Optimal. Leaf size=89 \[ \frac {10 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}-\frac {5 \sqrt {x} \sqrt {2-b x}}{b^3}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2-b x}}+\frac {2 x^{5/2}}{3 b (2-b x)^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 89, 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 = {47, 50, 54, 216} \[ -\frac {10 x^{3/2}}{3 b^2 \sqrt {2-b x}}-\frac {5 \sqrt {x} \sqrt {2-b x}}{b^3}+\frac {10 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}+\frac {2 x^{5/2}}{3 b (2-b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 54
Rule 216
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{(2-b x)^{5/2}} \, dx &=\frac {2 x^{5/2}}{3 b (2-b x)^{3/2}}-\frac {5 \int \frac {x^{3/2}}{(2-b x)^{3/2}} \, dx}{3 b}\\ &=\frac {2 x^{5/2}}{3 b (2-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2-b x}}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{b^2}\\ &=\frac {2 x^{5/2}}{3 b (2-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2-b x}}-\frac {5 \sqrt {x} \sqrt {2-b x}}{b^3}+\frac {5 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{b^3}\\ &=\frac {2 x^{5/2}}{3 b (2-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2-b x}}-\frac {5 \sqrt {x} \sqrt {2-b x}}{b^3}+\frac {10 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=\frac {2 x^{5/2}}{3 b (2-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2-b x}}-\frac {5 \sqrt {x} \sqrt {2-b x}}{b^3}+\frac {10 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.34 \[ \frac {x^{7/2} \, _2F_1\left (\frac {5}{2},\frac {7}{2};\frac {9}{2};\frac {b x}{2}\right )}{14 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 187, normalized size = 2.10 \[ \left [-\frac {15 \, {\left (b^{2} x^{2} - 4 \, b x + 4\right )} \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) + {\left (3 \, b^{3} x^{2} - 40 \, b^{2} x + 60 \, b\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, {\left (b^{6} x^{2} - 4 \, b^{5} x + 4 \, b^{4}\right )}}, -\frac {30 \, {\left (b^{2} x^{2} - 4 \, b x + 4\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) + {\left (3 \, b^{3} x^{2} - 40 \, b^{2} x + 60 \, b\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, {\left (b^{6} x^{2} - 4 \, b^{5} x + 4 \, b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 10.72, size = 200, normalized size = 2.25 \[ \frac {{\left (\frac {15 \, \log \left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{\sqrt {-b} b^{2}} - \frac {3 \, \sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2}}{b^{3}} - \frac {16 \, {\left (9 \, {\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{4} - 24 \, {\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} b + 28 \, b^{2}\right )}}{{\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}^{3} \sqrt {-b} b}\right )} {\left | b \right |}}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 168, normalized size = 1.89 \[ \frac {\left (\frac {5 \arctan \left (\frac {\left (x -\frac {1}{b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+2 x}}\right )}{b^{\frac {7}{2}}}+\frac {28 \sqrt {-\left (x -\frac {2}{b}\right )^{2} b -2 x +\frac {4}{b}}}{3 \left (x -\frac {2}{b}\right ) b^{4}}+\frac {8 \sqrt {-\left (x -\frac {2}{b}\right )^{2} b -2 x +\frac {4}{b}}}{3 \left (x -\frac {2}{b}\right )^{2} b^{5}}\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {-b x +2}\, \sqrt {x}}+\frac {\left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}\, \sqrt {x}}{\sqrt {-\left (b x -2\right ) x}\, \sqrt {-b x +2}\, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 86, normalized size = 0.97 \[ \frac {2 \, {\left (2 \, b^{2} + \frac {10 \, {\left (b x - 2\right )} b}{x} - \frac {15 \, {\left (b x - 2\right )}^{2}}{x^{2}}\right )}}{3 \, {\left (\frac {{\left (-b x + 2\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} + \frac {{\left (-b x + 2\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}}\right )}} - \frac {10 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {7}{2}}} \]
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 \[ \int \frac {x^{5/2}}{{\left (2-b\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.80, size = 753, normalized size = 8.46 \[ \begin {cases} - \frac {3 i b^{\frac {23}{2}} x^{15}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x - 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x - 2}} + \frac {40 i b^{\frac {21}{2}} x^{14}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x - 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x - 2}} - \frac {60 i b^{\frac {19}{2}} x^{13}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x - 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x - 2}} - \frac {30 i b^{10} x^{\frac {27}{2}} \sqrt {b x - 2} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x - 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x - 2}} + \frac {15 \pi b^{10} x^{\frac {27}{2}} \sqrt {b x - 2}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x - 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x - 2}} + \frac {60 i b^{9} x^{\frac {25}{2}} \sqrt {b x - 2} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x - 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x - 2}} - \frac {30 \pi b^{9} x^{\frac {25}{2}} \sqrt {b x - 2}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x - 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x - 2}} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\\frac {3 b^{\frac {23}{2}} x^{15}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {- b x + 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {- b x + 2}} - \frac {40 b^{\frac {21}{2}} x^{14}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {- b x + 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {- b x + 2}} + \frac {60 b^{\frac {19}{2}} x^{13}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {- b x + 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {- b x + 2}} + \frac {30 b^{10} x^{\frac {27}{2}} \sqrt {- b x + 2} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {- b x + 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {- b x + 2}} - \frac {60 b^{9} x^{\frac {25}{2}} \sqrt {- b x + 2} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {- b x + 2} - 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {- b x + 2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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