Optimal. Leaf size=89 \[ -\frac {15 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}+\frac {15 \sqrt {x} \sqrt {2-b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}+\frac {2 x^{5/2}}{b \sqrt {2-b x}} \]
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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} \[ \frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}+\frac {15 \sqrt {x} \sqrt {2-b x}}{2 b^3}-\frac {15 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}+\frac {2 x^{5/2}}{b \sqrt {2-b x}} \]
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)^{3/2}} \, dx &=\frac {2 x^{5/2}}{b \sqrt {2-b x}}-\frac {5 \int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx}{b}\\ &=\frac {2 x^{5/2}}{b \sqrt {2-b x}}+\frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}-\frac {15 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{2 b^2}\\ &=\frac {2 x^{5/2}}{b \sqrt {2-b x}}+\frac {15 \sqrt {x} \sqrt {2-b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}-\frac {15 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{2 b^3}\\ &=\frac {2 x^{5/2}}{b \sqrt {2-b x}}+\frac {15 \sqrt {x} \sqrt {2-b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}-\frac {15 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=\frac {2 x^{5/2}}{b \sqrt {2-b x}}+\frac {15 \sqrt {x} \sqrt {2-b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}-\frac {15 \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 \[ \frac {x^{7/2} \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};\frac {b x}{2}\right )}{7 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 155, normalized size = 1.74 \[ \left [-\frac {15 \, {\left (b x - 2\right )} \sqrt {-b} \log \left (-b x - \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) - {\left (b^{3} x^{2} + 5 \, b^{2} x - 30 \, b\right )} \sqrt {-b x + 2} \sqrt {x}}{2 \, {\left (b^{5} x - 2 \, b^{4}\right )}}, \frac {30 \, {\left (b x - 2\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) + {\left (b^{3} x^{2} + 5 \, b^{2} x - 30 \, b\right )} \sqrt {-b x + 2} \sqrt {x}}{2 \, {\left (b^{5} x - 2 \, b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 10.85, size = 136, normalized size = 1.53 \[ \frac {{\left (\sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2} {\left (\frac {b x - 2}{b^{3}} + \frac {9}{b^{3}}\right )} - \frac {15 \, \log \left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{\sqrt {-b} b^{2}} + \frac {64}{{\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} \sqrt {-b} b}\right )} {\left | b \right |}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 138, normalized size = 1.55 \[ -\frac {\left (\frac {15 \arctan \left (\frac {\left (x -\frac {1}{b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+2 x}}\right )}{2 b^{\frac {7}{2}}}+\frac {8 \sqrt {-\left (x -\frac {2}{b}\right )^{2} b -2 x +\frac {4}{b}}}{\left (x -\frac {2}{b}\right ) b^{4}}\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {-b x +2}\, \sqrt {x}}-\frac {\left (b x +7\right ) \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}\, \sqrt {x}}{2 \sqrt {-\left (b x -2\right ) x}\, \sqrt {-b x +2}\, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 101, normalized size = 1.13 \[ \frac {8 \, b^{2} - \frac {25 \, {\left (b x - 2\right )} b}{x} + \frac {15 \, {\left (b x - 2\right )}^{2}}{x^{2}}}{\frac {\sqrt {-b x + 2} b^{5}}{\sqrt {x}} + \frac {2 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} + \frac {{\left (-b x + 2\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}}} + \frac {15 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {7}{2}}} \]
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 \[ \int \frac {x^{5/2}}{{\left (2-b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.03, size = 173, normalized size = 1.94 \[ \begin {cases} \frac {i x^{\frac {5}{2}}}{2 b \sqrt {b x - 2}} + \frac {5 i x^{\frac {3}{2}}}{2 b^{2} \sqrt {b x - 2}} - \frac {15 i \sqrt {x}}{b^{3} \sqrt {b x - 2}} + \frac {15 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {7}{2}}} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\- \frac {x^{\frac {5}{2}}}{2 b \sqrt {- b x + 2}} - \frac {5 x^{\frac {3}{2}}}{2 b^{2} \sqrt {- b x + 2}} + \frac {15 \sqrt {x}}{b^{3} \sqrt {- b x + 2}} - \frac {15 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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