Optimal. Leaf size=53 \[ -\frac {2 \sqrt {b x+2}}{3 x^{3/2}}+\frac {1}{x^{3/2} \sqrt {b x+2}}+\frac {2 b \sqrt {b x+2}}{3 \sqrt {x}} \]
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Rubi [A] time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac {2 \sqrt {b x+2}}{3 x^{3/2}}+\frac {1}{x^{3/2} \sqrt {b x+2}}+\frac {2 b \sqrt {b x+2}}{3 \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} (2+b x)^{3/2}} \, dx &=\frac {1}{x^{3/2} \sqrt {2+b x}}+2 \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx\\ &=\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}{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*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.60 \[ \frac {2 b^2 x^2+2 b x-1}{3 x^{3/2} \sqrt {b x+2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 39, normalized size = 0.74 \[ \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x - 1\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b x^{3} + 2 \, x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.23, size = 86, normalized size = 1.62 \[ \frac {b^{\frac {7}{2}}}{{\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} {\left | b \right |}} + \frac {{\left (5 \, {\left (b x + 2\right )} b^{2} {\left | b \right |} - 12 \, b^{2} {\left | b \right |}\right )} \sqrt {b x + 2}}{12 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.51 \[ \frac {2 b^{2} x^{2}+2 b x -1}{3 \sqrt {b x +2}\, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 41, normalized size = 0.77 \[ \frac {b^{2} \sqrt {x}}{4 \, \sqrt {b x + 2}} + \frac {\sqrt {b x + 2} b}{2 \, \sqrt {x}} - \frac {{\left (b x + 2\right )}^{\frac {3}{2}}}{12 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 37, normalized size = 0.70 \[ \frac {\sqrt {b\,x+2}\,\left (\frac {2\,x}{3}+\frac {2\,b\,x^2}{3}-\frac {1}{3\,b}\right )}{x^{5/2}+\frac {2\,x^{3/2}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.86, size = 170, normalized size = 3.21 \[ \frac {2 b^{\frac {15}{2}} x^{3} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} + \frac {6 b^{\frac {13}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} + \frac {3 b^{\frac {11}{2}} x \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} - \frac {2 b^{\frac {9}{2}} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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