Optimal. Leaf size=109 \[ -\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{5/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {3 \text {ArcSin}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{5/2}}-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {1}{4} x^{5/2} \sqrt {2-b x}-\frac {x^{3/2} \sqrt {2-b x}}{8 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} (2-b x)^{3/2} \, dx &=\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3}{4} \int x^{3/2} \sqrt {2-b x} \, dx\\ &=\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {1}{4} \int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx\\ &=-\frac {x^{3/2} \sqrt {2-b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{8 b}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{8 b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 81, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {x} \sqrt {2-b x} \left (3+b x-6 b^2 x^2+2 b^3 x^3\right )}{8 b^2}-\frac {3 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{4 (-b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 122, normalized size = 1.12
method | result | size |
meijerg | \(-\frac {12 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {5}{2}} \left (10 b^{3} x^{3}-30 x^{2} b^{2}+5 b x +15\right ) \sqrt {-\frac {b x}{2}+1}}{480 b^{2}}+\frac {\sqrt {\pi }\, \left (-b \right )^{\frac {5}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{16 b^{\frac {5}{2}}}\right )}{\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, b}\) | \(89\) |
risch | \(\frac {\left (2 b^{3} x^{3}-6 x^{2} b^{2}+b x +3\right ) \sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{8 b^{2} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {3 \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{8 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {-b x +2}}\) | \(114\) |
default | \(-\frac {x^{\frac {3}{2}} \left (-b x +2\right )^{\frac {5}{2}}}{4 b}+\frac {-\frac {\sqrt {x}\, \left (-b x +2\right )^{\frac {5}{2}}}{4 b}+\frac {\frac {\left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 \sqrt {x}\, \sqrt {-b x +2}}{2}+\frac {3 \sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{2 \sqrt {-b x +2}\, \sqrt {x}\, \sqrt {b}}}{4 b}}{b}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 147, normalized size = 1.35 \begin {gather*} \frac {\frac {3 \, \sqrt {-b x + 2} b^{3}}{\sqrt {x}} + \frac {11 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} - \frac {11 \, {\left (-b x + 2\right )}^{\frac {5}{2}} b}{x^{\frac {5}{2}}} - \frac {3 \, {\left (-b x + 2\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}}}{4 \, {\left (b^{6} - \frac {4 \, {\left (b x - 2\right )} b^{5}}{x} + \frac {6 \, {\left (b x - 2\right )}^{2} b^{4}}{x^{2}} - \frac {4 \, {\left (b x - 2\right )}^{3} b^{3}}{x^{3}} + \frac {{\left (b x - 2\right )}^{4} b^{2}}{x^{4}}\right )}} - \frac {3 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{4 \, b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 139, normalized size = 1.28 \begin {gather*} \left [-\frac {{\left (2 \, b^{4} x^{3} - 6 \, 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 )}{8 \, b^{3}}, -\frac {{\left (2 \, b^{4} x^{3} - 6 \, 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 )}{8 \, 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 = 12.59, size = 250, normalized size = 2.29 \begin {gather*} \begin {cases} - \frac {i b^{2} x^{\frac {9}{2}}}{4 \sqrt {b x - 2}} + \frac {5 i b x^{\frac {7}{2}}}{4 \sqrt {b x - 2}} - \frac {13 i x^{\frac {5}{2}}}{8 \sqrt {b x - 2}} - \frac {i x^{\frac {3}{2}}}{8 b \sqrt {b x - 2}} + \frac {3 i \sqrt {x}}{4 b^{2} \sqrt {b x - 2}} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {5}{2}}} & \text {for}\: \left |{b x}\right | > 2 \\\frac {b^{2} x^{\frac {9}{2}}}{4 \sqrt {- b x + 2}} - \frac {5 b x^{\frac {7}{2}}}{4 \sqrt {- b x + 2}} + \frac {13 x^{\frac {5}{2}}}{8 \sqrt {- b x + 2}} + \frac {x^{\frac {3}{2}}}{8 b \sqrt {- b x + 2}} - \frac {3 \sqrt {x}}{4 b^{2} \sqrt {- b x + 2}} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 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}\,{\left (2-b\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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