Optimal. Leaf size=71 \[ -\frac{2 \sqrt{b x+2}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{b x+2}}+\frac{1}{3 x^{3/2} (b x+2)^{3/2}}+\frac{2 b \sqrt{b x+2}}{3 \sqrt{x}} \]
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Rubi [A] time = 0.0089598, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{2 \sqrt{b x+2}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{b x+2}}+\frac{1}{3 x^{3/2} (b x+2)^{3/2}}+\frac{2 b \sqrt{b x+2}}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{x^{5/2} (2+b x)^{5/2}} \, dx &=\frac{1}{3 x^{3/2} (2+b x)^{3/2}}+\int \frac{1}{x^{5/2} (2+b x)^{3/2}} \, dx\\ &=\frac{1}{3 x^{3/2} (2+b x)^{3/2}}+\frac{1}{x^{3/2} \sqrt{2+b x}}+2 \int \frac{1}{x^{5/2} \sqrt{2+b x}} \, dx\\ &=\frac{1}{3 x^{3/2} (2+b x)^{3/2}}+\frac{1}{x^{3/2} \sqrt{2+b x}}-\frac{2 \sqrt{2+b x}}{3 x^{3/2}}-\frac{1}{3} (2 b) \int \frac{1}{x^{3/2} \sqrt{2+b x}} \, dx\\ &=\frac{1}{3 x^{3/2} (2+b x)^{3/2}}+\frac{1}{x^{3/2} \sqrt{2+b x}}-\frac{2 \sqrt{2+b x}}{3 x^{3/2}}+\frac{2 b \sqrt{2+b x}}{3 \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0109285, size = 40, normalized size = 0.56 \[ \frac{2 b^3 x^3+6 b^2 x^2+3 b x-1}{3 x^{3/2} (b x+2)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 35, normalized size = 0.5 \begin{align*}{\frac{2\,{b}^{3}{x}^{3}+6\,{b}^{2}{x}^{2}+3\,bx-1}{3}{x}^{-{\frac{3}{2}}} \left ( bx+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977377, size = 74, normalized size = 1.04 \begin{align*} \frac{3 \, \sqrt{b x + 2} b}{8 \, \sqrt{x}} - \frac{{\left (b^{3} - \frac{9 \,{\left (b x + 2\right )} b^{2}}{x}\right )} x^{\frac{3}{2}}}{24 \,{\left (b x + 2\right )}^{\frac{3}{2}}} - \frac{{\left (b x + 2\right )}^{\frac{3}{2}}}{24 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51536, size = 123, normalized size = 1.73 \begin{align*} \frac{{\left (2 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 3 \, b x - 1\right )} \sqrt{b x + 2} \sqrt{x}}{3 \,{\left (b^{2} x^{4} + 4 \, b x^{3} + 4 \, x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 31.4723, size = 257, normalized size = 3.62 \begin{align*} \frac{2 b^{\frac{27}{2}} x^{4} \sqrt{1 + \frac{2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} + \frac{10 b^{\frac{25}{2}} x^{3} \sqrt{1 + \frac{2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} + \frac{15 b^{\frac{23}{2}} x^{2} \sqrt{1 + \frac{2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} + \frac{5 b^{\frac{21}{2}} x \sqrt{1 + \frac{2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} - \frac{2 b^{\frac{19}{2}} \sqrt{1 + \frac{2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13935, size = 213, normalized size = 3. \begin{align*} \frac{{\left (4 \,{\left (b x + 2\right )} b^{2}{\left | b \right |} - 9 \, b^{2}{\left | b \right |}\right )} \sqrt{b x + 2}}{12 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{3}{2}}} + \frac{3 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} b^{\frac{7}{2}} + 18 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} b^{\frac{9}{2}} + 16 \, b^{\frac{11}{2}}}{3 \,{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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