Optimal. Leaf size=106 \[ -\frac {5 \sqrt {x} \sqrt {2-b x}}{8 b}+\frac {5}{8} x^{3/2} \sqrt {2-b x}+\frac {5}{12} x^{3/2} (2-b x)^{3/2}+\frac {1}{4} x^{3/2} (2-b x)^{5/2}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 106, 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 {5 \text {ArcSin}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{3/2}}+\frac {1}{4} x^{3/2} (2-b x)^{5/2}+\frac {5}{12} x^{3/2} (2-b x)^{3/2}+\frac {5}{8} x^{3/2} \sqrt {2-b x}-\frac {5 \sqrt {x} \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 \sqrt {x} (2-b x)^{5/2} \, dx &=\frac {1}{4} x^{3/2} (2-b x)^{5/2}+\frac {5}{4} \int \sqrt {x} (2-b x)^{3/2} \, dx\\ &=\frac {5}{12} x^{3/2} (2-b x)^{3/2}+\frac {1}{4} x^{3/2} (2-b x)^{5/2}+\frac {5}{4} \int \sqrt {x} \sqrt {2-b x} \, dx\\ &=\frac {5}{8} x^{3/2} \sqrt {2-b x}+\frac {5}{12} x^{3/2} (2-b x)^{3/2}+\frac {1}{4} x^{3/2} (2-b x)^{5/2}+\frac {5}{8} \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx\\ &=-\frac {5 \sqrt {x} \sqrt {2-b x}}{8 b}+\frac {5}{8} x^{3/2} \sqrt {2-b x}+\frac {5}{12} x^{3/2} (2-b x)^{3/2}+\frac {1}{4} x^{3/2} (2-b x)^{5/2}+\frac {5 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{8 b}\\ &=-\frac {5 \sqrt {x} \sqrt {2-b x}}{8 b}+\frac {5}{8} x^{3/2} \sqrt {2-b x}+\frac {5}{12} x^{3/2} (2-b x)^{3/2}+\frac {1}{4} x^{3/2} (2-b x)^{5/2}+\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{4 b}\\ &=-\frac {5 \sqrt {x} \sqrt {2-b x}}{8 b}+\frac {5}{8} x^{3/2} \sqrt {2-b x}+\frac {5}{12} x^{3/2} (2-b x)^{3/2}+\frac {1}{4} x^{3/2} (2-b x)^{5/2}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 82, normalized size = 0.77 \begin {gather*} \frac {\sqrt {x} \sqrt {2-b x} \left (-15+59 b x-34 b^2 x^2+6 b^3 x^3\right )}{24 b}+\frac {5 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{4 (-b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 107, normalized size = 1.01
method | result | size |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {3}{2}} \left (-6 b^{3} x^{3}+34 x^{2} b^{2}-59 b x +15\right ) \sqrt {-\frac {b x}{2}+1}}{24 b}-\frac {5 \sqrt {\pi }\, \left (-b \right )^{\frac {3}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{4 b^{\frac {3}{2}}}}{\sqrt {-b}\, \sqrt {\pi }\, b}\) | \(89\) |
default | \(\frac {x^{\frac {3}{2}} \left (-b x +2\right )^{\frac {5}{2}}}{4}+\frac {5 x^{\frac {3}{2}} \left (-b x +2\right )^{\frac {3}{2}}}{12}+\frac {5 x^{\frac {3}{2}} \sqrt {-b x +2}}{8}-\frac {5 \sqrt {x}\, \sqrt {-b x +2}}{8 b}+\frac {5 \sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{8 b^{\frac {3}{2}} \sqrt {-b x +2}\, \sqrt {x}}\) | \(107\) |
risch | \(-\frac {\left (6 b^{3} x^{3}-34 x^{2} b^{2}+59 b x -15\right ) \sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{24 b \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {5 \sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{8 b^{\frac {3}{2}} \sqrt {-b x +2}\, \sqrt {x}}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 145, normalized size = 1.37 \begin {gather*} \frac {\frac {15 \, \sqrt {-b x + 2} b^{3}}{\sqrt {x}} + \frac {55 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} + \frac {73 \, {\left (-b x + 2\right )}^{\frac {5}{2}} b}{x^{\frac {5}{2}}} - \frac {15 \, {\left (-b x + 2\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}}}{12 \, {\left (b^{5} - \frac {4 \, {\left (b x - 2\right )} b^{4}}{x} + \frac {6 \, {\left (b x - 2\right )}^{2} b^{3}}{x^{2}} - \frac {4 \, {\left (b x - 2\right )}^{3} b^{2}}{x^{3}} + \frac {{\left (b x - 2\right )}^{4} b}{x^{4}}\right )}} - \frac {5 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{4 \, b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.73, size = 141, normalized size = 1.33 \begin {gather*} \left [\frac {{\left (6 \, b^{4} x^{3} - 34 \, b^{3} x^{2} + 59 \, b^{2} x - 15 \, b\right )} \sqrt {-b x + 2} \sqrt {x} - 15 \, \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{24 \, b^{2}}, \frac {{\left (6 \, b^{4} x^{3} - 34 \, b^{3} x^{2} + 59 \, b^{2} x - 15 \, b\right )} \sqrt {-b x + 2} \sqrt {x} - 30 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{24 \, b^{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 = 8.50, size = 253, normalized size = 2.39 \begin {gather*} \begin {cases} \frac {i b^{3} x^{\frac {9}{2}}}{4 \sqrt {b x - 2}} - \frac {23 i b^{2} x^{\frac {7}{2}}}{12 \sqrt {b x - 2}} + \frac {127 i b x^{\frac {5}{2}}}{24 \sqrt {b x - 2}} - \frac {133 i x^{\frac {3}{2}}}{24 \sqrt {b x - 2}} + \frac {5 i \sqrt {x}}{4 b \sqrt {b x - 2}} - \frac {5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {3}{2}}} & \text {for}\: \left |{b x}\right | > 2 \\- \frac {b^{3} x^{\frac {9}{2}}}{4 \sqrt {- b x + 2}} + \frac {23 b^{2} x^{\frac {7}{2}}}{12 \sqrt {- b x + 2}} - \frac {127 b x^{\frac {5}{2}}}{24 \sqrt {- b x + 2}} + \frac {133 x^{\frac {3}{2}}}{24 \sqrt {- b x + 2}} - \frac {5 \sqrt {x}}{4 b \sqrt {- b x + 2}} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {3}{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 \sqrt {x}\,{\left (2-b\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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