Optimal. Leaf size=39 \[ \frac {\sqrt {x}}{3 (2-b x)^{3/2}}+\frac {\sqrt {x}}{3 \sqrt {2-b x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 2, 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 {\sqrt {x}}{3 \sqrt {2-b x}}+\frac {\sqrt {x}}{3 (2-b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (2-b x)^{5/2}} \, dx &=\frac {\sqrt {x}}{3 (2-b x)^{3/2}}+\frac {1}{3} \int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx\\ &=\frac {\sqrt {x}}{3 (2-b x)^{3/2}}+\frac {\sqrt {x}}{3 \sqrt {2-b x}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 24, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {x} (-3+b x)}{3 (2-b x)^{3/2}} \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 = 3.48, size = 136, normalized size = 3.49 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\sqrt {b} x \left (3-b x\right ) \sqrt {\frac {2-b x}{b x}}}{3 \left (4-4 b x+b^2 x^2\right )},\frac {1}{\text {Abs}\left [b x\right ]}>\frac {1}{2}\right \}\right \},-\frac {I b x}{-6 \sqrt {b} \sqrt {1-\frac {2}{b x}}+3 b^{\frac {3}{2}} x \sqrt {1-\frac {2}{b x}}}+\frac {I 3}{-6 \sqrt {b} \sqrt {1-\frac {2}{b x}}+3 b^{\frac {3}{2}} x \sqrt {1-\frac {2}{b x}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.11, size = 28, normalized size = 0.72
method | result | size |
gosper | \(-\frac {\sqrt {x}\, \left (b x -3\right )}{3 \left (-b x +2\right )^{\frac {3}{2}}}\) | \(19\) |
meijerg | \(\frac {\sqrt {x}\, \sqrt {2}\, \left (-b x +3\right )}{12 \left (-\frac {b x}{2}+1\right )^{\frac {3}{2}}}\) | \(23\) |
default | \(\frac {\sqrt {x}}{3 \left (-b x +2\right )^{\frac {3}{2}}}+\frac {\sqrt {x}}{3 \sqrt {-b x +2}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 25, normalized size = 0.64 \begin {gather*} \frac {{\left (b - \frac {3 \, {\left (b x - 2\right )}}{x}\right )} x^{\frac {3}{2}}}{6 \, {\left (-b x + 2\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 33, normalized size = 0.85 \begin {gather*} -\frac {{\left (b x - 3\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{2} - 4 \, b x + 4\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 = 1.11, size = 165, normalized size = 4.23 \begin {gather*} \begin {cases} \frac {b^{2} x}{3 b^{\frac {5}{2}} x \sqrt {-1 + \frac {2}{b x}} - 6 b^{\frac {3}{2}} \sqrt {-1 + \frac {2}{b x}}} - \frac {3 b}{3 b^{\frac {5}{2}} x \sqrt {-1 + \frac {2}{b x}} - 6 b^{\frac {3}{2}} \sqrt {-1 + \frac {2}{b x}}} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- \frac {i b x}{3 b^{\frac {3}{2}} x \sqrt {1 - \frac {2}{b x}} - 6 \sqrt {b} \sqrt {1 - \frac {2}{b x}}} + \frac {3 i}{3 b^{\frac {3}{2}} x \sqrt {1 - \frac {2}{b x}} - 6 \sqrt {b} \sqrt {1 - \frac {2}{b 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. 90 vs.
\(2 (27) = 54\).
time = 0.01, size = 109, normalized size = 2.79 \begin {gather*} -\frac {32 b \sqrt {-b} b \left (-3 \left (\sqrt {-b \left (-b x+2\right )+2 b}-\sqrt {-b} \sqrt {-b x+2}\right )^{2}+2 b\right )}{2\cdot 6 \left |b\right | \left (\left (\sqrt {-b \left (-b x+2\right )+2 b}-\sqrt {-b} \sqrt {-b x+2}\right )^{2}-2 b\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 45, normalized size = 1.15 \begin {gather*} \frac {3\,\sqrt {x}\,\sqrt {2-b\,x}-b\,x^{3/2}\,\sqrt {2-b\,x}}{3\,b^2\,x^2-12\,b\,x+12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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