Optimal. Leaf size=61 \[ \frac{1}{2} \sqrt{x} (b x+2)^{3/2}+\frac{3}{2} \sqrt{x} \sqrt{b x+2}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0109375, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {50, 54, 215} \[ \frac{1}{2} \sqrt{x} (b x+2)^{3/2}+\frac{3}{2} \sqrt{x} \sqrt{b x+2}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{(2+b x)^{3/2}}{\sqrt{x}} \, dx &=\frac{1}{2} \sqrt{x} (2+b x)^{3/2}+\frac{3}{2} \int \frac{\sqrt{2+b x}}{\sqrt{x}} \, dx\\ &=\frac{3}{2} \sqrt{x} \sqrt{2+b x}+\frac{1}{2} \sqrt{x} (2+b x)^{3/2}+\frac{3}{2} \int \frac{1}{\sqrt{x} \sqrt{2+b x}} \, dx\\ &=\frac{3}{2} \sqrt{x} \sqrt{2+b x}+\frac{1}{2} \sqrt{x} (2+b x)^{3/2}+3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+b x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{3}{2} \sqrt{x} \sqrt{2+b x}+\frac{1}{2} \sqrt{x} (2+b x)^{3/2}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0240026, size = 48, normalized size = 0.79 \[ \frac{1}{2} \sqrt{x} \sqrt{b x+2} (b x+5)+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 72, normalized size = 1.2 \begin{align*}{\frac{1}{2} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3}{2}\sqrt{x}\sqrt{bx+2}}+{\frac{3}{2}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{b}}}} \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 = 1.73564, size = 279, normalized size = 4.57 \begin{align*} \left [\frac{{\left (b^{2} x + 5 \, b\right )} \sqrt{b x + 2} \sqrt{x} + 3 \, \sqrt{b} \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right )}{2 \, b}, \frac{{\left (b^{2} x + 5 \, b\right )} \sqrt{b x + 2} \sqrt{x} - 6 \, \sqrt{-b} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{2 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.14048, size = 76, normalized size = 1.25 \begin{align*} \frac{b^{2} x^{\frac{5}{2}}}{2 \sqrt{b x + 2}} + \frac{7 b x^{\frac{3}{2}}}{2 \sqrt{b x + 2}} + \frac{5 \sqrt{x}}{\sqrt{b x + 2}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \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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