Integrand size = 35, antiderivative size = 223 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {c^3 d^3}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac {4 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac {e^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^3}+\frac {3 c d e^2}{2 \left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (d+e x)}+\frac {10 c^3 d^3 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}-\frac {10 c^3 d^3 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^6} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 46} \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {4 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac {c^3 d^3}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac {10 c^3 d^3 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}-\frac {10 c^3 d^3 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^6}+\frac {6 c^2 d^2 e^2}{(d+e x) \left (c d^2-a e^2\right )^5}+\frac {3 c d e^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4}+\frac {e^2}{3 (d+e x)^3 \left (c d^2-a e^2\right )^3} \]
[In]
[Out]
Rule 46
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a e+c d x)^3 (d+e x)^4} \, dx \\ & = \int \left (\frac {c^4 d^4}{\left (c d^2-a e^2\right )^4 (a e+c d x)^3}-\frac {4 c^4 d^4 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)^2}+\frac {10 c^4 d^4 e^2}{\left (c d^2-a e^2\right )^6 (a e+c d x)}-\frac {e^3}{\left (c d^2-a e^2\right )^3 (d+e x)^4}-\frac {3 c d e^3}{\left (c d^2-a e^2\right )^4 (d+e x)^3}-\frac {6 c^2 d^2 e^3}{\left (c d^2-a e^2\right )^5 (d+e x)^2}-\frac {10 c^3 d^3 e^3}{\left (c d^2-a e^2\right )^6 (d+e x)}\right ) \, dx \\ & = -\frac {c^3 d^3}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac {4 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac {e^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^3}+\frac {3 c d e^2}{2 \left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (d+e x)}+\frac {10 c^3 d^3 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}-\frac {10 c^3 d^3 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^6} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {-\frac {3 c^3 d^3 \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {24 c^3 d^3 e \left (c d^2-a e^2\right )}{a e+c d x}-\frac {2 e^2 \left (-c d^2+a e^2\right )^3}{(d+e x)^3}+\frac {9 c d \left (c d^2 e-a e^3\right )^2}{(d+e x)^2}+\frac {36 c^2 d^2 e^2 \left (c d^2-a e^2\right )}{d+e x}+60 c^3 d^3 e^2 \log (a e+c d x)-60 c^3 d^3 e^2 \log (d+e x)}{6 \left (c d^2-a e^2\right )^6} \]
[In]
[Out]
Time = 2.70 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {c^{3} d^{3}}{2 \left (e^{2} a -c \,d^{2}\right )^{4} \left (c d x +a e \right )^{2}}+\frac {10 c^{3} d^{3} e^{2} \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{6}}-\frac {4 c^{3} d^{3} e}{\left (e^{2} a -c \,d^{2}\right )^{5} \left (c d x +a e \right )}-\frac {e^{2}}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{3}}-\frac {10 c^{3} d^{3} e^{2} \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{6}}-\frac {6 e^{2} c^{2} d^{2}}{\left (e^{2} a -c \,d^{2}\right )^{5} \left (e x +d \right )}+\frac {3 e^{2} c d}{2 \left (e^{2} a -c \,d^{2}\right )^{4} \left (e x +d \right )^{2}}\) | \(218\) |
risch | \(\frac {-\frac {10 c^{4} d^{4} e^{4} x^{4}}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}-\frac {5 c^{3} d^{3} e^{3} \left (3 e^{2} a +5 c \,d^{2}\right ) x^{3}}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}-\frac {5 \left (2 a^{2} e^{4}+23 a c \,d^{2} e^{2}+11 c^{2} d^{4}\right ) c^{2} d^{2} e^{2} x^{2}}{3 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}+\frac {5 \left (e^{6} a^{3}-11 d^{2} e^{4} a^{2} c -35 d^{4} e^{2} c^{2} a -3 c^{3} d^{6}\right ) c d e x}{6 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}-\frac {2 a^{4} e^{8}-13 a^{3} c \,d^{2} e^{6}+47 a^{2} c^{2} d^{4} e^{4}+27 a \,c^{3} d^{6} e^{2}-3 c^{4} d^{8}}{6 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}}{\left (e x +d \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{2}}+\frac {10 e^{2} c^{3} d^{3} \ln \left (-c d x -a e \right )}{a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}-\frac {10 e^{2} c^{3} d^{3} \ln \left (e x +d \right )}{a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}\) | \(777\) |
