Optimal. Leaf size=75 \[ \frac {1}{3 x^{3/2} (2-b x)^{3/2}}+\frac {1}{x^{3/2} \sqrt {2-b x}}-\frac {2 \sqrt {2-b x}}{3 x^{3/2}}-\frac {2 b \sqrt {2-b x}}{3 \sqrt {x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {1}{3 x^{3/2} (2-b x)^{3/2}}-\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)^{5/2}} \, dx &=\frac {1}{3 x^{3/2} (2-b x)^{3/2}}+\int \frac {1}{x^{5/2} (2-b x)^{3/2}} \, dx\\ &=\frac {1}{3 x^{3/2} (2-b x)^{3/2}}+\frac {1}{x^{3/2} \sqrt {2-b x}}+2 \int \frac {1}{x^{5/2} \sqrt {2-b x}} \, dx\\ &=\frac {1}{3 x^{3/2} (2-b x)^{3/2}}+\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}{3 x^{3/2} (2-b x)^{3/2}}+\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 = 0.09, size = 41, normalized size = 0.55 \begin {gather*} -\frac {1+3 b x-6 b^2 x^2+2 b^3 x^3}{3 x^{3/2} (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 = 7.28, size = 349, normalized size = 4.65 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\sqrt {b} \left (2+b x \left (5+10 b^2 x^2-2 b^3 x^3\right )-15 b^2 x^2\right ) \sqrt {\frac {2-b x}{b x}}}{3 x \left (-8+12 b x-6 b^2 x^2+b^3 x^3\right )},\frac {1}{\text {Abs}\left [b x\right ]}>\frac {1}{2}\right \}\right \},\frac {I 2 b^{\frac {19}{2}} \sqrt {1-\frac {2}{b x}}}{-24 b^9 x+36 b^{10} x^2-18 b^{11} x^3+3 b^{12} x^4}+\frac {I 5 b^{\frac {21}{2}} x \sqrt {1-\frac {2}{b x}}}{-24 b^9 x+36 b^{10} x^2-18 b^{11} x^3+3 b^{12} x^4}-\frac {15 I b^{\frac {23}{2}} x^2 \sqrt {1-\frac {2}{b x}}}{-24 b^9 x+36 b^{10} x^2-18 b^{11} x^3+3 b^{12} x^4}+\frac {I 10 b^{\frac {25}{2}} x^3 \sqrt {1-\frac {2}{b x}}}{-24 b^9 x+36 b^{10} x^2-18 b^{11} x^3+3 b^{12} x^4}-\frac {2 I b^{\frac {27}{2}} x^4 \sqrt {1-\frac {2}{b x}}}{-24 b^9 x+36 b^{10} x^2-18 b^{11} x^3+3 b^{12} x^4}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.12, size = 61, normalized size = 0.81
method | result | size |
gosper | \(-\frac {2 b^{3} x^{3}-6 x^{2} b^{2}+3 b x +1}{3 x^{\frac {3}{2}} \left (-b x +2\right )^{\frac {3}{2}}}\) | \(36\) |
meijerg | \(-\frac {\sqrt {2}\, \left (2 b^{3} x^{3}-6 x^{2} b^{2}+3 b x +1\right )}{12 x^{\frac {3}{2}} \left (-\frac {b x}{2}+1\right )^{\frac {3}{2}}}\) | \(39\) |
default | \(-\frac {1}{3 x^{\frac {3}{2}} \left (-b x +2\right )^{\frac {3}{2}}}+b \left (-\frac {1}{\left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}+2 b \left (\frac {\sqrt {x}}{3 \left (-b x +2\right )^{\frac {3}{2}}}+\frac {\sqrt {x}}{3 \sqrt {-b x +2}}\right )\right )\) | \(61\) |
