Integrand size = 32, antiderivative size = 40 \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx=\frac {c (d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {656, 622, 31} \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx=\frac {c (d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
[In]
[Out]
Rule 31
Rule 622
Rule 656
Rubi steps \begin{align*} \text {integral}& = c \int \frac {1}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx \\ & = \frac {\left (c \left (c d e+c e^2 x\right )\right ) \int \frac {1}{c d e+c e^2 x} \, dx}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ & = \frac {c (d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx=\frac {c (d+e x) \log (d+e x)}{e \sqrt {c (d+e x)^2}} \]
[In]
[Out]
Time = 2.97 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {\sqrt {c \left (e x +d \right )^{2}}\, \ln \left (e x +d \right )}{\left (e x +d \right ) e}\) | \(29\) |
default | \(\frac {\sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}\, \ln \left (e x +d \right )}{\left (e x +d \right ) e}\) | \(40\) |
[In]
[Out]
none
Time = 0.39 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{e^{2} x + d e} \]
[In]
[Out]
\[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx=\int \frac {\sqrt {c \left (d + e x\right )^{2}}}{\left (d + e x\right )^{2}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {c} \log \left ({\left | e x + d \right |}\right ) \mathrm {sgn}\left (e x + d\right )}{e} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx=\int \frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{{\left (d+e\,x\right )}^2} \,d x \]
[In]
[Out]