norman | \(\frac {-\frac {10 c^{4} d^{4} e^{4} x^{4}}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {\left (-15 a \,c^{5} d^{5} e^{7}-25 c^{6} d^{7} e^{5}\right ) x^{3}}{e^{2} d^{2} c^{2} \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}+\frac {-2 a^{4} c^{2} e^{9}+13 a^{3} c^{3} e^{7} d^{2}-47 a^{2} c^{4} e^{5} d^{4}-27 a \,c^{5} e^{3} d^{6}+3 c^{6} e \,d^{8}}{6 e \,c^{2} \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}+\frac {\left (5 c^{3} a^{3} e^{9} d -55 a^{2} c^{4} d^{3} e^{7}-175 a \,c^{5} d^{5} e^{5}-15 c^{6} d^{7} e^{3}\right ) x}{6 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) c^{2} e^{2}}+\frac {\left (-10 a^{2} c^{4} d^{3} e^{9}-115 a \,c^{5} d^{5} e^{7}-55 c^{6} d^{7} e^{5}\right ) x^{2}}{3 d \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) c^{2} e^{3}}}{\left (e x +d \right )^{3} \left (c d x +a e \right )^{2}}-\frac {10 e^{2} c^{3} d^{3} \ln \left (e x +d \right )}{a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}+\frac {10 e^{2} c^{3} d^{3} \ln \left (c d x +a e \right )}{a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}\) | \(805\) |
parallelrisch | \(-\frac {-15 x \,c^{7} d^{11} e^{4}+2 a^{5} c^{2} d^{2} e^{13}-15 a^{4} c^{3} d^{4} e^{11}+60 a^{3} c^{4} d^{6} e^{9}-20 a^{2} c^{5} d^{8} e^{7}-30 a \,c^{6} d^{10} e^{5}-60 \ln \left (c d x +a e \right ) x^{5} c^{7} d^{7} e^{8}+180 \ln \left (e x +d \right ) x^{4} c^{7} d^{8} e^{7}-180 \ln \left (c d x +a e \right ) x^{4} c^{7} d^{8} e^{7}+180 \ln \left (e x +d \right ) x^{3} c^{7} d^{9} e^{6}-180 \ln \left (c d x +a e \right ) x^{3} c^{7} d^{9} e^{6}+60 \ln \left (e x +d \right ) x^{2} c^{7} d^{10} e^{5}-60 \ln \left (c d x +a e \right ) x^{2} c^{7} d^{10} e^{5}+3 e^{3} d^{12} c^{7}+120 \ln \left (e x +d \right ) x^{4} a \,c^{6} d^{6} e^{9}-110 x^{2} c^{7} d^{10} e^{5}+60 \ln \left (e x +d \right ) x^{3} a^{2} c^{5} d^{5} e^{10}+120 \ln \left (e x +d \right ) x a \,c^{6} d^{9} e^{6}-180 \ln \left (c d x +a e \right ) x \,a^{2} c^{5} d^{7} e^{8}+180 \ln \left (e x +d \right ) x^{2} a^{2} c^{5} d^{6} e^{9}+360 \ln \left (e x +d \right ) x^{2} a \,c^{6} d^{8} e^{7}-180 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{5} d^{6} e^{9}-360 \ln \left (c d x +a e \right ) x^{2} a \,c^{6} d^{8} e^{7}-120 \ln \left (c d x +a e \right ) x^{4} a \,c^{6} d^{6} e^{9}-360 \ln \left (c d x +a e \right ) x^{3} a \,c^{6} d^{7} e^{8}-150 x^{3} c^{7} d^{9} e^{6}+360 \ln \left (e x +d \right ) x^{3} a \,c^{6} d^{7} e^{8}-60 \ln \left (c d x +a e \right ) x^{3} a^{2} c^{5} d^{5} e^{10}-120 \ln \left (c d x +a e \right ) x a \,c^{6} d^{9} e^{6}+180 \ln \left (e x +d \right ) x \,a^{2} c^{5} d^{7} e^{8}+60 \ln \left (e x +d \right ) a^{2} c^{5} d^{8} e^{7}-60 \ln \left (c d x +a e \right ) a^{2} c^{5} d^{8} e^{7}+60 x^{4} a \,c^{6} d^{6} e^{9}+90 x^{3} a^{2} c^{5} d^{5} e^{10}+60 x^{3} a \,c^{6} d^{7} e^{8}+20 x^{2} a^{3} c^{4} d^{4} e^{11}+210 x^{2} a^{2} c^{5} d^{6} e^{9}-120 x^{2} a \,c^{6} d^{8} e^{7}-5 x \,a^{4} c^{3} d^{3} e^{12}+60 x \,a^{3} c^{4} d^{5} e^{10}+120 x \,a^{2} c^{5} d^{7} e^{8}-160 x a \,c^{6} d^{9} e^{6}+60 \ln \left (e x +d \right ) x^{5} c^{7} d^{7} e^{8}-60 x^{4} c^{7} d^{8} e^{7}}{6 \left (a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{2} \left (e x +d \right ) c^{2} d^{2} e^{3}}\) | \(950\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1222 vs. \(2 (217) = 434\).
