Integrand size = 16, antiderivative size = 71 \[ \int \frac {x^{3/2}}{(a-b x)^{3/2}} \, dx=\frac {2 x^{3/2}}{b \sqrt {a-b x}}+\frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}-\frac {3 a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{5/2}} \]
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Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {49, 52, 65, 223, 209} \[ \int \frac {x^{3/2}}{(a-b x)^{3/2}} \, dx=-\frac {3 a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{5/2}}+\frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}+\frac {2 x^{3/2}}{b \sqrt {a-b x}} \]
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Rule 49
Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{3/2}}{b \sqrt {a-b x}}-\frac {3 \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{b} \\ & = \frac {2 x^{3/2}}{b \sqrt {a-b x}}+\frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}-\frac {(3 a) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{2 b^2} \\ & = \frac {2 x^{3/2}}{b \sqrt {a-b x}}+\frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{b^2} \\ & = \frac {2 x^{3/2}}{b \sqrt {a-b x}}+\frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{b^2} \\ & = \frac {2 x^{3/2}}{b \sqrt {a-b x}}+\frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}-\frac {3 a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96 \[ \int \frac {x^{3/2}}{(a-b x)^{3/2}} \, dx=\frac {\sqrt {x} (3 a-b x)}{b^2 \sqrt {a-b x}}+\frac {6 a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a-b x}}\right )}{b^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(113\) vs. \(2(55)=110\).
Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.61
method | result | size |
risch | \(\frac {\sqrt {x}\, \sqrt {-b x +a}}{b^{2}}+\frac {\left (-\frac {3 a \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 b^{\frac {5}{2}}}-\frac {2 a \sqrt {-b \left (-\frac {a}{b}+x \right )^{2}-\left (-\frac {a}{b}+x \right ) a}}{b^{3} \left (-\frac {a}{b}+x \right )}\right ) \sqrt {x \left (-b x +a \right )}}{\sqrt {x}\, \sqrt {-b x +a}}\) | \(114\) |
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none
Time = 0.24 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.14 \[ \int \frac {x^{3/2}}{(a-b x)^{3/2}} \, dx=\left [-\frac {3 \, {\left (a b x - a^{2}\right )} \sqrt {-b} \log \left (-2 \, b x - 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (b^{2} x - 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{2 \, {\left (b^{4} x - a b^{3}\right )}}, \frac {3 \, {\left (a b x - a^{2}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (b^{2} x - 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{b^{4} x - a b^{3}}\right ] \]
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Result contains complex when optimal does not.
Time = 2.33 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.18 \[ \int \frac {x^{3/2}}{(a-b x)^{3/2}} \, dx=\begin {cases} - \frac {3 i \sqrt {a} \sqrt {x}}{b^{2} \sqrt {-1 + \frac {b x}{a}}} + \frac {3 i a \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {5}{2}}} + \frac {i x^{\frac {3}{2}}}{\sqrt {a} b \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {3 \sqrt {a} \sqrt {x}}{b^{2} \sqrt {1 - \frac {b x}{a}}} - \frac {3 a \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {5}{2}}} - \frac {x^{\frac {3}{2}}}{\sqrt {a} b \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.06 \[ \int \frac {x^{3/2}}{(a-b x)^{3/2}} \, dx=\frac {2 \, a b - \frac {3 \, {\left (b x - a\right )} a}{x}}{\frac {\sqrt {-b x + a} b^{3}}{\sqrt {x}} + \frac {{\left (-b x + a\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}}} + \frac {3 \, a \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {5}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (55) = 110\).
Time = 15.78 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.83 \[ \int \frac {x^{3/2}}{(a-b x)^{3/2}} \, dx=-\frac {{\left (\frac {8 \, a^{2} \sqrt {-b}}{{\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b} + \frac {3 \, a \log \left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt {-b}} - \frac {2 \, \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}}{b}\right )} {\left | b \right |}}{2 \, b^{3}} \]
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Timed out. \[ \int \frac {x^{3/2}}{(a-b x)^{3/2}} \, dx=\int \frac {x^{3/2}}{{\left (a-b\,x\right )}^{3/2}} \,d x \]
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