Optimal. Leaf size=87 \[ -\frac {\sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{6 b}+\frac {1}{3} x^{5/2} \sqrt {2-b x}+\frac {\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {52, 56, 222}
\begin {gather*} \frac {\text {ArcSin}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}-\frac {\sqrt {x} \sqrt {2-b x}}{2 b^2}+\frac {1}{3} x^{5/2} \sqrt {2-b x}-\frac {x^{3/2} \sqrt {2-b x}}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int x^{3/2} \sqrt {2-b x} \, dx &=\frac {1}{3} x^{5/2} \sqrt {2-b x}+\frac {1}{3} \int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx\\ &=-\frac {x^{3/2} \sqrt {2-b x}}{6 b}+\frac {1}{3} x^{5/2} \sqrt {2-b x}+\frac {\int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{2 b}\\ &=-\frac {\sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{6 b}+\frac {1}{3} x^{5/2} \sqrt {2-b x}+\frac {\int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{2 b^2}\\ &=-\frac {\sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{6 b}+\frac {1}{3} x^{5/2} \sqrt {2-b x}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {\sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{6 b}+\frac {1}{3} x^{5/2} \sqrt {2-b x}+\frac {\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 72, normalized size = 0.83 \begin {gather*} \frac {\sqrt {x} \sqrt {2-b x} \left (-3-b x+2 b^2 x^2\right )}{6 b^2}+\frac {b \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{(-b)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 106, normalized size = 1.22
method | result | size |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {5}{2}} \left (-10 x^{2} b^{2}+5 b x +15\right ) \sqrt {-\frac {b x}{2}+1}}{30 b^{2}}-\frac {\sqrt {\pi }\, \left (-b \right )^{\frac {5}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {5}{2}}}}{\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, b}\) | \(81\) |
default | \(-\frac {x^{\frac {3}{2}} \left (-b x +2\right )^{\frac {3}{2}}}{3 b}+\frac {-\frac {\sqrt {x}\, \left (-b x +2\right )^{\frac {3}{2}}}{2 b}+\frac {\sqrt {x}\, \sqrt {-b x +2}+\frac {\sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{\sqrt {-b x +2}\, \sqrt {x}\, \sqrt {b}}}{2 b}}{b}\) | \(106\) |
risch | \(-\frac {\left (2 x^{2} b^{2}-b x -3\right ) \sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{6 b^{2} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {\arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{2 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {-b x +2}}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 117, normalized size = 1.34 \begin {gather*} \frac {\frac {3 \, \sqrt {-b x + 2} b^{2}}{\sqrt {x}} - \frac {8 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} - \frac {3 \, {\left (-b x + 2\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}}{3 \, {\left (b^{5} - \frac {3 \, {\left (b x - 2\right )} b^{4}}{x} + \frac {3 \, {\left (b x - 2\right )}^{2} b^{3}}{x^{2}} - \frac {{\left (b x - 2\right )}^{3} b^{2}}{x^{3}}\right )}} - \frac {\arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 125, normalized size = 1.44 \begin {gather*} \left [\frac {{\left (2 \, b^{3} x^{2} - b^{2} x - 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x} - 3 \, \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{6 \, b^{3}}, \frac {{\left (2 \, b^{3} x^{2} - b^{2} x - 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x} - 6 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{6 \, b^{3}}\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 = 5.37, size = 194, normalized size = 2.23 \begin {gather*} \begin {cases} \frac {i b x^{\frac {7}{2}}}{3 \sqrt {b x - 2}} - \frac {5 i x^{\frac {5}{2}}}{6 \sqrt {b x - 2}} - \frac {i x^{\frac {3}{2}}}{6 b \sqrt {b x - 2}} + \frac {i \sqrt {x}}{b^{2} \sqrt {b x - 2}} - \frac {i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} & \text {for}\: \left |{b x}\right | > 2 \\- \frac {b x^{\frac {7}{2}}}{3 \sqrt {- b x + 2}} + \frac {5 x^{\frac {5}{2}}}{6 \sqrt {- b x + 2}} + \frac {x^{\frac {3}{2}}}{6 b \sqrt {- b x + 2}} - \frac {\sqrt {x}}{b^{2} \sqrt {- b x + 2}} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \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 x^{3/2}\,\sqrt {2-b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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