Integrand size = 20, antiderivative size = 56 \[ \int \frac {d+e x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {1}{4} e \sqrt {9+12 x+4 x^2}+\frac {(2 d-3 e) (3+2 x) \log (3+2 x)}{4 \sqrt {9+12 x+4 x^2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {654, 622, 31} \[ \int \frac {d+e x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {(2 x+3) (2 d-3 e) \log (2 x+3)}{4 \sqrt {4 x^2+12 x+9}}+\frac {1}{4} e \sqrt {4 x^2+12 x+9} \]
[In]
[Out]
Rule 31
Rule 622
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} e \sqrt {9+12 x+4 x^2}+\frac {1}{2} (2 d-3 e) \int \frac {1}{\sqrt {9+12 x+4 x^2}} \, dx \\ & = \frac {1}{4} e \sqrt {9+12 x+4 x^2}+\frac {((2 d-3 e) (6+4 x)) \int \frac {1}{6+4 x} \, dx}{2 \sqrt {9+12 x+4 x^2}} \\ & = \frac {1}{4} e \sqrt {9+12 x+4 x^2}+\frac {(2 d-3 e) (3+2 x) \log (3+2 x)}{4 \sqrt {9+12 x+4 x^2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.73 \[ \int \frac {d+e x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {(3+2 x) \left (\frac {e x}{2}+\frac {1}{4} (2 d-3 e) \log (3+2 x)\right )}{\sqrt {(3+2 x)^2}} \]
[In]
[Out]
Time = 2.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.46
method | result | size |
meijerg | \(\frac {3 e \left (\frac {2 x}{3}-\ln \left (1+\frac {2 x}{3}\right )\right )}{4}+\frac {d \ln \left (1+\frac {2 x}{3}\right )}{2}\) | \(26\) |
default | \(\frac {\left (2 x +3\right ) \left (2 \ln \left (2 x +3\right ) d -3 e \ln \left (2 x +3\right )+2 e x \right )}{4 \sqrt {\left (2 x +3\right )^{2}}}\) | \(40\) |
risch | \(\frac {\sqrt {\left (2 x +3\right )^{2}}\, e x}{4 x +6}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (-\frac {3 e}{2}+d \right ) \ln \left (2 x +3\right )}{4 x +6}\) | \(51\) |
[In]
[Out]
none
Time = 0.55 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.36 \[ \int \frac {d+e x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {1}{2} \, e x + \frac {1}{4} \, {\left (2 \, d - 3 \, e\right )} \log \left (2 \, x + 3\right ) \]
[In]
[Out]
Time = 0.55 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {d+e x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {e \sqrt {4 x^{2} + 12 x + 9}}{4} + \frac {\left (d - \frac {3 e}{2}\right ) \left (x + \frac {3}{2}\right ) \log {\left (x + \frac {3}{2} \right )}}{2 \sqrt {\left (x + \frac {3}{2}\right )^{2}}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.54 \[ \int \frac {d+e x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {1}{2} \, d \log \left (x + \frac {3}{2}\right ) - \frac {3}{4} \, e \log \left (x + \frac {3}{2}\right ) + \frac {1}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} e \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.66 \[ \int \frac {d+e x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {e x}{2 \, \mathrm {sgn}\left (2 \, x + 3\right )} + \frac {{\left (2 \, d - 3 \, e\right )} \log \left ({\left | 2 \, x + 3 \right |}\right )}{4 \, \mathrm {sgn}\left (2 \, x + 3\right )} \]
[In]
[Out]
Time = 10.48 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {d+e x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {e\,\sqrt {4\,x^2+12\,x+9}}{4}-\frac {3\,e\,\ln \left (x+\frac {\left |2\,x+3\right |}{2}+\frac {3}{2}\right )}{4}+\frac {d\,\ln \left (4\,x+6\right )\,\mathrm {sign}\left (8\,x+12\right )}{2} \]
[In]
[Out]