Integrand size = 24, antiderivative size = 48 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx=-\frac {2 \log (b+2 c x)}{\left (b^2-4 a c\right ) d}+\frac {\log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right ) d} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {695, 31, 642} \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx=\frac {\log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )}-\frac {2 \log (b+2 c x)}{d \left (b^2-4 a c\right )} \]
[In]
[Out]
Rule 31
Rule 642
Rule 695
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) d^2}-\frac {(4 c) \int \frac {1}{b+2 c x} \, dx}{\left (b^2-4 a c\right ) d} \\ & = -\frac {2 \log (b+2 c x)}{\left (b^2-4 a c\right ) d}+\frac {\log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right ) d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx=\frac {-2 \log (b+2 c x)+\log (a+x (b+c x))}{\left (b^2-4 a c\right ) d} \]
[In]
[Out]
Time = 2.73 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85
method | result | size |
parallelrisch | \(\frac {2 \ln \left (\frac {b}{2}+c x \right )-\ln \left (c \,x^{2}+b x +a \right )}{d \left (4 a c -b^{2}\right )}\) | \(41\) |
default | \(\frac {-\frac {\ln \left (c \,x^{2}+b x +a \right )}{4 a c -b^{2}}+\frac {2 \ln \left (2 c x +b \right )}{4 a c -b^{2}}}{d}\) | \(52\) |
norman | \(\frac {2 \ln \left (2 c x +b \right )}{d \left (4 a c -b^{2}\right )}-\frac {\ln \left (c \,x^{2}+b x +a \right )}{d \left (4 a c -b^{2}\right )}\) | \(54\) |
risch | \(\frac {2 \ln \left (2 c x +b \right )}{d \left (4 a c -b^{2}\right )}-\frac {\ln \left (c \,x^{2}+b x +a \right )}{d \left (4 a c -b^{2}\right )}\) | \(54\) |
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx=\frac {\log \left (c x^{2} + b x + a\right ) - 2 \, \log \left (2 \, c x + b\right )}{{\left (b^{2} - 4 \, a c\right )} d} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx=\frac {2 \log {\left (\frac {b}{2 c} + x \right )}}{d \left (4 a c - b^{2}\right )} - \frac {\log {\left (\frac {a}{c} + \frac {b x}{c} + x^{2} \right )}}{d \left (4 a c - b^{2}\right )} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx=\frac {\log \left (c x^{2} + b x + a\right )}{{\left (b^{2} - 4 \, a c\right )} d} - \frac {2 \, \log \left (2 \, c x + b\right )}{{\left (b^{2} - 4 \, a c\right )} d} \]
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx=-\frac {2 \, c^{2} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{2} c^{2} d - 4 \, a c^{3} d} + \frac {\log \left (c x^{2} + b x + a\right )}{b^{2} d - 4 \, a c d} \]
[In]
[Out]
Time = 9.63 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx=\frac {2\,c\,d\,\ln \left (\frac {{\left (b+2\,c\,x\right )}^2}{c\,x^2+b\,x+a}\right )}{8\,a\,c^2\,d^2-2\,b^2\,c\,d^2} \]
[In]
[Out]