Integrand size = 17, antiderivative size = 27 \[ \int \frac {x^5}{b x^2+c x^4} \, dx=\frac {x^2}{2 c}-\frac {b \log \left (b+c x^2\right )}{2 c^2} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 272, 45} \[ \int \frac {x^5}{b x^2+c x^4} \, dx=\frac {x^2}{2 c}-\frac {b \log \left (b+c x^2\right )}{2 c^2} \]
[In]
[Out]
Rule 45
Rule 272
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{b+c x^2} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x}{b+c x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{c}-\frac {b}{c (b+c x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {x^2}{2 c}-\frac {b \log \left (b+c x^2\right )}{2 c^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{b x^2+c x^4} \, dx=\frac {x^2}{2 c}-\frac {b \log \left (b+c x^2\right )}{2 c^2} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
parallelrisch | \(-\frac {-c \,x^{2}+b \ln \left (c \,x^{2}+b \right )}{2 c^{2}}\) | \(23\) |
default | \(\frac {x^{2}}{2 c}-\frac {b \ln \left (c \,x^{2}+b \right )}{2 c^{2}}\) | \(24\) |
norman | \(\frac {x^{2}}{2 c}-\frac {b \ln \left (c \,x^{2}+b \right )}{2 c^{2}}\) | \(24\) |
risch | \(\frac {x^{2}}{2 c}-\frac {b \ln \left (c \,x^{2}+b \right )}{2 c^{2}}\) | \(24\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{b x^2+c x^4} \, dx=\frac {c x^{2} - b \log \left (c x^{2} + b\right )}{2 \, c^{2}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {x^5}{b x^2+c x^4} \, dx=- \frac {b \log {\left (b + c x^{2} \right )}}{2 c^{2}} + \frac {x^{2}}{2 c} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {x^5}{b x^2+c x^4} \, dx=\frac {x^{2}}{2 \, c} - \frac {b \log \left (c x^{2} + b\right )}{2 \, c^{2}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {x^5}{b x^2+c x^4} \, dx=\frac {x^{2}}{2 \, c} - \frac {b \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{2}} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{b x^2+c x^4} \, dx=-\frac {b\,\ln \left (c\,x^2+b\right )-c\,x^2}{2\,c^2} \]
[In]
[Out]