Integrand size = 16, antiderivative size = 50 \[ \int \frac {\sqrt {x}}{(a-b x)^{3/2}} \, dx=\frac {2 \sqrt {x}}{b \sqrt {a-b x}}-\frac {2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {49, 65, 223, 209} \[ \int \frac {\sqrt {x}}{(a-b x)^{3/2}} \, dx=\frac {2 \sqrt {x}}{b \sqrt {a-b x}}-\frac {2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}} \]
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Rule 49
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {x}}{b \sqrt {a-b x}}-\frac {\int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{b} \\ & = \frac {2 \sqrt {x}}{b \sqrt {a-b x}}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{b} \\ & = \frac {2 \sqrt {x}}{b \sqrt {a-b x}}-\frac {2 \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{b} \\ & = \frac {2 \sqrt {x}}{b \sqrt {a-b x}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {x}}{(a-b x)^{3/2}} \, dx=\frac {2 \sqrt {x}}{b \sqrt {a-b x}}+\frac {4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a-b x}}\right )}{b^{3/2}} \]
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\[\int \frac {\sqrt {x}}{\left (-b x +a \right )^{\frac {3}{2}}}d x\]
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none
Time = 0.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.56 \[ \int \frac {\sqrt {x}}{(a-b x)^{3/2}} \, dx=\left [-\frac {{\left (b x - a\right )} \sqrt {-b} \log \left (-2 \, b x - 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, \sqrt {-b x + a} b \sqrt {x}}{b^{3} x - a b^{2}}, \frac {2 \, {\left ({\left (b x - a\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - \sqrt {-b x + a} b \sqrt {x}\right )}}{b^{3} x - a b^{2}}\right ] \]
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Result contains complex when optimal does not.
Time = 1.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.04 \[ \int \frac {\sqrt {x}}{(a-b x)^{3/2}} \, dx=\begin {cases} \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {2 i \sqrt {x}}{\sqrt {a} b \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {2 \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} + \frac {2 \sqrt {x}}{\sqrt {a} b \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {x}}{(a-b x)^{3/2}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, \sqrt {x}}{\sqrt {-b x + a} b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (38) = 76\).
Time = 16.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.96 \[ \int \frac {\sqrt {x}}{(a-b x)^{3/2}} \, dx=-\frac {{\left (\frac {4 \, a \sqrt {-b}}{{\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b} + \frac {\log \left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt {-b}}\right )} {\left | b \right |}}{b^{2}} \]
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Timed out. \[ \int \frac {\sqrt {x}}{(a-b x)^{3/2}} \, dx=\int \frac {\sqrt {x}}{{\left (a-b\,x\right )}^{3/2}} \,d x \]
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