Time = 0.31 (sec) , antiderivative size = 1222, normalized size of antiderivative = 5.48 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Too large to display} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1357 vs. \(2 (206) = 412\).
Time = 9.93 (sec) , antiderivative size = 1357, normalized size of antiderivative = 6.09 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Too large to display} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 947 vs. \(2 (217) = 434\).
Time = 0.23 (sec) , antiderivative size = 947, normalized size of antiderivative = 4.25 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {10 \, c^{3} d^{3} e^{2} \log \left (c d x + a e\right )}{c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}} - \frac {10 \, c^{3} d^{3} e^{2} \log \left (e x + d\right )}{c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}} + \frac {60 \, c^{4} d^{4} e^{4} x^{4} - 3 \, c^{4} d^{8} + 27 \, a c^{3} d^{6} e^{2} + 47 \, a^{2} c^{2} d^{4} e^{4} - 13 \, a^{3} c d^{2} e^{6} + 2 \, a^{4} e^{8} + 30 \, {\left (5 \, c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 10 \, {\left (11 \, c^{4} d^{6} e^{2} + 23 \, a c^{3} d^{4} e^{4} + 2 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 5 \, {\left (3 \, c^{4} d^{7} e + 35 \, a c^{3} d^{5} e^{3} + 11 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x}{6 \, {\left (a^{2} c^{5} d^{13} e^{2} - 5 \, a^{3} c^{4} d^{11} e^{4} + 10 \, a^{4} c^{3} d^{9} e^{6} - 10 \, a^{5} c^{2} d^{7} e^{8} + 5 \, a^{6} c d^{5} e^{10} - a^{7} d^{3} e^{12} + {\left (c^{7} d^{12} e^{3} - 5 \, a c^{6} d^{10} e^{5} + 10 \, a^{2} c^{5} d^{8} e^{7} - 10 \, a^{3} c^{4} d^{6} e^{9} + 5 \, a^{4} c^{3} d^{4} e^{11} - a^{5} c^{2} d^{2} e^{13}\right )} x^{5} + {\left (3 \, c^{7} d^{13} e^{2} - 13 \, a c^{6} d^{11} e^{4} + 20 \, a^{2} c^{5} d^{9} e^{6} - 10 \, a^{3} c^{4} d^{7} e^{8} - 5 \, a^{4} c^{3} d^{5} e^{10} + 7 \, a^{5} c^{2} d^{3} e^{12} - 2 \, a^{6} c d e^{14}\right )} x^{4} + {\left (3 \, c^{7} d^{14} e - 9 \, a c^{6} d^{12} e^{3} + a^{2} c^{5} d^{10} e^{5} + 25 \, a^{3} c^{4} d^{8} e^{7} - 35 \, a^{4} c^{3} d^{6} e^{9} + 17 \, a^{5} c^{2} d^{4} e^{11} - a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} x^{3} + {\left (c^{7} d^{15} + a c^{6} d^{13} e^{2} - 17 \, a^{2} c^{5} d^{11} e^{4} + 35 \, a^{3} c^{4} d^{9} e^{6} - 25 \, a^{4} c^{3} d^{7} e^{8} - a^{5} c^{2} d^{5} e^{10} + 9 \, a^{6} c d^{3} e^{12} - 3 \, a^{7} d e^{14}\right )} x^{2} + {\left (2 \, a c^{6} d^{14} e - 7 \, a^{2} c^{5} d^{12} e^{3} + 5 \, a^{3} c^{4} d^{10} e^{5} + 10 \, a^{4} c^{3} d^{8} e^{7} - 20 \, a^{5} c^{2} d^{6} e^{9} + 13 \, a^{6} c d^{4} e^{11} - 3 \, a^{7} d^{2} e^{13}\right )} x\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (217) = 434\).
