Optimal. Leaf size=86 \[ \frac{5 x^{3/2} \sqrt{b x+2}}{2 b^2}-\frac{15 \sqrt{x} \sqrt{b x+2}}{2 b^3}+\frac{15 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{2 x^{5/2}}{b \sqrt{b x+2}} \]
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Rubi [A] time = 0.0204427, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {47, 50, 54, 215} \[ \frac{5 x^{3/2} \sqrt{b x+2}}{2 b^2}-\frac{15 \sqrt{x} \sqrt{b x+2}}{2 b^3}+\frac{15 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{2 x^{5/2}}{b \sqrt{b x+2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 54
Rule 215
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 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0058571, size = 30, normalized size = 0.35 \[ \frac{x^{7/2} \, _2F_1\left (\frac{3}{2},\frac{7}{2};\frac{9}{2};-\frac{b x}{2}\right )}{7 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 106, normalized size = 1.2 \begin{align*}{\frac{bx-7}{2\,{b}^{3}}\sqrt{x}\sqrt{bx+2}}+{ \left ({\frac{15}{2}\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\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{x \left ( bx+2 \right ) }{\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 = 1.96873, size = 383, normalized size = 4.45 \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}}{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 = 13.4935, size = 80, normalized size = 0.93 \begin{align*} \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{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} \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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