Optimal. Leaf size=82 \[ \frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {5}{2} \sqrt {x} \sqrt {2-b x}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
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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 = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {50, 54, 216} \begin {gather*} \frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {5}{2} \sqrt {x} \sqrt {2-b x}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin {align*} \int \frac {(2-b x)^{5/2}}{\sqrt {x}} \, dx &=\frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5}{3} \int \frac {(2-b x)^{3/2}}{\sqrt {x}} \, dx\\ &=\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5}{2} \int \frac {\sqrt {2-b x}}{\sqrt {x}} \, dx\\ &=\frac {5}{2} \sqrt {x} \sqrt {2-b x}+\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5}{2} \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx\\ &=\frac {5}{2} \sqrt {x} \sqrt {2-b x}+\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {1}{3} \sqrt {x} (2-b x)^{5/2}+5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5}{2} \sqrt {x} \sqrt {2-b x}+\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 0.71 \begin {gather*} \frac {1}{6} \sqrt {x} \sqrt {2-b x} \left (2 b^2 x^2-13 b x+33\right )+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 79, normalized size = 0.96 \begin {gather*} \frac {1}{6} \sqrt {2-b x} \left (2 b^2 x^{5/2}-13 b x^{3/2}+33 \sqrt {x}\right )+\frac {5 \sqrt {-b} \log \left (\sqrt {2-b x}-\sqrt {-b} \sqrt {x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 125, normalized size = 1.52 \begin {gather*} \left [\frac {{\left (2 \, b^{3} x^{2} - 13 \, b^{2} x + 33 \, 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}, \frac {{\left (2 \, b^{3} x^{2} - 13 \, b^{2} x + 33 \, 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}\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 = 91, normalized size = 1.11 \begin {gather*} \frac {\left (-b x +2\right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 \left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}{6}+\frac {5 \sqrt {-b x +2}\, \sqrt {x}}{2}+\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}\, \sqrt {b}\, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 112, normalized size = 1.37 \begin {gather*} -\frac {5 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} + \frac {\frac {15 \, \sqrt {-b x + 2} b^{2}}{\sqrt {x}} + \frac {40 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {33 \, {\left (-b x + 2\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}}{3 \, {\left (b^{3} - \frac {3 \, {\left (b x - 2\right )} b^{2}}{x} + \frac {3 \, {\left (b x - 2\right )}^{2} b}{x^{2}} - \frac {{\left (b x - 2\right )}^{3}}{x^{3}}\right )}} \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 {{\left (2-b\,x\right )}^{5/2}}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.52, size = 209, normalized size = 2.55 \begin {gather*} \begin {cases} \frac {i b^{3} x^{\frac {7}{2}}}{3 \sqrt {b x - 2}} - \frac {17 i b^{2} x^{\frac {5}{2}}}{6 \sqrt {b x - 2}} + \frac {59 i b x^{\frac {3}{2}}}{6 \sqrt {b x - 2}} - \frac {11 i \sqrt {x}}{\sqrt {b x - 2}} - \frac {5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\- \frac {b^{3} x^{\frac {7}{2}}}{3 \sqrt {- b x + 2}} + \frac {17 b^{2} x^{\frac {5}{2}}}{6 \sqrt {- b x + 2}} - \frac {59 b x^{\frac {3}{2}}}{6 \sqrt {- b x + 2}} + \frac {11 \sqrt {x}}{\sqrt {- b x + 2}} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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