Optimal. Leaf size=65 \[ \frac {2 x^{3/2}}{b \sqrt {2-b x}}+\frac {3 \sqrt {x} \sqrt {2-b x}}{b^2}-\frac {6 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {6 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}+\frac {3 \sqrt {x} \sqrt {2-b x}}{b^2}+\frac {2 x^{3/2}}{b \sqrt {2-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{(2-b x)^{3/2}} \, dx &=\frac {2 x^{3/2}}{b \sqrt {2-b x}}-\frac {3 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{b}\\ &=\frac {2 x^{3/2}}{b \sqrt {2-b x}}+\frac {3 \sqrt {x} \sqrt {2-b x}}{b^2}-\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{b^2}\\ &=\frac {2 x^{3/2}}{b \sqrt {2-b x}}+\frac {3 \sqrt {x} \sqrt {2-b x}}{b^2}-\frac {6 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=\frac {2 x^{3/2}}{b \sqrt {2-b x}}+\frac {3 \sqrt {x} \sqrt {2-b x}}{b^2}-\frac {6 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 61, normalized size = 0.94 \begin {gather*} \frac {\sqrt {x} (6-b x)}{b^2 \sqrt {2-b x}}+\frac {6 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{(-b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 73, normalized size = 1.12
method | result | size |
meijerg | \(-\frac {4 \left (\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {5}{2}} \left (-\frac {5 b x}{2}+15\right )}{20 b^{2} \sqrt {-\frac {b x}{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-b \right )^{\frac {5}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{2 b^{\frac {5}{2}}}\right )}{\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, b}\) | \(73\) |
risch | \(-\frac {\sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{b^{2} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}-\frac {\left (\frac {3 \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{b^{\frac {5}{2}}}+\frac {4 \sqrt {-\left (x -\frac {2}{b}\right )^{2} b -2 x +\frac {4}{b}}}{b^{3} \left (x -\frac {2}{b}\right )}\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {x}\, \sqrt {-b x +2}}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 71, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (2 \, b - \frac {3 \, {\left (b x - 2\right )}}{x}\right )}}{\frac {\sqrt {-b x + 2} b^{3}}{\sqrt {x}} + \frac {{\left (-b x + 2\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}}} + \frac {6 \, \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 = 0.32, size = 138, normalized size = 2.12 \begin {gather*} \left [-\frac {3 \, {\left (b x - 2\right )} \sqrt {-b} \log \left (-b x - \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) - {\left (b^{2} x - 6 \, b\right )} \sqrt {-b x + 2} \sqrt {x}}{b^{4} x - 2 \, b^{3}}, \frac {6 \, {\left (b x - 2\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) + {\left (b^{2} x - 6 \, b\right )} \sqrt {-b x + 2} \sqrt {x}}{b^{4} x - 2 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.83, size = 126, normalized size = 1.94 \begin {gather*} \begin {cases} \frac {i x^{\frac {3}{2}}}{b \sqrt {b x - 2}} - \frac {6 i \sqrt {x}}{b^{2} \sqrt {b x - 2}} + \frac {6 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 {x^{\frac {3}{2}}}{b \sqrt {- b x + 2}} + \frac {6 \sqrt {x}}{b^{2} \sqrt {- b x + 2}} - \frac {6 \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 [A]
time = 0.01, size = 100, normalized size = 1.54 \begin {gather*} -2 \left (\frac {2 \left (\frac {\frac {1}{4} b^{2} \sqrt {x} \sqrt {x}}{b^{3}}-\frac {\frac {1}{4}\cdot 6 b}{b^{3}}\right ) \sqrt {x} \sqrt {-b x+2}}{-b x+2}-\frac {3 \ln \left (\sqrt {-b x+2}-\sqrt {-b} \sqrt {x}\right )}{b^{2} \sqrt {-b}}\right ) \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 = -0.02 \begin {gather*} \int \frac {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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