Optimal. Leaf size=102 \[ -\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{3/2}}+\frac{1}{4} x^{3/2} (b x+2)^{5/2}+\frac{5}{12} x^{3/2} (b x+2)^{3/2}+\frac{5}{8} x^{3/2} \sqrt{b x+2}+\frac{5 \sqrt{x} \sqrt{b x+2}}{8 b} \]
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Rubi [A] time = 0.0221546, antiderivative size = 102, 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} \[ -\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{3/2}}+\frac{1}{4} x^{3/2} (b x+2)^{5/2}+\frac{5}{12} x^{3/2} (b x+2)^{3/2}+\frac{5}{8} x^{3/2} \sqrt{b x+2}+\frac{5 \sqrt{x} \sqrt{b x+2}}{8 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \sqrt{x} (2+b x)^{5/2} \, dx &=\frac{1}{4} x^{3/2} (2+b x)^{5/2}+\frac{5}{4} \int \sqrt{x} (2+b x)^{3/2} \, dx\\ &=\frac{5}{12} x^{3/2} (2+b x)^{3/2}+\frac{1}{4} x^{3/2} (2+b x)^{5/2}+\frac{5}{4} \int \sqrt{x} \sqrt{2+b x} \, dx\\ &=\frac{5}{8} x^{3/2} \sqrt{2+b x}+\frac{5}{12} x^{3/2} (2+b x)^{3/2}+\frac{1}{4} x^{3/2} (2+b x)^{5/2}+\frac{5}{8} \int \frac{\sqrt{x}}{\sqrt{2+b x}} \, dx\\ &=\frac{5 \sqrt{x} \sqrt{2+b x}}{8 b}+\frac{5}{8} x^{3/2} \sqrt{2+b x}+\frac{5}{12} x^{3/2} (2+b x)^{3/2}+\frac{1}{4} x^{3/2} (2+b x)^{5/2}-\frac{5 \int \frac{1}{\sqrt{x} \sqrt{2+b x}} \, dx}{8 b}\\ &=\frac{5 \sqrt{x} \sqrt{2+b x}}{8 b}+\frac{5}{8} x^{3/2} \sqrt{2+b x}+\frac{5}{12} x^{3/2} (2+b x)^{3/2}+\frac{1}{4} x^{3/2} (2+b x)^{5/2}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+b x^2}} \, dx,x,\sqrt{x}\right )}{4 b}\\ &=\frac{5 \sqrt{x} \sqrt{2+b x}}{8 b}+\frac{5}{8} x^{3/2} \sqrt{2+b x}+\frac{5}{12} x^{3/2} (2+b x)^{3/2}+\frac{1}{4} x^{3/2} (2+b x)^{5/2}-\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0403282, size = 70, normalized size = 0.69 \[ \frac{\sqrt{x} \sqrt{b x+2} \left (6 b^3 x^3+34 b^2 x^2+59 b x+15\right )}{24 b}-\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 99, normalized size = 1. \begin{align*}{\frac{1}{4}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}}+{\frac{5}{12}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}}+{\frac{5}{8}{x}^{{\frac{3}{2}}}\sqrt{bx+2}}+{\frac{5}{8\,b}\sqrt{x}\sqrt{bx+2}}-{\frac{5}{8}\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 = 1.81017, size = 369, normalized size = 3.62 \begin{align*} \left [\frac{{\left (6 \, b^{4} x^{3} + 34 \, b^{3} x^{2} + 59 \, b^{2} x + 15 \, b\right )} \sqrt{b x + 2} \sqrt{x} + 15 \, \sqrt{b} \log \left (b x - \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right )}{24 \, b^{2}}, \frac{{\left (6 \, b^{4} x^{3} + 34 \, b^{3} x^{2} + 59 \, b^{2} x + 15 \, b\right )} \sqrt{b x + 2} \sqrt{x} + 30 \, \sqrt{-b} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{24 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.3658, size = 119, normalized size = 1.17 \begin{align*} \frac{b^{3} x^{\frac{9}{2}}}{4 \sqrt{b x + 2}} + \frac{23 b^{2} x^{\frac{7}{2}}}{12 \sqrt{b x + 2}} + \frac{127 b x^{\frac{5}{2}}}{24 \sqrt{b x + 2}} + \frac{133 x^{\frac{3}{2}}}{24 \sqrt{b x + 2}} + \frac{5 \sqrt{x}}{4 b \sqrt{b x + 2}} - \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 b^{\frac{3}{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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