Optimal. Leaf size=62 \[ -\frac {2 b^2 (2-b x)^{3/2}}{105 x^{3/2}}-\frac {2 b (2-b x)^{3/2}}{35 x^{5/2}}-\frac {(2-b x)^{3/2}}{7 x^{7/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 62, 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} \[ -\frac {2 b^2 (2-b x)^{3/2}}{105 x^{3/2}}-\frac {2 b (2-b x)^{3/2}}{35 x^{5/2}}-\frac {(2-b x)^{3/2}}{7 x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {\sqrt {2-b x}}{x^{9/2}} \, dx &=-\frac {(2-b x)^{3/2}}{7 x^{7/2}}+\frac {1}{7} (2 b) \int \frac {\sqrt {2-b x}}{x^{7/2}} \, dx\\ &=-\frac {(2-b x)^{3/2}}{7 x^{7/2}}-\frac {2 b (2-b x)^{3/2}}{35 x^{5/2}}+\frac {1}{35} \left (2 b^2\right ) \int \frac {\sqrt {2-b x}}{x^{5/2}} \, dx\\ &=-\frac {(2-b x)^{3/2}}{7 x^{7/2}}-\frac {2 b (2-b x)^{3/2}}{35 x^{5/2}}-\frac {2 b^2 (2-b x)^{3/2}}{105 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.53 \[ -\frac {(2-b x)^{3/2} \left (2 b^2 x^2+6 b x+15\right )}{105 x^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 35, normalized size = 0.56 \[ \frac {{\left (2 \, b^{3} x^{3} + 2 \, b^{2} x^{2} + 3 \, b x - 30\right )} \sqrt {-b x + 2}}{105 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 61, normalized size = 0.98 \[ \frac {{\left (35 \, b^{7} + 2 \, {\left ({\left (b x - 2\right )} b^{7} + 7 \, b^{7}\right )} {\left (b x - 2\right )}\right )} {\left (b x - 2\right )} \sqrt {-b x + 2} b}{105 \, {\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac {7}{2}} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.45 \[ -\frac {\left (2 b^{2} x^{2}+6 b x +15\right ) \left (-b x +2\right )^{\frac {3}{2}}}{105 x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 44, normalized size = 0.71 \[ -\frac {{\left (-b x + 2\right )}^{\frac {3}{2}} b^{2}}{12 \, x^{\frac {3}{2}}} - \frac {{\left (-b x + 2\right )}^{\frac {5}{2}} b}{10 \, x^{\frac {5}{2}}} - \frac {{\left (-b x + 2\right )}^{\frac {7}{2}}}{28 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 34, normalized size = 0.55 \[ \frac {\sqrt {2-b\,x}\,\left (\frac {2\,b^3\,x^3}{105}+\frac {2\,b^2\,x^2}{105}+\frac {b\,x}{35}-\frac {2}{7}\right )}{x^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 24.60, size = 554, normalized size = 8.94 \[ \begin {cases} - \frac {2 b^{\frac {19}{2}} x^{5} \sqrt {-1 + \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} + \frac {6 b^{\frac {17}{2}} x^{4} \sqrt {-1 + \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} - \frac {3 b^{\frac {15}{2}} x^{3} \sqrt {-1 + \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} + \frac {34 b^{\frac {13}{2}} x^{2} \sqrt {-1 + \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} - \frac {132 b^{\frac {11}{2}} x \sqrt {-1 + \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} + \frac {120 b^{\frac {9}{2}} \sqrt {-1 + \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} & \text {for}\: \frac {2}{\left |{b x}\right |} > 1 \\- \frac {2 i b^{\frac {19}{2}} x^{5} \sqrt {1 - \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} + \frac {6 i b^{\frac {17}{2}} x^{4} \sqrt {1 - \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} - \frac {3 i b^{\frac {15}{2}} x^{3} \sqrt {1 - \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} + \frac {34 i b^{\frac {13}{2}} x^{2} \sqrt {1 - \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} - \frac {132 i b^{\frac {11}{2}} x \sqrt {1 - \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} + \frac {120 i b^{\frac {9}{2}} \sqrt {1 - \frac {2}{b x}}}{- 105 b^{6} x^{5} + 420 b^{5} x^{4} - 420 b^{4} x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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