Optimal. Leaf size=89 \[ \frac {2 x^{5/2}}{b \sqrt {2-b x}}+\frac {15 \sqrt {x} \sqrt {2-b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}-\frac {15 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 89, 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 = {49, 52, 56, 222}
\begin {gather*} -\frac {15 \text {ArcSin}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}+\frac {15 \sqrt {x} \sqrt {2-b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}+\frac {2 x^{5/2}}{b \sqrt {2-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{(2-b x)^{3/2}} \, dx &=\frac {2 x^{5/2}}{b \sqrt {2-b x}}-\frac {5 \int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx}{b}\\ &=\frac {2 x^{5/2}}{b \sqrt {2-b x}}+\frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}-\frac {15 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{2 b^2}\\ &=\frac {2 x^{5/2}}{b \sqrt {2-b x}}+\frac {15 \sqrt {x} \sqrt {2-b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}-\frac {15 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{2 b^3}\\ &=\frac {2 x^{5/2}}{b \sqrt {2-b x}}+\frac {15 \sqrt {x} \sqrt {2-b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}-\frac {15 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=\frac {2 x^{5/2}}{b \sqrt {2-b x}}+\frac {15 \sqrt {x} \sqrt {2-b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2-b x}}{2 b^2}-\frac {15 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 71, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {x} \left (-30+5 b x+b^2 x^2\right )}{2 b^3 \sqrt {2-b x}}-\frac {15 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{(-b)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 81, normalized size = 0.91
method | result | size |
meijerg | \(-\frac {8 \left (\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {7}{2}} \left (-\frac {7}{2} x^{2} b^{2}-\frac {35}{2} b x +105\right )}{112 b^{3} \sqrt {-\frac {b x}{2}+1}}-\frac {15 \sqrt {\pi }\, \left (-b \right )^{\frac {7}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{8 b^{\frac {7}{2}}}\right )}{\left (-b \right )^{\frac {5}{2}} \sqrt {\pi }\, b}\) | \(81\) |
risch | \(-\frac {\left (b x +7\right ) \sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{2 b^{3} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}-\frac {\left (\frac {15 \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{2 b^{\frac {7}{2}}}+\frac {8 \sqrt {-\left (x -\frac {2}{b}\right )^{2} b -2 x +\frac {4}{b}}}{b^{4} \left (x -\frac {2}{b}\right )}\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {x}\, \sqrt {-b x +2}}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 101, normalized size = 1.13 \begin {gather*} \frac {8 \, b^{2} - \frac {25 \, {\left (b x - 2\right )} b}{x} + \frac {15 \, {\left (b x - 2\right )}^{2}}{x^{2}}}{\frac {\sqrt {-b x + 2} b^{5}}{\sqrt {x}} + \frac {2 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} + \frac {{\left (-b x + 2\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}}} + \frac {15 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.38, size = 155, normalized size = 1.74 \begin {gather*} \left [-\frac {15 \, {\left (b x - 2\right )} \sqrt {-b} \log \left (-b x - \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) - {\left (b^{3} x^{2} + 5 \, b^{2} x - 30 \, b\right )} \sqrt {-b x + 2} \sqrt {x}}{2 \, {\left (b^{5} x - 2 \, b^{4}\right )}}, \frac {30 \, {\left (b x - 2\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) + {\left (b^{3} x^{2} + 5 \, b^{2} x - 30 \, b\right )} \sqrt {-b x + 2} \sqrt {x}}{2 \, {\left (b^{5} x - 2 \, b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 5.08, size = 172, normalized size = 1.93 \begin {gather*} \begin {cases} \frac {i x^{\frac {5}{2}}}{2 b \sqrt {b x - 2}} + \frac {5 i x^{\frac {3}{2}}}{2 b^{2} \sqrt {b x - 2}} - \frac {15 i \sqrt {x}}{b^{3} \sqrt {b x - 2}} + \frac {15 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {7}{2}}} & \text {for}\: \left |{b x}\right | > 2 \\- \frac {x^{\frac {5}{2}}}{2 b \sqrt {- b x + 2}} - \frac {5 x^{\frac {3}{2}}}{2 b^{2} \sqrt {- b x + 2}} + \frac {15 \sqrt {x}}{b^{3} \sqrt {- b x + 2}} - \frac {15 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (66) = 132\).
time = 4.56, size = 136, normalized size = 1.53 \begin {gather*} \frac {{\left (\sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2} {\left (\frac {b x - 2}{b^{3}} + \frac {9}{b^{3}}\right )} - \frac {15 \, \log \left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{\sqrt {-b} b^{2}} + \frac {64}{{\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} \sqrt {-b} b}\right )} {\left | b \right |}}{2 \, b^{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^{5/2}}{{\left (2-b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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