\frac{3}{2}}}{2 b^{2} \sqrt{- b x + 2}} + \frac{15 \sqrt{x}}{b^{3} \sqrt{- b x + 2}} - \frac{15 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{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(-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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