Optimal. Leaf size=81 \[ 5 b^2 \sqrt{x} \sqrt{b x+2}+10 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 (b x+2)^{5/2}}{3 x^{3/2}}-\frac{10 b (b x+2)^{3/2}}{3 \sqrt{x}} \]
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Rubi [A] time = 0.0171751, antiderivative size = 81, 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} \[ 5 b^2 \sqrt{x} \sqrt{b x+2}+10 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 (b x+2)^{5/2}}{3 x^{3/2}}-\frac{10 b (b x+2)^{3/2}}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{(2+b x)^{5/2}}{x^{5/2}} \, dx &=-\frac{2 (2+b x)^{5/2}}{3 x^{3/2}}+\frac{1}{3} (5 b) \int \frac{(2+b x)^{3/2}}{x^{3/2}} \, dx\\ &=-\frac{10 b (2+b x)^{3/2}}{3 \sqrt{x}}-\frac{2 (2+b x)^{5/2}}{3 x^{3/2}}+\left (5 b^2\right ) \int \frac{\sqrt{2+b x}}{\sqrt{x}} \, dx\\ &=5 b^2 \sqrt{x} \sqrt{2+b x}-\frac{10 b (2+b x)^{3/2}}{3 \sqrt{x}}-\frac{2 (2+b x)^{5/2}}{3 x^{3/2}}+\left (5 b^2\right ) \int \frac{1}{\sqrt{x} \sqrt{2+b x}} \, dx\\ &=5 b^2 \sqrt{x} \sqrt{2+b x}-\frac{10 b (2+b x)^{3/2}}{3 \sqrt{x}}-\frac{2 (2+b x)^{5/2}}{3 x^{3/2}}+\left (10 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+b x^2}} \, dx,x,\sqrt{x}\right )\\ &=5 b^2 \sqrt{x} \sqrt{2+b x}-\frac{10 b (2+b x)^{3/2}}{3 \sqrt{x}}-\frac{2 (2+b x)^{5/2}}{3 x^{3/2}}+10 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0070435, size = 30, normalized size = 0.37 \[ -\frac{8 \sqrt{2} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x}{2}\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 82, normalized size = 1. \begin{align*}{\frac{3\,{b}^{3}{x}^{3}-22\,{b}^{2}{x}^{2}-64\,bx-16}{3}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}}+5\,{\frac{{b}^{3/2}\sqrt{x \left ( bx+2 \right ) }}{\sqrt{x}\sqrt{bx+2}}\ln \left ({\frac{bx+1}{\sqrt{b}}}+\sqrt{b{x}^{2}+2\,x} \right ) } \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 = 1.80796, size = 332, normalized size = 4.1 \begin{align*} \left [\frac{15 \, b^{\frac{3}{2}} x^{2} \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right ) +{\left (3 \, b^{2} x^{2} - 28 \, b x - 8\right )} \sqrt{b x + 2} \sqrt{x}}{3 \, x^{2}}, -\frac{30 \, \sqrt{-b} b x^{2} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (3 \, b^{2} x^{2} - 28 \, b x - 8\right )} \sqrt{b x + 2} \sqrt{x}}{3 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.9004, size = 88, normalized size = 1.09 \begin{align*} b^{\frac{5}{2}} x \sqrt{1 + \frac{2}{b x}} - \frac{28 b^{\frac{3}{2}} \sqrt{1 + \frac{2}{b x}}}{3} - 5 b^{\frac{3}{2}} \log{\left (\frac{1}{b x} \right )} + 10 b^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{2}{b x}} + 1 \right )} - \frac{8 \sqrt{b} \sqrt{1 + \frac{2}{b x}}}{3 x} \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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