Optimal. Leaf size=144 \[ -\frac{5 x^{3/2} \sqrt{b x+2}}{48 b^2}+\frac{5 \sqrt{x} \sqrt{b x+2}}{16 b^3}-\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{8 b^{7/2}}+\frac{1}{6} x^{7/2} (b x+2)^{5/2}+\frac{1}{6} x^{7/2} (b x+2)^{3/2}+\frac{1}{8} x^{7/2} \sqrt{b x+2}+\frac{x^{5/2} \sqrt{b x+2}}{24 b} \]
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Rubi [A] time = 0.0450716, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {50, 54, 215} \[ -\frac{5 x^{3/2} \sqrt{b x+2}}{48 b^2}+\frac{5 \sqrt{x} \sqrt{b x+2}}{16 b^3}-\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{8 b^{7/2}}+\frac{1}{6} x^{7/2} (b x+2)^{5/2}+\frac{1}{6} x^{7/2} (b x+2)^{3/2}+\frac{1}{8} x^{7/2} \sqrt{b x+2}+\frac{x^{5/2} \sqrt{b x+2}}{24 b} \]
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)^{5/2} \, dx &=\frac{1}{6} x^{7/2} (2+b x)^{5/2}+\frac{5}{6} \int x^{5/2} (2+b x)^{3/2} \, dx\\ &=\frac{1}{6} x^{7/2} (2+b x)^{3/2}+\frac{1}{6} x^{7/2} (2+b x)^{5/2}+\frac{1}{2} \int x^{5/2} \sqrt{2+b x} \, dx\\ &=\frac{1}{8} x^{7/2} \sqrt{2+b x}+\frac{1}{6} x^{7/2} (2+b x)^{3/2}+\frac{1}{6} x^{7/2} (2+b x)^{5/2}+\frac{1}{8} \int \frac{x^{5/2}}{\sqrt{2+b x}} \, dx\\ &=\frac{x^{5/2} \sqrt{2+b x}}{24 b}+\frac{1}{8} x^{7/2} \sqrt{2+b x}+\frac{1}{6} x^{7/2} (2+b x)^{3/2}+\frac{1}{6} x^{7/2} (2+b x)^{5/2}-\frac{5 \int \frac{x^{3/2}}{\sqrt{2+b x}} \, dx}{24 b}\\ &=-\frac{5 x^{3/2} \sqrt{2+b x}}{48 b^2}+\frac{x^{5/2} \sqrt{2+b x}}{24 b}+\frac{1}{8} x^{7/2} \sqrt{2+b x}+\frac{1}{6} x^{7/2} (2+b x)^{3/2}+\frac{1}{6} x^{7/2} (2+b x)^{5/2}+\frac{5 \int \frac{\sqrt{x}}{\sqrt{2+b x}} \, dx}{16 b^2}\\ &=\frac{5 \sqrt{x} \sqrt{2+b x}}{16 b^3}-\frac{5 x^{3/2} \sqrt{2+b x}}{48 b^2}+\frac{x^{5/2} \sqrt{2+b x}}{24 b}+\frac{1}{8} x^{7/2} \sqrt{2+b x}+\frac{1}{6} x^{7/2} (2+b x)^{3/2}+\frac{1}{6} x^{7/2} (2+b x)^{5/2}-\frac{5 \int \frac{1}{\sqrt{x} \sqrt{2+b x}} \, dx}{16 b^3}\\ &=\frac{5 \sqrt{x} \sqrt{2+b x}}{16 b^3}-\frac{5 x^{3/2} \sqrt{2+b x}}{48 b^2}+\frac{x^{5/2} \sqrt{2+b x}}{24 b}+\frac{1}{8} x^{7/2} \sqrt{2+b x}+\frac{1}{6} x^{7/2} (2+b x)^{3/2}+\frac{1}{6} x^{7/2} (2+b x)^{5/2}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+b x^2}} \, dx,x,\sqrt{x}\right )}{8 b^3}\\ &=\frac{5 \sqrt{x} \sqrt{2+b x}}{16 b^3}-\frac{5 x^{3/2} \sqrt{2+b x}}{48 b^2}+\frac{x^{5/2} \sqrt{2+b x}}{24 b}+\frac{1}{8} x^{7/2} \sqrt{2+b x}+\frac{1}{6} x^{7/2} (2+b x)^{3/2}+\frac{1}{6} x^{7/2} (2+b x)^{5/2}-\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{8 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0562539, size = 86, normalized size = 0.6 \[ \frac{\sqrt{x} \sqrt{b x+2} \left (8 b^5 x^5+40 b^4 x^4+54 b^3 x^3+2 b^2 x^2-5 b x+15\right )}{48 b^3}-\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{8 b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 138, normalized size = 1. \begin{align*}{\frac{1}{6\,b}{x}^{{\frac{5}{2}}} \left ( bx+2 \right ) ^{{\frac{7}{2}}}}-{\frac{1}{6\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{7}{2}}}}+{\frac{1}{8\,{b}^{3}}\sqrt{x} \left ( bx+2 \right ) ^{{\frac{7}{2}}}}-{\frac{1}{24\,{b}^{3}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}-{\frac{5}{48\,{b}^{3}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}-{\frac{5}{16\,{b}^{3}}\sqrt{x}\sqrt{bx+2}}-{\frac{5}{16}\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 = 1.90134, size = 433, normalized size = 3.01 \begin{align*} \left [\frac{{\left (8 \, b^{6} x^{5} + 40 \, b^{5} x^{4} + 54 \, 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 )}{48 \, b^{4}}, \frac{{\left (8 \, b^{6} x^{5} + 40 \, b^{5} x^{4} + 54 \, 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 )}{48 \, b^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 67.264, size = 158, normalized size = 1.1 \begin{align*} \frac{b^{3} x^{\frac{13}{2}}}{6 \sqrt{b x + 2}} + \frac{7 b^{2} x^{\frac{11}{2}}}{6 \sqrt{b x + 2}} + \frac{67 b x^{\frac{9}{2}}}{24 \sqrt{b x + 2}} + \frac{55 x^{\frac{7}{2}}}{24 \sqrt{b x + 2}} - \frac{x^{\frac{5}{2}}}{48 b \sqrt{b x + 2}} + \frac{5 x^{\frac{3}{2}}}{48 b^{2} \sqrt{b x + 2}} + \frac{5 \sqrt{x}}{8 b^{3} \sqrt{b x + 2}} - \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{8 b^{\frac{7}{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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