Integrand size = 16, antiderivative size = 32 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \sqrt {x}+\frac {2 \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2436, 2332} \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=\frac {2 \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b}-2 \sqrt {x} \]
[In]
[Out]
Rule 2332
Rule 2436
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \log (a+b x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 \text {Subst}\left (\int \log (x) \, dx,x,a+b \sqrt {x}\right )}{b} \\ & = -2 \sqrt {x}+\frac {2 \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=2 \left (-\sqrt {x}+\frac {\left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b}\right ) \]
[In]
[Out]
Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {2 \left (a +b \sqrt {x}\right ) \ln \left (a +b \sqrt {x}\right )-2 b \sqrt {x}-2 a}{b}\) | \(32\) |
default | \(\frac {2 \left (a +b \sqrt {x}\right ) \ln \left (a +b \sqrt {x}\right )-2 b \sqrt {x}-2 a}{b}\) | \(32\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=\frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x}\right )}}{b} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (27) = 54\).
Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 4.88 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=\begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: \left (a = 0 \vee a = - b \sqrt {x}\right ) \wedge \left (a = - b \sqrt {x} \vee b = 0\right ) \\2 \sqrt {x} \log {\left (a \right )} & \text {for}\: b = 0 \\\frac {2 a^{2} \log {\left (a + b \sqrt {x} \right )}}{a b + b^{2} \sqrt {x}} + \frac {2 a^{2}}{a b + b^{2} \sqrt {x}} + \frac {4 a b \sqrt {x} \log {\left (a + b \sqrt {x} \right )}}{a b + b^{2} \sqrt {x}} + \frac {2 b^{2} x \log {\left (a + b \sqrt {x} \right )}}{a b + b^{2} \sqrt {x}} - \frac {2 b^{2} x}{a b + b^{2} \sqrt {x}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=\frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x} - a\right )}}{b} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=\frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x} - a\right )}}{b} \]
[In]
[Out]
Time = 1.46 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=2\,\sqrt {x}\,\ln \left (a+b\,\sqrt {x}\right )-2\,\sqrt {x}+\frac {2\,a\,\ln \left (a+b\,\sqrt {x}\right )}{b} \]
[In]
[Out]