Optimal. Leaf size=79 \[ \frac{1}{3} \sqrt{x} (b x+2)^{5/2}+\frac{5}{6} \sqrt{x} (b x+2)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{b x+2}+\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0159358, antiderivative size = 79, 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} \[ \frac{1}{3} \sqrt{x} (b x+2)^{5/2}+\frac{5}{6} \sqrt{x} (b x+2)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{b x+2}+\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{(2+b x)^{5/2}}{\sqrt{x}} \, dx &=\frac{1}{3} \sqrt{x} (2+b x)^{5/2}+\frac{5}{3} \int \frac{(2+b x)^{3/2}}{\sqrt{x}} \, dx\\ &=\frac{5}{6} \sqrt{x} (2+b x)^{3/2}+\frac{1}{3} \sqrt{x} (2+b x)^{5/2}+\frac{5}{2} \int \frac{\sqrt{2+b x}}{\sqrt{x}} \, dx\\ &=\frac{5}{2} \sqrt{x} \sqrt{2+b x}+\frac{5}{6} \sqrt{x} (2+b x)^{3/2}+\frac{1}{3} \sqrt{x} (2+b x)^{5/2}+\frac{5}{2} \int \frac{1}{\sqrt{x} \sqrt{2+b x}} \, dx\\ &=\frac{5}{2} \sqrt{x} \sqrt{2+b x}+\frac{5}{6} \sqrt{x} (2+b x)^{3/2}+\frac{1}{3} \sqrt{x} (2+b x)^{5/2}+5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+b x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{5}{2} \sqrt{x} \sqrt{2+b x}+\frac{5}{6} \sqrt{x} (2+b x)^{3/2}+\frac{1}{3} \sqrt{x} (2+b x)^{5/2}+\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0303438, size = 57, normalized size = 0.72 \[ \frac{1}{6} \sqrt{x} \sqrt{b x+2} \left (2 b^2 x^2+13 b x+33\right )+\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 84, normalized size = 1.1 \begin{align*}{\frac{1}{3} \left ( bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{5}{6} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5}{2}\sqrt{x}\sqrt{bx+2}}+{\frac{5}{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 = 1.75174, size = 325, normalized size = 4.11 \begin{align*} \left [\frac{{\left (2 \, b^{3} x^{2} + 13 \, b^{2} x + 33 \, 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 )}{6 \, b}, \frac{{\left (2 \, b^{3} x^{2} + 13 \, b^{2} x + 33 \, b\right )} \sqrt{b x + 2} \sqrt{x} - 30 \, \sqrt{-b} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{6 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.1798, size = 97, normalized size = 1.23 \begin{align*} \frac{b^{3} x^{\frac{7}{2}}}{3 \sqrt{b x + 2}} + \frac{17 b^{2} x^{\frac{5}{2}}}{6 \sqrt{b x + 2}} + \frac{59 b x^{\frac{3}{2}}}{6 \sqrt{b x + 2}} + \frac{11 \sqrt{x}}{\sqrt{b x + 2}} + \frac{5 \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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