Integrand size = 19, antiderivative size = 134 \[ \int \frac {1}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=-\frac {e}{2 d (c d-b e) (d+e x)^2}-\frac {e (2 c d-b e)}{d^2 (c d-b e)^2 (d+e x)}+\frac {\log (x)}{b d^3}-\frac {c^3 \log (b+c x)}{b (c d-b e)^3}+\frac {e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) \log (d+e x)}{d^3 (c d-b e)^3} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 134, 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 = {712} \[ \int \frac {1}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=\frac {e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right ) \log (d+e x)}{d^3 (c d-b e)^3}-\frac {c^3 \log (b+c x)}{b (c d-b e)^3}-\frac {e (2 c d-b e)}{d^2 (d+e x) (c d-b e)^2}-\frac {e}{2 d (d+e x)^2 (c d-b e)}+\frac {\log (x)}{b d^3} \]
[In]
[Out]
Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{b d^3 x}+\frac {c^4}{b (-c d+b e)^3 (b+c x)}+\frac {e^2}{d (c d-b e) (d+e x)^3}+\frac {e^2 (2 c d-b e)}{d^2 (c d-b e)^2 (d+e x)^2}+\frac {e^2 \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx \\ & = -\frac {e}{2 d (c d-b e) (d+e x)^2}-\frac {e (2 c d-b e)}{d^2 (c d-b e)^2 (d+e x)}+\frac {\log (x)}{b d^3}-\frac {c^3 \log (b+c x)}{b (c d-b e)^3}+\frac {e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) \log (d+e x)}{d^3 (c d-b e)^3} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=\frac {\log (x)}{b d^3}+\frac {\frac {2 c^3 \log (b+c x)}{b}+\frac {e \left (\frac {d (c d-b e) (-b e (3 d+2 e x)+c d (5 d+4 e x))}{(d+e x)^2}-2 \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) \log (d+e x)\right )}{d^3}}{2 (-c d+b e)^3} \]
[In]
[Out]
Time = 2.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {\ln \left (x \right )}{b \,d^{3}}+\frac {c^{3} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b}+\frac {e}{2 d \left (b e -c d \right ) \left (e x +d \right )^{2}}+\frac {e \left (b e -2 c d \right )}{d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )}-\frac {e \left (b^{2} e^{2}-3 b c d e +3 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{d^{3} \left (b e -c d \right )^{3}}\) | \(131\) |
norman | \(\frac {\frac {\left (-2 e^{2} b +3 c d e \right ) e x}{d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (-3 e^{2} b +5 c d e \right ) e^{2} x^{2}}{2 d^{3} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}}{\left (e x +d \right )^{2}}+\frac {\ln \left (x \right )}{b \,d^{3}}+\frac {c^{3} \ln \left (c x +b \right )}{b \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {e \left (b^{2} e^{2}-3 b c d e +3 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\) | \(230\) |
risch | \(\frac {\frac {e^{2} \left (b e -2 c d \right ) x}{d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {e \left (3 b e -5 c d \right )}{2 d \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}}{\left (e x +d \right )^{2}}+\frac {c^{3} \ln \left (c x +b \right )}{b \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {\ln \left (-x \right )}{d^{3} b}-\frac {e^{3} \ln \left (-e x -d \right ) b^{2}}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {3 e^{2} \ln \left (-e x -d \right ) b c}{d^{2} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {3 e \ln \left (-e x -d \right ) c^{2}}{d \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\) | \(321\) |
parallelrisch | \(\frac {-2 \ln \left (x \right ) c^{3} d^{5}+6 \ln \left (e x +d \right ) b^{2} c \,d^{3} e^{2}-6 \ln \left (e x +d \right ) b \,c^{2} d^{4} e -3 x^{2} b^{3} e^{5}+2 \ln \left (c x +b \right ) c^{3} d^{5}-6 \ln \left (x \right ) x^{2} b^{2} c d \,e^{4}+6 \ln \left (x \right ) x^{2} b \,c^{2} d^{2} e^{3}+6 \ln \left (e x +d \right ) x^{2} b^{2} c d \,e^{4}-6 \ln \left (e x +d \right ) x^{2} b \,c^{2} d^{2} e^{3}-12 \ln \left (x \right ) x \,b^{2} c \,d^{2} e^{3}+12 \ln \left (x \right ) x b \,c^{2} d^{3} e^{2}+12 \ln \left (e x +d \right ) x \,b^{2} c \,d^{2} e^{3}-12 \ln \left (e x +d \right ) x b \,c^{2} d^{3} e^{2}-4 x \,b^{3} d \,e^{4}+2 \ln \left (x \right ) x^{2} b^{3} e^{5}-2 \ln \left (e x +d \right ) x^{2} b^{3} e^{5}+2 \ln \left (x \right ) b^{3} d^{2} e^{3}-2 \ln \left (e x +d \right ) b^{3} d^{2} e^{3}+8 x^{2} b^{2} c d \,e^{4}-5 x^{2} b \,c^{2} d^{2} e^{3}+10 x \,b^{2} c \,d^{2} e^{3}-6 x b \,c^{2} d^{3} e^{2}-2 \ln \left (x \right ) x^{2} c^{3} d^{3} e^{2}+2 \ln \left (c x +b \right ) x^{2} c^{3} d^{3} e^{2}+4 \ln \left (x \right ) x \,b^{3} d \,e^{4}-4 \ln \left (x \right ) x \,c^{3} d^{4} e +4 \ln \left (c x +b \right ) x \,c^{3} d^{4} e -4 \ln \left (e x +d \right ) x \,b^{3} d \,e^{4}-6 \ln \left (x \right ) b^{2} c \,d^{3} e^{2}+6 \ln \left (x \right ) b \,c^{2} d^{4} e}{2 \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) b \left (e x +d \right )^{2} d^{3}}\) | \(506\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (132) = 264\).
