Optimal. Leaf size=69 \[ \frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}-\frac {3 \sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{2 b} \]
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Rubi [A] time = 0.01, antiderivative size = 69, 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 = {50, 54, 216} \begin {gather*} -\frac {3 \sqrt {x} \sqrt {2-b x}}{2 b^2}+\frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}-\frac {x^{3/2} \sqrt {2-b x}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx &=-\frac {x^{3/2} \sqrt {2-b x}}{2 b}+\frac {3 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{2 b}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{2 b}+\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{2 b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{2 b}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{2 b}+\frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 52, normalized size = 0.75 \begin {gather*} \frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}-\frac {\sqrt {x} \sqrt {2-b x} (b x+3)}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 72, normalized size = 1.04 \begin {gather*} \frac {3 \sqrt {-b} \log \left (\sqrt {2-b x}-\sqrt {-b} \sqrt {x}\right )}{b^3}+\frac {\sqrt {2-b x} \left (-b x^{3/2}-3 \sqrt {x}\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.34, size = 107, normalized size = 1.55 \begin {gather*} \left [-\frac {{\left (b^{2} x + 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 3 \, \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{2 \, b^{3}}, -\frac {{\left (b^{2} x + 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 6 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{2 \, b^{3}}\right ] \end {gather*}
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 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 84, normalized size = 1.22 \begin {gather*} -\frac {\sqrt {-b x +2}\, x^{\frac {3}{2}}}{2 b}-\frac {3 \sqrt {-b x +2}\, \sqrt {x}}{2 b^{2}}+\frac {3 \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 {5}{2}} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 85, normalized size = 1.23 \begin {gather*} -\frac {\frac {5 \, \sqrt {-b x + 2} b}{\sqrt {x}} + \frac {3 \, {\left (-b x + 2\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}}{b^{4} - \frac {2 \, {\left (b x - 2\right )} b^{3}}{x} + \frac {{\left (b x - 2\right )}^{2} b^{2}}{x^{2}}} - \frac {3 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {5}{2}}} \end {gather*}
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 \begin {gather*} \int \frac {x^{3/2}}{\sqrt {2-b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.59, size = 163, normalized size = 2.36 \begin {gather*} \begin {cases} - \frac {i x^{\frac {5}{2}}}{2 \sqrt {b x - 2}} - \frac {i x^{\frac {3}{2}}}{2 b \sqrt {b x - 2}} + \frac {3 i \sqrt {x}}{b^{2} \sqrt {b x - 2}} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\\frac {x^{\frac {5}{2}}}{2 \sqrt {- b x + 2}} + \frac {x^{\frac {3}{2}}}{2 b \sqrt {- b x + 2}} - \frac {3 \sqrt {x}}{b^{2} \sqrt {- b x + 2}} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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