Integrand size = 16, antiderivative size = 105 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=-\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {5 a^3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{7/2}} \]
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Time = 0.02 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223, 209} \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=\frac {5 a^3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{7/2}}-\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b} \]
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Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {(5 a) \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx}{6 b} \\ & = -\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {\left (5 a^2\right ) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{8 b^2} \\ & = -\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {\left (5 a^3\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{16 b^3} \\ & = -\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {\left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^3} \\ & = -\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {\left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^3} \\ & = -\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {5 a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{7/2}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.82 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=-\frac {\sqrt {x} \sqrt {a-b x} \left (15 a^2+10 a b x+8 b^2 x^2\right )}{24 b^3}+\frac {5 a^3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{4 b^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {\left (8 b^{2} x^{2}+10 a b x +15 a^{2}\right ) \sqrt {x}\, \sqrt {-b x +a}}{24 b^{3}}+\frac {5 a^{3} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{16 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(91\) |
default | \(-\frac {x^{\frac {5}{2}} \sqrt {-b x +a}}{3 b}+\frac {5 a \left (-\frac {x^{\frac {3}{2}} \sqrt {-b x +a}}{2 b}+\frac {3 a \left (-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\right )}{4 b}\right )}{6 b}\) | \(116\) |
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Time = 0.23 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.34 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=\left [-\frac {15 \, a^{3} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (8 \, b^{3} x^{2} + 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{48 \, b^{4}}, -\frac {15 \, a^{3} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (8 \, b^{3} x^{2} + 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{24 \, b^{4}}\right ] \]
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Result contains complex when optimal does not.
Time = 10.15 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.57 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=\begin {cases} \frac {5 i a^{\frac {5}{2}} \sqrt {x}}{8 b^{3} \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i \sqrt {a} x^{\frac {5}{2}}}{12 b \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{3} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} - \frac {i x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {5 a^{\frac {5}{2}} \sqrt {x}}{8 b^{3} \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {\sqrt {a} x^{\frac {5}{2}}}{12 b \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{3} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} + \frac {x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.29 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=-\frac {5 \, a^{3} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{8 \, b^{\frac {7}{2}}} - \frac {\frac {33 \, \sqrt {-b x + a} a^{3} b^{2}}{\sqrt {x}} + \frac {40 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {3}{2}}} + \frac {15 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{6} - \frac {3 \, {\left (b x - a\right )} b^{5}}{x} + \frac {3 \, {\left (b x - a\right )}^{2} b^{4}}{x^{2}} - \frac {{\left (b x - a\right )}^{3} b^{3}}{x^{3}}\right )}} \]
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Time = 78.89 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.10 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=\frac {{\left (\frac {15 \, a^{3} \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b} b} - \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a} {\left (2 \, {\left (b x - a\right )} {\left (\frac {4 \, {\left (b x - a\right )}}{b^{2}} + \frac {13 \, a}{b^{2}}\right )} + \frac {33 \, a^{2}}{b^{2}}\right )}\right )} {\left | b \right |}}{24 \, b^{3}} \]
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Timed out. \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=\int \frac {x^{5/2}}{\sqrt {a-b\,x}} \,d x \]
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