Optimal. Leaf size=82 \[ -\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{3} x^{3/2} (b x+2)^{3/2}+\frac{1}{2} x^{3/2} \sqrt{b x+2}+\frac{\sqrt{x} \sqrt{b x+2}}{2 b} \]
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Rubi [A] time = 0.0163102, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {50, 54, 215} \[ -\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{3} x^{3/2} (b x+2)^{3/2}+\frac{1}{2} x^{3/2} \sqrt{b x+2}+\frac{\sqrt{x} \sqrt{b x+2}}{2 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \sqrt{x} (2+b x)^{3/2} \, dx &=\frac{1}{3} x^{3/2} (2+b x)^{3/2}+\int \sqrt{x} \sqrt{2+b x} \, dx\\ &=\frac{1}{2} x^{3/2} \sqrt{2+b x}+\frac{1}{3} x^{3/2} (2+b x)^{3/2}+\frac{1}{2} \int \frac{\sqrt{x}}{\sqrt{2+b x}} \, dx\\ &=\frac{\sqrt{x} \sqrt{2+b x}}{2 b}+\frac{1}{2} x^{3/2} \sqrt{2+b x}+\frac{1}{3} x^{3/2} (2+b x)^{3/2}-\frac{\int \frac{1}{\sqrt{x} \sqrt{2+b x}} \, dx}{2 b}\\ &=\frac{\sqrt{x} \sqrt{2+b x}}{2 b}+\frac{1}{2} x^{3/2} \sqrt{2+b x}+\frac{1}{3} x^{3/2} (2+b x)^{3/2}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{2+b x^2}} \, dx,x,\sqrt{x}\right )}{b}\\ &=\frac{\sqrt{x} \sqrt{2+b x}}{2 b}+\frac{1}{2} x^{3/2} \sqrt{2+b x}+\frac{1}{3} x^{3/2} (2+b x)^{3/2}-\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0323364, size = 60, normalized size = 0.73 \[ \frac{\sqrt{x} \sqrt{b x+2} \left (2 b^2 x^2+7 b x+3\right )}{6 b}-\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 87, normalized size = 1.1 \begin{align*}{\frac{1}{3}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}}+{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{bx+2}}+{\frac{1}{2\,b}\sqrt{x}\sqrt{bx+2}}-{\frac{1}{2}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{3}{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 = 1.79548, size = 323, normalized size = 3.94 \begin{align*} \left [\frac{{\left (2 \, b^{3} x^{2} + 7 \, b^{2} x + 3 \, 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 )}{6 \, b^{2}}, \frac{{\left (2 \, b^{3} x^{2} + 7 \, b^{2} x + 3 \, b\right )} \sqrt{b x + 2} \sqrt{x} + 6 \, \sqrt{-b} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{6 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.32213, size = 92, normalized size = 1.12 \begin{align*} \frac{b^{2} x^{\frac{7}{2}}}{3 \sqrt{b x + 2}} + \frac{11 b x^{\frac{5}{2}}}{6 \sqrt{b x + 2}} + \frac{17 x^{\frac{3}{2}}}{6 \sqrt{b x + 2}} + \frac{\sqrt{x}}{b \sqrt{b x + 2}} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \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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