Optimal. Leaf size=82 \[ -\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-15 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 82, 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} \[ -\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-15 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 54
Rule 216
Rubi steps
\begin {align*} \int \frac {(2-b x)^{5/2}}{x^{3/2}} \, dx &=-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-(5 b) \int \frac {(2-b x)^{3/2}}{\sqrt {x}} \, dx\\ &=-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-\frac {1}{2} (15 b) \int \frac {\sqrt {2-b x}}{\sqrt {x}} \, dx\\ &=-\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-\frac {1}{2} (15 b) \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx\\ &=-\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-(15 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-15 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 28, normalized size = 0.34 \[ -\frac {8 \sqrt {2} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};\frac {b x}{2}\right )}{\sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 117, normalized size = 1.43 \[ \left [\frac {15 \, \sqrt {-b} x \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) + {\left (b^{2} x^{2} - 9 \, b x - 16\right )} \sqrt {-b x + 2} \sqrt {x}}{2 \, x}, \frac {30 \, \sqrt {b} x \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) + {\left (b^{2} x^{2} - 9 \, b x - 16\right )} \sqrt {-b x + 2} \sqrt {x}}{2 \, x}\right ] \]
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 [A] time = 0.02, size = 106, normalized size = 1.29 \[ -\frac {15 \sqrt {\left (-b x +2\right ) x}\, \sqrt {b}\, \arctan \left (\frac {\left (x -\frac {1}{b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+2 x}}\right )}{2 \sqrt {-b x +2}\, \sqrt {x}}-\frac {\left (b^{3} x^{3}-11 b^{2} x^{2}+2 b x +32\right ) \sqrt {\left (-b x +2\right ) x}}{2 \sqrt {-\left (b x -2\right ) x}\, \sqrt {-b x +2}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.92, size = 96, normalized size = 1.17 \[ 15 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - \frac {\frac {7 \, \sqrt {-b x + 2} b^{2}}{\sqrt {x}} + \frac {9 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}}}{b^{2} - \frac {2 \, {\left (b x - 2\right )} b}{x} + \frac {{\left (b x - 2\right )}^{2}}{x^{2}}} - \frac {8 \, \sqrt {-b x + 2}}{\sqrt {x}} \]
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 \[ \int \frac {{\left (2-b\,x\right )}^{5/2}}{x^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.62, size = 202, normalized size = 2.46 \[ \begin {cases} 15 i \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} + \frac {i b^{3} x^{\frac {5}{2}}}{2 \sqrt {b x - 2}} - \frac {11 i b^{2} x^{\frac {3}{2}}}{2 \sqrt {b x - 2}} + \frac {i b \sqrt {x}}{\sqrt {b x - 2}} + \frac {16 i}{\sqrt {x} \sqrt {b x - 2}} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\- 15 \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} - \frac {b^{3} x^{\frac {5}{2}}}{2 \sqrt {- b x + 2}} + \frac {11 b^{2} x^{\frac {3}{2}}}{2 \sqrt {- b x + 2}} - \frac {b \sqrt {x}}{\sqrt {- b x + 2}} - \frac {16}{\sqrt {x} \sqrt {- b x + 2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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