Integrand size = 15, antiderivative size = 18 \[ \int \frac {x}{a x^2+b x^3} \, dx=\frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1598, 36, 29, 31} \[ \int \frac {x}{a x^2+b x^3} \, dx=\frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x (a+b x)} \, dx \\ & = \frac {\int \frac {1}{x} \, dx}{a}-\frac {b \int \frac {1}{a+b x} \, dx}{a} \\ & = \frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x}{a x^2+b x^3} \, dx=\frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \]
[In]
[Out]
Time = 2.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89
method | result | size |
parallelrisch | \(\frac {\ln \left (x \right )-\ln \left (b x +a \right )}{a}\) | \(16\) |
default | \(\frac {\ln \left (x \right )}{a}-\frac {\ln \left (b x +a \right )}{a}\) | \(19\) |
norman | \(\frac {\ln \left (x \right )}{a}-\frac {\ln \left (b x +a \right )}{a}\) | \(19\) |
risch | \(\frac {\ln \left (-x \right )}{a}-\frac {\ln \left (b x +a \right )}{a}\) | \(21\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {x}{a x^2+b x^3} \, dx=-\frac {\log \left (b x + a\right ) - \log \left (x\right )}{a} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.56 \[ \int \frac {x}{a x^2+b x^3} \, dx=\frac {\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}}{a} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x}{a x^2+b x^3} \, dx=-\frac {\log \left (b x + a\right )}{a} + \frac {\log \left (x\right )}{a} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x}{a x^2+b x^3} \, dx=-\frac {\log \left ({\left | b x + a \right |}\right )}{a} + \frac {\log \left ({\left | x \right |}\right )}{a} \]
[In]
[Out]
Time = 8.90 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {x}{a x^2+b x^3} \, dx=-\frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a} \]
[In]
[Out]