risch | \(\frac {\left (4 x^{2} b^{2}-7 b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{12 x^{\frac {3}{2}} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {b^{2} \left (4 b x -9\right ) \sqrt {x}\, \sqrt {\left (-b x +2\right ) x}}{12 \sqrt {-x \left (b x -2\right )}\, \left (b x -2\right ) \sqrt {-b x +2}}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 58, normalized size = 0.77 \begin {gather*} -\frac {3 \, \sqrt {-b x + 2} b}{8 \, \sqrt {x}} + \frac {{\left (b^{3} - \frac {9 \, {\left (b x - 2\right )} b^{2}}{x}\right )} x^{\frac {3}{2}}}{24 \, {\left (-b x + 2\right )}^{\frac {3}{2}}} - \frac {{\left (-b x + 2\right )}^{\frac {3}{2}}}{24 \, x^{\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 = 56, normalized size = 0.75 \begin {gather*} -\frac {{\left (2 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 3 \, b x + 1\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{4} - 4 \, b x^{3} + 4 \, 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 = 5.78, size = 530, normalized size = 7.07 \begin {gather*} \begin {cases} - \frac {2 b^{\frac {27}{2}} x^{4} \sqrt {-1 + \frac {2}{b x}}}{3 b^{12} x^{4} - 18 b^{11} x^{3} + 36 b^{10} x^{2} - 24 b^{9} x} + \frac {10 b^{\frac {25}{2}} x^{3} \sqrt {-1 + \frac {2}{b x}}}{3 b^{12} x^{4} - 18 b^{11} x^{3} + 36 b^{10} x^{2} - 24 b^{9} x} - \frac {15 b^{\frac {23}{2}} x^{2} \sqrt {-1 + \frac {2}{b x}}}{3 b^{12} x^{4} - 18 b^{11} x^{3} + 36 b^{10} x^{2} - 24 b^{9} x} + \frac {5 b^{\frac {21}{2}} x \sqrt {-1 + \frac {2}{b x}}}{3 b^{12} x^{4} - 18 b^{11} x^{3} + 36 b^{10} x^{2} - 24 b^{9} x} + \frac {2 b^{\frac {19}{2}} \sqrt {-1 + \frac {2}{b x}}}{3 b^{12} x^{4} - 18 b^{11} x^{3} + 36 b^{10} x^{2} - 24 b^{9} x} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- \frac {2 i b^{\frac {27}{2}} x^{4} \sqrt {1 - \frac {2}{b x}}}{3 b^{12} x^{4} - 18 b^{11} x^{3} + 36 b^{10} x^{2} - 24 b^{9} x} + \frac {10 i b^{\frac {25}{2}} x^{3} \sqrt {1 - \frac {2}{b x}}}{3 b^{12} x^{4} - 18 b^{11} x^{3} + 36 b^{10} x^{2} - 24 b^{9} x} - \frac {15 i b^{\frac {23}{2}} x^{2} \sqrt {1 - \frac {2}{b x}}}{3 b^{12} x^{4} - 18 b^{11} x^{3} + 36 b^{10} x^{2} - 24 b^{9} x} + \frac {5 i b^{\frac {21}{2}} x \sqrt {1 - \frac {2}{b x}}}{3 b^{12} x^{4} - 18 b^{11} x^{3} + 36 b^{10} x^{2} - 24 b^{9} x} + \frac {2 i b^{\frac {19}{2}} \sqrt {1 - \frac {2}{b x}}}{3 b^{12} x^{4} - 18 b^{11} x^{3} + 36 b^{10} x^{2} - 24 b^{9} 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. 130 vs.
\(2 (53) = 106\).
time = 0.02, size = 177, normalized size = 2.36 \begin {gather*} 2 \left (\frac {2 \left (-\frac {\frac {1}{2304}\cdot 192 b^{4} \sqrt {x} \sqrt {x}}{b}+\frac {\frac {1}{2304}\cdot 432 b^{3}}{b}\right ) \sqrt {x} \sqrt {-b x+2}}{\left (-b x+2\right )^{2}}+\frac {2 \left (3 b \sqrt {-b} \left (\sqrt {-b x+2}-\sqrt {-b} \sqrt {x}\right )^{4}-18 b \sqrt {-b} \left (\sqrt {-b x+2}-\sqrt {-b} \sqrt {x}\right )^{2}+16 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 = 0.44, size = 73, normalized size = 0.97 \begin {gather*} \frac {\sqrt {2-b\,x}+3\,b\,x\,\sqrt {2-b\,x}-6\,b^2\,x^2\,\sqrt {2-b\,x}+2\,b^3\,x^3\,\sqrt {2-b\,x}}{x^{3/2}\,\left (x\,\left (12\,b-3\,b^2\,x\right )-12\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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