Optimal. Leaf size=126 \[ -\frac{x^{3/2} \sqrt{b x+2}}{8 b^2}+\frac{3 \sqrt{x} \sqrt{b x+2}}{8 b^3}-\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}+\frac{1}{5} x^{7/2} (b x+2)^{3/2}+\frac{3}{20} x^{7/2} \sqrt{b x+2}+\frac{x^{5/2} \sqrt{b x+2}}{20 b} \]
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Rubi [A] time = 0.034079, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {50, 54, 215} \[ -\frac{x^{3/2} \sqrt{b x+2}}{8 b^2}+\frac{3 \sqrt{x} \sqrt{b x+2}}{8 b^3}-\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}+\frac{1}{5} x^{7/2} (b x+2)^{3/2}+\frac{3}{20} x^{7/2} \sqrt{b x+2}+\frac{x^{5/2} \sqrt{b x+2}}{20 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int x^{5/2} (2+b x)^{3/2} \, dx &=\frac{1}{5} x^{7/2} (2+b x)^{3/2}+\frac{3}{5} \int x^{5/2} \sqrt{2+b x} \, dx\\ &=\frac{3}{20} x^{7/2} \sqrt{2+b x}+\frac{1}{5} x^{7/2} (2+b x)^{3/2}+\frac{3}{20} \int \frac{x^{5/2}}{\sqrt{2+b x}} \, dx\\ &=\frac{x^{5/2} \sqrt{2+b x}}{20 b}+\frac{3}{20} x^{7/2} \sqrt{2+b x}+\frac{1}{5} x^{7/2} (2+b x)^{3/2}-\frac{\int \frac{x^{3/2}}{\sqrt{2+b x}} \, dx}{4 b}\\ &=-\frac{x^{3/2} \sqrt{2+b x}}{8 b^2}+\frac{x^{5/2} \sqrt{2+b x}}{20 b}+\frac{3}{20} x^{7/2} \sqrt{2+b x}+\frac{1}{5} x^{7/2} (2+b x)^{3/2}+\frac{3 \int \frac{\sqrt{x}}{\sqrt{2+b x}} \, dx}{8 b^2}\\ &=\frac{3 \sqrt{x} \sqrt{2+b x}}{8 b^3}-\frac{x^{3/2} \sqrt{2+b x}}{8 b^2}+\frac{x^{5/2} \sqrt{2+b x}}{20 b}+\frac{3}{20} x^{7/2} \sqrt{2+b x}+\frac{1}{5} x^{7/2} (2+b x)^{3/2}-\frac{3 \int \frac{1}{\sqrt{x} \sqrt{2+b x}} \, dx}{8 b^3}\\ &=\frac{3 \sqrt{x} \sqrt{2+b x}}{8 b^3}-\frac{x^{3/2} \sqrt{2+b x}}{8 b^2}+\frac{x^{5/2} \sqrt{2+b x}}{20 b}+\frac{3}{20} x^{7/2} \sqrt{2+b x}+\frac{1}{5} x^{7/2} (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^3}\\ &=\frac{3 \sqrt{x} \sqrt{2+b x}}{8 b^3}-\frac{x^{3/2} \sqrt{2+b x}}{8 b^2}+\frac{x^{5/2} \sqrt{2+b x}}{20 b}+\frac{3}{20} x^{7/2} \sqrt{2+b x}+\frac{1}{5} x^{7/2} (2+b x)^{3/2}-\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0479891, size = 78, normalized size = 0.62 \[ \frac{\sqrt{x} \sqrt{b x+2} \left (8 b^4 x^4+22 b^3 x^3+2 b^2 x^2-5 b x+15\right )}{40 b^3}-\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 123, normalized size = 1. \begin{align*}{\frac{1}{5\,b}{x}^{{\frac{5}{2}}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}}-{\frac{1}{4\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}}+{\frac{1}{4\,{b}^{3}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}-{\frac{1}{8\,{b}^{3}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}-{\frac{3}{8\,{b}^{3}}\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{7}{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 = 1.8483, size = 398, normalized size = 3.16 \begin{align*} \left [\frac{{\left (8 \, b^{5} x^{4} + 22 \, b^{4} x^{3} + 2 \, b^{3} x^{2} - 5 \, 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 )}{40 \, b^{4}}, \frac{{\left (8 \, b^{5} x^{4} + 22 \, b^{4} x^{3} + 2 \, b^{3} x^{2} - 5 \, 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 )}{40 \, b^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.5666, size = 136, normalized size = 1.08 \begin{align*} \frac{b^{2} x^{\frac{11}{2}}}{5 \sqrt{b x + 2}} + \frac{19 b x^{\frac{9}{2}}}{20 \sqrt{b x + 2}} + \frac{23 x^{\frac{7}{2}}}{20 \sqrt{b x + 2}} - \frac{x^{\frac{5}{2}}}{40 b \sqrt{b x + 2}} + \frac{x^{\frac{3}{2}}}{8 b^{2} \sqrt{b x + 2}} + \frac{3 \sqrt{x}}{4 b^{3} \sqrt{b x + 2}} - \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 b^{\frac{7}{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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