Optimal. Leaf size=56 \[ \frac {1}{x^{3/2} \sqrt {2-b x}}-\frac {2 \sqrt {2-b x}}{3 x^{3/2}}-\frac {2 b \sqrt {2-b x}}{3 \sqrt {x}} \]
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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 = {47, 37}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
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 = 0.08, size = 33, normalized size = 0.59 \begin {gather*} \frac {-1-2 b x+2 b^2 x^2}{3 x^{3/2} \sqrt {2-b x}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 4.90, size = 245, normalized size = 4.38 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\sqrt {b} \left (-2+3 b x \left (-1+2 b x\right )-2 b^3 x^3\right ) \sqrt {\frac {2-b x}{b x}}}{3 x \left (4-4 b x+b^2 x^2\right )},\frac {1}{\text {Abs}\left [b x\right ]}>\frac {1}{2}\right \}\right \},\frac {-2 I b^{\frac {9}{2}} \sqrt {1-\frac {2}{b x}}}{12 b^4 x-12 b^5 x^2+3 b^6 x^3}-\frac {3 I b^{\frac {11}{2}} x \sqrt {1-\frac {2}{b x}}}{12 b^4 x-12 b^5 x^2+3 b^6 x^3}+\frac {I 6 b^{\frac {13}{2}} x^2 \sqrt {1-\frac {2}{b x}}}{12 b^4 x-12 b^5 x^2+3 b^6 x^3}-\frac {2 I b^{\frac {15}{2}} x^3 \sqrt {1-\frac {2}{b x}}}{12 b^4 x-12 b^5 x^2+3 b^6 x^3}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 45, normalized size = 0.80
method | result | size |
gosper | \(\frac {2 x^{2} b^{2}-2 b x -1}{3 x^{\frac {3}{2}} \sqrt {-b x +2}}\) | \(28\) |
meijerg | \(-\frac {\sqrt {2}\, \left (-2 x^{2} b^{2}+2 b x +1\right )}{6 x^{\frac {3}{2}} \sqrt {-\frac {b x}{2}+1}}\) | \(31\) |
default | \(-\frac {1}{3 x^{\frac {3}{2}} \sqrt {-b x +2}}+\frac {2 b \left (-\frac {1}{\sqrt {x}\, \sqrt {-b x +2}}+\frac {b \sqrt {x}}{\sqrt {-b x +2}}\right )}{3}\) | \(45\) |
risch | \(\frac {\left (5 x^{2} b^{2}-8 b x -4\right ) \sqrt {\left (-b x +2\right ) x}}{12 x^{\frac {3}{2}} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {b^{2} \sqrt {x}\, \sqrt {\left (-b x +2\right ) x}}{4 \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 44, normalized size = 0.79 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 40, normalized size = 0.71 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.63, size = 355, normalized size = 6.34 \begin {gather*} \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 {1}{\left |{b x}\right |} > \frac {1}{2} \\- \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (40) = 80\).
time = 0.01, size = 151, normalized size = 2.70 \begin {gather*} -2 \left (-\frac {2 b^{2} \sqrt {x} \sqrt {-b x+2}}{16 \left (-b x+2\right )}+\frac {2 \left (-3 b \sqrt {-b} \left (\sqrt {-b x+2}-\sqrt {-b} \sqrt {x}\right )^{4}+24 b \sqrt {-b} \left (\sqrt {-b x+2}-\sqrt {-b} \sqrt {x}\right )^{2}-20 b \sqrt {-b}\right )}{12 \left (\left (\sqrt {-b x+2}-\sqrt {-b} \sqrt {x}\right )^{2}-2\right )^{3}}\right ) \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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