Optimal. Leaf size=82 \[ -\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-15 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {49, 52, 56, 222}
\begin {gather*} -\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-15 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {(2-b x)^{5/2}}{x^{3/2}} \, dx &=-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-(5 b) \int \frac {(2-b x)^{3/2}}{\sqrt {x}} \, dx\\ &=-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-\frac {1}{2} (15 b) \int \frac {\sqrt {2-b x}}{\sqrt {x}} \, dx\\ &=-\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-\frac {1}{2} (15 b) \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx\\ &=-\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-(15 b) \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-15 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 68, normalized size = 0.83 \begin {gather*} \frac {\sqrt {2-b x} \left (-16-9 b x+b^2 x^2\right )}{2 \sqrt {x}}-15 \sqrt {-b} \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 6.99, size = 156, normalized size = 1.90 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (30 \sqrt {b} \sqrt {x} \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ] \left (-2+b x\right )+b x \left (2-11 b x+b^2 x^2\right ) \sqrt {-2+b x}+32 \sqrt {-2+b x}\right )}{2 \sqrt {x} \left (-2+b x\right )},\text {Abs}\left [b x\right ]>2\right \}\right \},-15 \sqrt {b} \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ]-\frac {16}{\sqrt {x} \sqrt {2-b x}}-\frac {b \sqrt {x}}{\sqrt {2-b x}}+\frac {11 b^2 x^{\frac {3}{2}}}{2 \sqrt {2-b x}}-\frac {b^3 x^{\frac {5}{2}}}{2 \sqrt {2-b x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.12, size = 78, normalized size = 0.95
method | result | size |
meijerg | \(\frac {15 \left (-b \right )^{\frac {3}{2}} \left (\frac {16 \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {1}{16} x^{2} b^{2}+\frac {9}{16} b x +1\right ) \sqrt {-\frac {b x}{2}+1}}{15 \sqrt {x}\, \sqrt {-b}}+\frac {2 \sqrt {\pi }\, \sqrt {b}\, \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {-b}}\right )}{2 \sqrt {\pi }\, b}\) | \(78\) |
risch | \(-\frac {\left (b^{3} x^{3}-11 x^{2} b^{2}+2 b x +32\right ) \sqrt {\left (-b x +2\right ) x}}{2 \sqrt {-x \left (b x -2\right )}\, \sqrt {x}\, \sqrt {-b x +2}}-\frac {15 \sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{2 \sqrt {x}\, \sqrt {-b x +2}}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 96, normalized size = 1.17 \begin {gather*} 15 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - \frac {\frac {7 \, \sqrt {-b x + 2} b^{2}}{\sqrt {x}} + \frac {9 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}}}{b^{2} - \frac {2 \, {\left (b x - 2\right )} b}{x} + \frac {{\left (b x - 2\right )}^{2}}{x^{2}}} - \frac {8 \, \sqrt {-b x + 2}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 117, normalized size = 1.43 \begin {gather*} \left [\frac {15 \, \sqrt {-b} x \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) + {\left (b^{2} x^{2} - 9 \, b x - 16\right )} \sqrt {-b x + 2} \sqrt {x}}{2 \, x}, \frac {30 \, \sqrt {b} x \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) + {\left (b^{2} x^{2} - 9 \, b x - 16\right )} \sqrt {-b x + 2} \sqrt {x}}{2 \, x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.69, size = 201, normalized size = 2.45 \begin {gather*} \begin {cases} 15 i \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} + \frac {i b^{3} x^{\frac {5}{2}}}{2 \sqrt {b x - 2}} - \frac {11 i b^{2} x^{\frac {3}{2}}}{2 \sqrt {b x - 2}} + \frac {i b \sqrt {x}}{\sqrt {b x - 2}} + \frac {16 i}{\sqrt {x} \sqrt {b x - 2}} & \text {for}\: \left |{b x}\right | > 2 \\- 15 \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} - \frac {b^{3} x^{\frac {5}{2}}}{2 \sqrt {- b x + 2}} + \frac {11 b^{2} x^{\frac {3}{2}}}{2 \sqrt {- b x + 2}} - \frac {b \sqrt {x}}{\sqrt {- b x + 2}} - \frac {16}{\sqrt {x} \sqrt {- b x + 2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.26, size = 157, normalized size = 1.91 \begin {gather*} -\frac {b b^{2} \left (\frac {2 \left (\left (-\frac {5}{4}-\frac {1}{4} \sqrt {-b x+2} \sqrt {-b x+2}\right ) \sqrt {-b x+2} \sqrt {-b x+2}+\frac {15}{2}\right ) \sqrt {-b x+2} \sqrt {-b \left (-b x+2\right )+2 b}}{-b \left (-b x+2\right )+2 b}+\frac {15 \ln \left |\sqrt {-b \left (-b x+2\right )+2 b}-\sqrt {-b} \sqrt {-b x+2}\right |}{\sqrt {-b}}\right )}{\left |b\right | b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (2-b\,x\right )}^{5/2}}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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