Optimal. Leaf size=62 \[ 2 b^{3/2} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )-\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}+\frac {2 b \sqrt {2-b x}}{\sqrt {x}} \]
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Rubi [A] time = 0.01, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {47, 54, 216} \[ 2 b^{3/2} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )-\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}+\frac {2 b \sqrt {2-b x}}{\sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 54
Rule 216
Rubi steps
\begin {align*} \int \frac {(2-b x)^{3/2}}{x^{5/2}} \, dx &=-\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}-b \int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx\\ &=\frac {2 b \sqrt {2-b x}}{\sqrt {x}}-\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}+b^2 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx\\ &=\frac {2 b \sqrt {2-b x}}{\sqrt {x}}-\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 b \sqrt {2-b x}}{\sqrt {x}}-\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}+2 b^{3/2} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.48 \[ -\frac {4 \sqrt {2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {b x}{2}\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 111, normalized size = 1.79 \[ \left [\frac {3 \, \sqrt {-b} b x^{2} \log \left (-b x - \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) + 4 \, {\left (2 \, b x - 1\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, x^{2}}, -\frac {2 \, {\left (3 \, b^{\frac {3}{2}} x^{2} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - 2 \, {\left (2 \, b x - 1\right )} \sqrt {-b x + 2} \sqrt {x}\right )}}{3 \, x^{2}}\right ] \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 98, normalized size = 1.58 \[ \frac {\sqrt {\left (-b x +2\right ) x}\, b^{\frac {3}{2}} \arctan \left (\frac {\left (x -\frac {1}{b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+2 x}}\right )}{\sqrt {-b x +2}\, \sqrt {x}}-\frac {4 \left (2 b^{2} x^{2}-5 b x +2\right ) \sqrt {\left (-b x +2\right ) x}}{3 \sqrt {-\left (b x -2\right ) x}\, \sqrt {-b x +2}\, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 49, normalized size = 0.79 \[ -2 \, b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) + \frac {2 \, \sqrt {-b x + 2} b}{\sqrt {x}} - \frac {2 \, {\left (-b x + 2\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (2-b\,x\right )}^{3/2}}{x^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.92, size = 182, normalized size = 2.94 \[ \begin {cases} \frac {8 b^{\frac {3}{2}} \sqrt {-1 + \frac {2}{b x}}}{3} + i b^{\frac {3}{2}} \log {\left (\frac {1}{b x} \right )} - 2 i b^{\frac {3}{2}} \log {\left (\frac {1}{\sqrt {b} \sqrt {x}} \right )} + 2 b^{\frac {3}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} - \frac {4 \sqrt {b} \sqrt {-1 + \frac {2}{b x}}}{3 x} & \text {for}\: \frac {2}{\left |{b x}\right |} > 1 \\\frac {8 i b^{\frac {3}{2}} \sqrt {1 - \frac {2}{b x}}}{3} + i b^{\frac {3}{2}} \log {\left (\frac {1}{b x} \right )} - 2 i b^{\frac {3}{2}} \log {\left (\sqrt {1 - \frac {2}{b x}} + 1 \right )} - \frac {4 i \sqrt {b} \sqrt {1 - \frac {2}{b x}}}{3 x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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