Optimal. Leaf size=105 \[ -\frac{3 \sqrt{x} \sqrt{b x+2}}{8 b^2}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{5/2}}+\frac{1}{4} x^{5/2} (b x+2)^{3/2}+\frac{1}{4} x^{5/2} \sqrt{b x+2}+\frac{x^{3/2} \sqrt{b x+2}}{8 b} \]
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Rubi [A] time = 0.0255435, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {50, 54, 215} \[ -\frac{3 \sqrt{x} \sqrt{b x+2}}{8 b^2}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{5/2}}+\frac{1}{4} x^{5/2} (b x+2)^{3/2}+\frac{1}{4} x^{5/2} \sqrt{b x+2}+\frac{x^{3/2} \sqrt{b x+2}}{8 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int x^{3/2} (2+b x)^{3/2} \, dx &=\frac{1}{4} x^{5/2} (2+b x)^{3/2}+\frac{3}{4} \int x^{3/2} \sqrt{2+b x} \, dx\\ &=\frac{1}{4} x^{5/2} \sqrt{2+b x}+\frac{1}{4} x^{5/2} (2+b x)^{3/2}+\frac{1}{4} \int \frac{x^{3/2}}{\sqrt{2+b x}} \, dx\\ &=\frac{x^{3/2} \sqrt{2+b x}}{8 b}+\frac{1}{4} x^{5/2} \sqrt{2+b x}+\frac{1}{4} x^{5/2} (2+b x)^{3/2}-\frac{3 \int \frac{\sqrt{x}}{\sqrt{2+b x}} \, dx}{8 b}\\ &=-\frac{3 \sqrt{x} \sqrt{2+b x}}{8 b^2}+\frac{x^{3/2} \sqrt{2+b x}}{8 b}+\frac{1}{4} x^{5/2} \sqrt{2+b x}+\frac{1}{4} x^{5/2} (2+b x)^{3/2}+\frac{3 \int \frac{1}{\sqrt{x} \sqrt{2+b x}} \, dx}{8 b^2}\\ &=-\frac{3 \sqrt{x} \sqrt{2+b x}}{8 b^2}+\frac{x^{3/2} \sqrt{2+b x}}{8 b}+\frac{1}{4} x^{5/2} \sqrt{2+b x}+\frac{1}{4} x^{5/2} (2+b x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+b x^2}} \, dx,x,\sqrt{x}\right )}{4 b^2}\\ &=-\frac{3 \sqrt{x} \sqrt{2+b x}}{8 b^2}+\frac{x^{3/2} \sqrt{2+b x}}{8 b}+\frac{1}{4} x^{5/2} \sqrt{2+b x}+\frac{1}{4} x^{5/2} (2+b x)^{3/2}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0347363, size = 70, normalized size = 0.67 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{b x+2} \left (2 b^3 x^3+6 b^2 x^2+b x-3\right )+6 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{8 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 108, normalized size = 1. \begin{align*}{\frac{1}{4\,b}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}}-{\frac{1}{4\,{b}^{2}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{1}{8\,{b}^{2}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3}{8\,{b}^{2}}\sqrt{x}\sqrt{bx+2}}+{\frac{3}{8}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{5}{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 = 1.91894, size = 350, normalized size = 3.33 \begin{align*} \left [\frac{{\left (2 \, b^{4} x^{3} + 6 \, b^{3} x^{2} + 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 )}{8 \, b^{3}}, \frac{{\left (2 \, b^{4} x^{3} + 6 \, b^{3} x^{2} + 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 )}{8 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.39, size = 117, normalized size = 1.11 \begin{align*} \frac{b^{2} x^{\frac{9}{2}}}{4 \sqrt{b x + 2}} + \frac{5 b x^{\frac{7}{2}}}{4 \sqrt{b x + 2}} + \frac{13 x^{\frac{5}{2}}}{8 \sqrt{b x + 2}} - \frac{x^{\frac{3}{2}}}{8 b \sqrt{b x + 2}} - \frac{3 \sqrt{x}}{4 b^{2} \sqrt{b x + 2}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 b^{\frac{5}{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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