Time = 4.25 (sec) , antiderivative size = 506, normalized size of antiderivative = 3.78 \[ \int \frac {1}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=-\frac {5 \, b c^{2} d^{4} e - 8 \, b^{2} c d^{3} e^{2} + 3 \, b^{3} d^{2} e^{3} + 2 \, {\left (2 \, b c^{2} d^{3} e^{2} - 3 \, b^{2} c d^{2} e^{3} + b^{3} d e^{4}\right )} x + 2 \, {\left (c^{3} d^{3} e^{2} x^{2} + 2 \, c^{3} d^{4} e x + c^{3} d^{5}\right )} \log \left (c x + b\right ) - 2 \, {\left (3 \, b c^{2} d^{4} e - 3 \, b^{2} c d^{3} e^{2} + b^{3} d^{2} e^{3} + {\left (3 \, b c^{2} d^{2} e^{3} - 3 \, b^{2} c d e^{4} + b^{3} e^{5}\right )} x^{2} + 2 \, {\left (3 \, b c^{2} d^{3} e^{2} - 3 \, b^{2} c d^{2} e^{3} + b^{3} d e^{4}\right )} x\right )} \log \left (e x + d\right ) - 2 \, {\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e + 3 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3} + {\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} + 3 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{2} + 2 \, {\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x\right )} \log \left (x\right )}{2 \, {\left (b c^{3} d^{8} - 3 \, b^{2} c^{2} d^{7} e + 3 \, b^{3} c d^{6} e^{2} - b^{4} d^{5} e^{3} + {\left (b c^{3} d^{6} e^{2} - 3 \, b^{2} c^{2} d^{5} e^{3} + 3 \, b^{3} c d^{4} e^{4} - b^{4} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (b c^{3} d^{7} e - 3 \, b^{2} c^{2} d^{6} e^{2} + 3 \, b^{3} c d^{5} e^{3} - b^{4} d^{4} e^{4}\right )} x\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (132) = 264\).
Time = 0.22 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.99 \[ \int \frac {1}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=-\frac {c^{3} \log \left (c x + b\right )}{b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}} + \frac {{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {5 \, c d^{2} e - 3 \, b d e^{2} + 2 \, {\left (2 \, c d e^{2} - b e^{3}\right )} x}{2 \, {\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} + {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}} + \frac {\log \left (x\right )}{b d^{3}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.75 \[ \int \frac {1}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=-\frac {c^{4} \log \left ({\left | c x + b \right |}\right )}{b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} - b^{4} c e^{3}} + \frac {{\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} + \frac {\log \left ({\left | x \right |}\right )}{b d^{3}} - \frac {5 \, c^{2} d^{4} e - 8 \, b c d^{3} e^{2} + 3 \, b^{2} d^{2} e^{3} + 2 \, {\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4}\right )} x}{2 \, {\left (c d - b e\right )}^{3} {\left (e x + d\right )}^{2} d^{3}} \]
[In]
[Out]
Time = 10.01 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.75 \[ \int \frac {1}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=\frac {\frac {3\,b\,e^2-5\,c\,d\,e}{2\,d\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {e^2\,x\,\left (b\,e-2\,c\,d\right )}{d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{d^2+2\,d\,e\,x+e^2\,x^2}+\frac {c^3\,\ln \left (b+c\,x\right )}{b^4\,e^3-3\,b^3\,c\,d\,e^2+3\,b^2\,c^2\,d^2\,e-b\,c^3\,d^3}+\frac {\ln \left (d+e\,x\right )\,\left (b^2\,e^3-3\,b\,c\,d\,e^2+3\,c^2\,d^2\,e\right )}{-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}+\frac {\ln \left (x\right )}{b\,d^3} \]
[In]
[Out]