Optimal. Leaf size=56 \[ -\frac {2 \sqrt {2-b x}}{3 x^{3/2}}+\frac {1}{x^{3/2} \sqrt {2-b x}}-\frac {2 b \sqrt {2-b x}}{3 \sqrt {x}} \]
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Rubi [A] time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {45, 37} \[ -\frac {2 \sqrt {2-b x}}{3 x^{3/2}}+\frac {1}{x^{3/2} \sqrt {2-b x}}-\frac {2 b \sqrt {2-b x}}{3 \sqrt {x}} \]
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 = 33, normalized size = 0.59 \[ \frac {2 b^2 x^2-2 b x-1}{3 x^{3/2} \sqrt {2-b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 40, normalized size = 0.71 \[ -\frac {{\left (2 \, b^{2} x^{2} - 2 \, b x - 1\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, {\left (b x^{3} - 2 \, x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.17, size = 96, normalized size = 1.71 \[ -\frac {\sqrt {-b} b^{3}}{{\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} {\left | b \right |}} - \frac {{\left (5 \, {\left (b x - 2\right )} b^{2} {\left | b \right |} + 12 \, b^{2} {\left | b \right |}\right )} \sqrt {-b x + 2}}{12 \, {\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 0.50 \[ \frac {2 b^{2} x^{2}-2 b x -1}{3 \sqrt {-b x +2}\, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 44, normalized size = 0.79 \[ \frac {b^{2} \sqrt {x}}{4 \, \sqrt {-b x + 2}} - \frac {\sqrt {-b x + 2} b}{2 \, \sqrt {x}} - \frac {{\left (-b x + 2\right )}^{\frac {3}{2}}}{12 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 38, normalized size = 0.68 \[ \frac {\sqrt {2-b\,x}\,\left (\frac {2\,x}{3}-\frac {2\,b\,x^2}{3}+\frac {1}{3\,b}\right )}{x^{5/2}-\frac {2\,x^{3/2}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.29, size = 354, normalized size = 6.32 \[ \begin {cases} \frac {2 b^{\frac {15}{2}} x^{3} \sqrt {-1 + \frac {2}{b x}}}{- 3 b^{6} x^{3} + 12 b^{5} x^{2} - 12 b^{4} x} - \frac {6 b^{\frac {13}{2}} x^{2} \sqrt {-1 + \frac {2}{b x}}}{- 3 b^{6} x^{3} + 12 b^{5} x^{2} - 12 b^{4} x} + \frac {3 b^{\frac {11}{2}} x \sqrt {-1 + \frac {2}{b x}}}{- 3 b^{6} x^{3} + 12 b^{5} x^{2} - 12 b^{4} x} + \frac {2 b^{\frac {9}{2}} \sqrt {-1 + \frac {2}{b x}}}{- 3 b^{6} x^{3} + 12 b^{5} x^{2} - 12 b^{4} x} & \text {for}\: \frac {2}{\left |{b x}\right |} > 1 \\\frac {2 i b^{\frac {15}{2}} x^{3} \sqrt {1 - \frac {2}{b x}}}{- 3 b^{6} x^{3} + 12 b^{5} x^{2} - 12 b^{4} x} - \frac {6 i b^{\frac {13}{2}} x^{2} \sqrt {1 - \frac {2}{b x}}}{- 3 b^{6} x^{3} + 12 b^{5} x^{2} - 12 b^{4} x} + \frac {3 i b^{\frac {11}{2}} x \sqrt {1 - \frac {2}{b x}}}{- 3 b^{6} x^{3} + 12 b^{5} x^{2} - 12 b^{4} x} + \frac {2 i b^{\frac {9}{2}} \sqrt {1 - \frac {2}{b x}}}{- 3 b^{6} x^{3} + 12 b^{5} x^{2} - 12 b^{4} x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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