Integrand size = 13, antiderivative size = 40 \[ \int \frac {1}{x^{3/2} (a+b x)} \, dx=-\frac {2}{a \sqrt {x}}-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {53, 65, 211} \[ \int \frac {1}{x^{3/2} (a+b x)} \, dx=-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}} \]
[In]
[Out]
Rule 53
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{a \sqrt {x}}-\frac {b \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{a} \\ & = -\frac {2}{a \sqrt {x}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a} \\ & = -\frac {2}{a \sqrt {x}}-\frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{3/2} (a+b x)} \, dx=-\frac {2}{a \sqrt {x}}-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {2}{a \sqrt {x}}-\frac {2 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}\) | \(32\) |
default | \(-\frac {2}{a \sqrt {x}}-\frac {2 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}\) | \(32\) |
risch | \(-\frac {2}{a \sqrt {x}}-\frac {2 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}\) | \(32\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.32 \[ \int \frac {1}{x^{3/2} (a+b x)} \, dx=\left [\frac {x \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) - 2 \, \sqrt {x}}{a x}, \frac {2 \, {\left (x \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - \sqrt {x}\right )}}{a x}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (37) = 74\).
Time = 0.95 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.12 \[ \int \frac {1}{x^{3/2} (a+b x)} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 b x^{\frac {3}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a \sqrt {x}} & \text {for}\: b = 0 \\- \frac {\log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{a \sqrt {- \frac {a}{b}}} + \frac {\log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{a \sqrt {- \frac {a}{b}}} - \frac {2}{a \sqrt {x}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^{3/2} (a+b x)} \, dx=-\frac {2 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {2}{a \sqrt {x}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^{3/2} (a+b x)} \, dx=-\frac {2 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {2}{a \sqrt {x}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^{3/2} (a+b x)} \, dx=-\frac {2}{a\,\sqrt {x}}-\frac {2\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \]
[In]
[Out]