Integrand size = 19, antiderivative size = 83 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^3} \, dx=\frac {1}{8 a^2 b c^3 (a-b x)^2}+\frac {1}{4 a^3 b c^3 (a-b x)}-\frac {1}{8 a^3 b c^3 (a+b x)}+\frac {3 \text {arctanh}\left (\frac {b x}{a}\right )}{8 a^4 b c^3} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {46, 214} \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^3} \, dx=\frac {3 \text {arctanh}\left (\frac {b x}{a}\right )}{8 a^4 b c^3}+\frac {1}{4 a^3 b c^3 (a-b x)}-\frac {1}{8 a^3 b c^3 (a+b x)}+\frac {1}{8 a^2 b c^3 (a-b x)^2} \]
[In]
[Out]
Rule 46
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4 a^2 c^3 (a-b x)^3}+\frac {1}{4 a^3 c^3 (a-b x)^2}+\frac {1}{8 a^3 c^3 (a+b x)^2}+\frac {3}{8 a^3 c^3 \left (a^2-b^2 x^2\right )}\right ) \, dx \\ & = \frac {1}{8 a^2 b c^3 (a-b x)^2}+\frac {1}{4 a^3 b c^3 (a-b x)}-\frac {1}{8 a^3 b c^3 (a+b x)}+\frac {3 \int \frac {1}{a^2-b^2 x^2} \, dx}{8 a^3 c^3} \\ & = \frac {1}{8 a^2 b c^3 (a-b x)^2}+\frac {1}{4 a^3 b c^3 (a-b x)}-\frac {1}{8 a^3 b c^3 (a+b x)}+\frac {3 \tanh ^{-1}\left (\frac {b x}{a}\right )}{8 a^4 b c^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^3} \, dx=\frac {\frac {2 a \left (2 a^2+3 a b x-3 b^2 x^2\right )}{(a-b x)^2 (a+b x)}-3 \log (a-b x)+3 \log (a+b x)}{16 a^4 b c^3} \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {-\frac {3 b \,x^{2}}{8 a^{3}}+\frac {3 x}{8 a^{2}}+\frac {1}{4 b a}}{\left (b x +a \right ) c^{3} \left (-b x +a \right )^{2}}-\frac {3 \ln \left (-b x +a \right )}{16 a^{4} c^{3} b}+\frac {3 \ln \left (b x +a \right )}{16 a^{4} c^{3} b}\) | \(80\) |
default | \(\frac {\frac {3 \ln \left (b x +a \right )}{16 a^{4} b}-\frac {1}{8 a^{3} b \left (b x +a \right )}-\frac {3 \ln \left (-b x +a \right )}{16 a^{4} b}+\frac {1}{4 a^{3} b \left (-b x +a \right )}+\frac {1}{8 a^{2} b \left (-b x +a \right )^{2}}}{c^{3}}\) | \(82\) |
norman | \(\frac {\frac {1}{4 a c b}+\frac {3 x}{8 a^{2} c}-\frac {3 b \,x^{2}}{8 a^{3} c}}{\left (b x +a \right ) c^{2} \left (-b x +a \right )^{2}}-\frac {3 \ln \left (-b x +a \right )}{16 a^{4} c^{3} b}+\frac {3 \ln \left (b x +a \right )}{16 a^{4} c^{3} b}\) | \(89\) |
parallelrisch | \(\frac {-3 \ln \left (b x -a \right ) x^{3} b^{5}+3 \ln \left (b x +a \right ) x^{3} b^{5}+3 \ln \left (b x -a \right ) x^{2} a \,b^{4}-3 \ln \left (b x +a \right ) x^{2} a \,b^{4}+3 \ln \left (b x -a \right ) x \,a^{2} b^{3}-3 \ln \left (b x +a \right ) x \,a^{2} b^{3}-6 a \,b^{4} x^{2}-3 \ln \left (b x -a \right ) a^{3} b^{2}+3 \ln \left (b x +a \right ) a^{3} b^{2}+6 a^{2} b^{3} x +4 a^{3} b^{2}}{16 a^{4} b^{3} c^{3} \left (b x +a \right ) \left (b x -a \right )^{2}}\) | \(179\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^3} \, dx=-\frac {6 \, a b^{2} x^{2} - 6 \, a^{2} b x - 4 \, a^{3} - 3 \, {\left (b^{3} x^{3} - a b^{2} x^{2} - a^{2} b x + a^{3}\right )} \log \left (b x + a\right ) + 3 \, {\left (b^{3} x^{3} - a b^{2} x^{2} - a^{2} b x + a^{3}\right )} \log \left (b x - a\right )}{16 \, {\left (a^{4} b^{4} c^{3} x^{3} - a^{5} b^{3} c^{3} x^{2} - a^{6} b^{2} c^{3} x + a^{7} b c^{3}\right )}} \]
[In]
[Out]
Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^3} \, dx=- \frac {- 2 a^{2} - 3 a b x + 3 b^{2} x^{2}}{8 a^{6} b c^{3} - 8 a^{5} b^{2} c^{3} x - 8 a^{4} b^{3} c^{3} x^{2} + 8 a^{3} b^{4} c^{3} x^{3}} - \frac {\frac {3 \log {\left (- \frac {a}{b} + x \right )}}{16} - \frac {3 \log {\left (\frac {a}{b} + x \right )}}{16}}{a^{4} b c^{3}} \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^3} \, dx=-\frac {3 \, b^{2} x^{2} - 3 \, a b x - 2 \, a^{2}}{8 \, {\left (a^{3} b^{4} c^{3} x^{3} - a^{4} b^{3} c^{3} x^{2} - a^{5} b^{2} c^{3} x + a^{6} b c^{3}\right )}} + \frac {3 \, \log \left (b x + a\right )}{16 \, a^{4} b c^{3}} - \frac {3 \, \log \left (b x - a\right )}{16 \, a^{4} b c^{3}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^3} \, dx=-\frac {3 \, \log \left ({\left | -\frac {2 \, a}{b x + a} + 1 \right |}\right )}{16 \, a^{4} b c^{3}} - \frac {1}{8 \, {\left (b x + a\right )} a^{3} b c^{3}} + \frac {\frac {12 \, a}{b x + a} - 5}{32 \, a^{4} b c^{3} {\left (\frac {2 \, a}{b x + a} - 1\right )}^{2}} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^3} \, dx=\frac {\frac {3\,x}{8\,a^2}+\frac {1}{4\,a\,b}-\frac {3\,b\,x^2}{8\,a^3}}{a^3\,c^3-a^2\,b\,c^3\,x-a\,b^2\,c^3\,x^2+b^3\,c^3\,x^3}+\frac {3\,\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{8\,a^4\,b\,c^3} \]
[In]
[Out]