Integrand size = 19, antiderivative size = 43 \[ \int \frac {(a+b x)^2}{a c-b c x} \, dx=-\frac {2 a x}{c}-\frac {(a+b x)^2}{2 b c}-\frac {4 a^2 \log (a-b x)}{b c} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {45} \[ \int \frac {(a+b x)^2}{a c-b c x} \, dx=-\frac {4 a^2 \log (a-b x)}{b c}-\frac {(a+b x)^2}{2 b c}-\frac {2 a x}{c} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 a}{c}-\frac {a+b x}{c}+\frac {4 a^2}{a c-b c x}\right ) \, dx \\ & = -\frac {2 a x}{c}-\frac {(a+b x)^2}{2 b c}-\frac {4 a^2 \log (a-b x)}{b c} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^2}{a c-b c x} \, dx=-\frac {3 a x}{c}-\frac {b x^2}{2 c}-\frac {4 a^2 \log (a-b x)}{b c} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {-\frac {b \,x^{2}}{2}-3 a x -\frac {4 a^{2} \ln \left (-b x +a \right )}{b}}{c}\) | \(31\) |
norman | \(-\frac {3 a x}{c}-\frac {b \,x^{2}}{2 c}-\frac {4 a^{2} \ln \left (-b x +a \right )}{b c}\) | \(36\) |
risch | \(-\frac {3 a x}{c}-\frac {b \,x^{2}}{2 c}-\frac {4 a^{2} \ln \left (-b x +a \right )}{b c}\) | \(36\) |
parallelrisch | \(\frac {-b^{2} x^{2}-8 a^{2} \ln \left (b x -a \right )-6 a b x}{2 b c}\) | \(36\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \frac {(a+b x)^2}{a c-b c x} \, dx=-\frac {b^{2} x^{2} + 6 \, a b x + 8 \, a^{2} \log \left (b x - a\right )}{2 \, b c} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.72 \[ \int \frac {(a+b x)^2}{a c-b c x} \, dx=- \frac {4 a^{2} \log {\left (- a + b x \right )}}{b c} - \frac {3 a x}{c} - \frac {b x^{2}}{2 c} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^2}{a c-b c x} \, dx=-\frac {4 \, a^{2} \log \left (b x - a\right )}{b c} - \frac {b x^{2} + 6 \, a x}{2 \, c} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x)^2}{a c-b c x} \, dx=-\frac {4 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b c} - \frac {b^{3} c x^{2} + 6 \, a b^{2} c x}{2 \, b^{2} c^{2}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \frac {(a+b x)^2}{a c-b c x} \, dx=-\frac {8\,a^2\,\ln \left (b\,x-a\right )+b^2\,x^2+6\,a\,b\,x}{2\,b\,c} \]
[In]
[Out]