Optimal. Leaf size=91 \[ \frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}-\frac {5 \sqrt {x} \sqrt {2-b x}}{2 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{6 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{3 b} \]
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Rubi [A] time = 0.02, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {50, 54, 216} \[ -\frac {5 x^{3/2} \sqrt {2-b x}}{6 b^2}-\frac {5 \sqrt {x} \sqrt {2-b x}}{2 b^3}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}-\frac {x^{5/2} \sqrt {2-b x}}{3 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\sqrt {2-b x}} \, dx &=-\frac {x^{5/2} \sqrt {2-b x}}{3 b}+\frac {5 \int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx}{3 b}\\ &=-\frac {5 x^{3/2} \sqrt {2-b x}}{6 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{3 b}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{2 b^2}\\ &=-\frac {5 \sqrt {x} \sqrt {2-b x}}{2 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{6 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{3 b}+\frac {5 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{2 b^3}\\ &=-\frac {5 \sqrt {x} \sqrt {2-b x}}{2 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{6 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{3 b}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {5 \sqrt {x} \sqrt {2-b x}}{2 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{6 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{3 b}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 61, normalized size = 0.67 \[ \frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}-\frac {\sqrt {x} \sqrt {2-b x} \left (2 b^2 x^2+5 b x+15\right )}{6 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 125, normalized size = 1.37 \[ \left [-\frac {{\left (2 \, b^{3} x^{2} + 5 \, b^{2} x + 15 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 15 \, \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{6 \, b^{4}}, -\frac {{\left (2 \, b^{3} x^{2} + 5 \, b^{2} x + 15 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 30 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{6 \, b^{4}}\right ] \]
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 = 0.00, size = 100, normalized size = 1.10 \[ -\frac {\sqrt {-b x +2}\, x^{\frac {5}{2}}}{3 b}-\frac {5 \sqrt {-b x +2}\, x^{\frac {3}{2}}}{6 b^{2}}-\frac {5 \sqrt {-b x +2}\, \sqrt {x}}{2 b^{3}}+\frac {5 \sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\left (x -\frac {1}{b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+2 x}}\right )}{2 \sqrt {-b x +2}\, b^{\frac {7}{2}} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.03, size = 117, normalized size = 1.29 \[ -\frac {\frac {33 \, \sqrt {-b x + 2} b^{2}}{\sqrt {x}} + \frac {40 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {15 \, {\left (-b x + 2\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}}{3 \, {\left (b^{6} - \frac {3 \, {\left (b x - 2\right )} b^{5}}{x} + \frac {3 \, {\left (b x - 2\right )}^{2} b^{4}}{x^{2}} - \frac {{\left (b x - 2\right )}^{3} b^{3}}{x^{3}}\right )}} - \frac {5 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {7}{2}}} \]
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 \[ \int \frac {x^{5/2}}{\sqrt {2-b\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.51, size = 206, normalized size = 2.26 \[ \begin {cases} - \frac {i x^{\frac {7}{2}}}{3 \sqrt {b x - 2}} - \frac {i x^{\frac {5}{2}}}{6 b \sqrt {b x - 2}} - \frac {5 i x^{\frac {3}{2}}}{6 b^{2} \sqrt {b x - 2}} + \frac {5 i \sqrt {x}}{b^{3} \sqrt {b x - 2}} - \frac {5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {7}{2}}} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\\frac {x^{\frac {7}{2}}}{3 \sqrt {- b x + 2}} + \frac {x^{\frac {5}{2}}}{6 b \sqrt {- b x + 2}} + \frac {5 x^{\frac {3}{2}}}{6 b^{2} \sqrt {- b x + 2}} - \frac {5 \sqrt {x}}{b^{3} \sqrt {- b x + 2}} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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