Integrand size = 15, antiderivative size = 42 \[ \int \frac {\sqrt {-a+b x}}{x^2} \, dx=-\frac {\sqrt {-a+b x}}{x}+\frac {b \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {43, 65, 211} \[ \int \frac {\sqrt {-a+b x}}{x^2} \, dx=\frac {b \arctan \left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {b x-a}}{x} \]
[In]
[Out]
Rule 43
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {-a+b x}}{x}+\frac {1}{2} b \int \frac {1}{x \sqrt {-a+b x}} \, dx \\ & = -\frac {\sqrt {-a+b x}}{x}+\text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right ) \\ & = -\frac {\sqrt {-a+b x}}{x}+\frac {b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-a+b x}}{x^2} \, dx=-\frac {\sqrt {-a+b x}}{x}+\frac {b \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {-b x +a}{x \sqrt {b x -a}}+\frac {b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{\sqrt {a}}\) | \(40\) |
pseudoelliptic | \(\frac {b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right ) x -\sqrt {b x -a}\, \sqrt {a}}{\sqrt {a}\, x}\) | \(40\) |
derivativedivides | \(2 b \left (-\frac {\sqrt {b x -a}}{2 b x}+\frac {\arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\) | \(41\) |
default | \(2 b \left (-\frac {\sqrt {b x -a}}{2 b x}+\frac {\arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\) | \(41\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.33 \[ \int \frac {\sqrt {-a+b x}}{x^2} \, dx=\left [-\frac {\sqrt {-a} b x \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, \sqrt {b x - a} a}{2 \, a x}, \frac {\sqrt {a} b x \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) - \sqrt {b x - a} a}{a x}\right ] \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.45 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.79 \[ \int \frac {\sqrt {-a+b x}}{x^2} \, dx=\begin {cases} - \frac {i \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{\sqrt {x}} + \frac {i b \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {a}{\sqrt {b} x^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}} - \frac {\sqrt {b}}{\sqrt {x} \sqrt {- \frac {a}{b x} + 1}} - \frac {b \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {-a+b x}}{x^2} \, dx=\frac {b \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}} - \frac {\sqrt {b x - a}}{x} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {-a+b x}}{x^2} \, dx=\frac {\frac {b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}} - \frac {\sqrt {b x - a} b}{x}}{b} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {-a+b x}}{x^2} \, dx=\frac {b\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {b\,x-a}}{x} \]
[In]
[Out]