Integrand size = 16, antiderivative size = 50 \[ \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx=-\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}} \]
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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 = {52, 65, 223, 209} \[ \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx=\frac {a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {a-b x}}{b} \]
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Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{2 b} \\ & = -\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{b} \\ & = -\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{b} \\ & = -\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx=-\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {2 a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{b^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.40
method | result | size |
default | \(-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(70\) |
risch | \(-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(70\) |
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Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx=\left [-\frac {a \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, \sqrt {-b x + a} b \sqrt {x}}{2 \, b^{2}}, -\frac {a \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + \sqrt {-b x + a} b \sqrt {x}}{b^{2}}\right ] \]
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Result contains complex when optimal does not.
Time = 2.59 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.42 \[ \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx=\begin {cases} \frac {i \sqrt {a} \sqrt {x}}{b \sqrt {-1 + \frac {b x}{a}}} - \frac {i a \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {i x^{\frac {3}{2}}}{\sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {\sqrt {a} \sqrt {x} \sqrt {1 - \frac {b x}{a}}}{b} + \frac {a \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx=-\frac {a \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {3}{2}}} - \frac {\sqrt {-b x + a} a}{{\left (b^{2} - \frac {{\left (b x - a\right )} b}{x}\right )} \sqrt {x}} \]
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Time = 77.54 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx=\frac {{\left (\frac {a b \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} - \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}\right )} {\left | b \right |}}{b^{3}} \]
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Time = 0.54 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx=\frac {2\,a\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a-b\,x}-\sqrt {a}}\right )}{b^{3/2}}-\frac {\sqrt {x}\,\sqrt {a-b\,x}}{b} \]
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