Time = 0.27 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.26 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {10 \, c^{4} d^{4} e^{2} \log \left ({\left | c d x + a e \right |}\right )}{c^{7} d^{13} - 6 \, a c^{6} d^{11} e^{2} + 15 \, a^{2} c^{5} d^{9} e^{4} - 20 \, a^{3} c^{4} d^{7} e^{6} + 15 \, a^{4} c^{3} d^{5} e^{8} - 6 \, a^{5} c^{2} d^{3} e^{10} + a^{6} c d e^{12}} - \frac {10 \, c^{3} d^{3} e^{3} \log \left ({\left | e x + d \right |}\right )}{c^{6} d^{12} e - 6 \, a c^{5} d^{10} e^{3} + 15 \, a^{2} c^{4} d^{8} e^{5} - 20 \, a^{3} c^{3} d^{6} e^{7} + 15 \, a^{4} c^{2} d^{4} e^{9} - 6 \, a^{5} c d^{2} e^{11} + a^{6} e^{13}} - \frac {3 \, c^{5} d^{10} - 30 \, a c^{4} d^{8} e^{2} - 20 \, a^{2} c^{3} d^{6} e^{4} + 60 \, a^{3} c^{2} d^{4} e^{6} - 15 \, a^{4} c d^{2} e^{8} + 2 \, a^{5} e^{10} - 60 \, {\left (c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} - 30 \, {\left (5 \, c^{5} d^{7} e^{3} - 2 \, a c^{4} d^{5} e^{5} - 3 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{3} - 10 \, {\left (11 \, c^{5} d^{8} e^{2} + 12 \, a c^{4} d^{6} e^{4} - 21 \, a^{2} c^{3} d^{4} e^{6} - 2 \, a^{3} c^{2} d^{2} e^{8}\right )} x^{2} - 5 \, {\left (3 \, c^{5} d^{9} e + 32 \, a c^{4} d^{7} e^{3} - 24 \, a^{2} c^{3} d^{5} e^{5} - 12 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x}{6 \, {\left (c d^{2} - a e^{2}\right )}^{6} {\left (c d x + a e\right )}^{2} {\left (e x + d\right )}^{3}} \]
[In]
[Out]
Time = 10.35 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.94 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {20\,c^3\,d^3\,e^2\,\mathrm {atanh}\left (\frac {a^6\,e^{12}-4\,a^5\,c\,d^2\,e^{10}+5\,a^4\,c^2\,d^4\,e^8-5\,a^2\,c^4\,d^8\,e^4+4\,a\,c^5\,d^{10}\,e^2-c^6\,d^{12}}{{\left (a\,e^2-c\,d^2\right )}^6}+\frac {2\,c\,d\,e\,x\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}{{\left (a\,e^2-c\,d^2\right )}^6}\right )}{{\left (a\,e^2-c\,d^2\right )}^6}-\frac {\frac {2\,a^4\,e^8-13\,a^3\,c\,d^2\,e^6+47\,a^2\,c^2\,d^4\,e^4+27\,a\,c^3\,d^6\,e^2-3\,c^4\,d^8}{6\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}+\frac {5\,c^2\,d\,x^3\,\left (5\,c^2\,d^4\,e^3+3\,a\,c\,d^2\,e^5\right )}{a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}}+\frac {5\,c^2\,d^2\,x^2\,\left (2\,a^2\,e^6+23\,a\,c\,d^2\,e^4+11\,c^2\,d^4\,e^2\right )}{3\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}+\frac {10\,c^4\,d^4\,e^4\,x^4}{a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}}+\frac {5\,c\,d\,e\,x\,\left (-a^3\,e^6+11\,a^2\,c\,d^2\,e^4+35\,a\,c^2\,d^4\,e^2+3\,c^3\,d^6\right )}{6\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}}{x\,\left (3\,a^2\,d^2\,e^3+2\,c\,a\,d^4\,e\right )+x^2\,\left (3\,a^2\,d\,e^4+6\,a\,c\,d^3\,e^2+c^2\,d^5\right )+x^3\,\left (a^2\,e^5+6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,e\right )+x^4\,\left (3\,c^2\,d^3\,e^2+2\,a\,c\,d\,e^4\right )+a^2\,d^3\,e^2+c^2\,d^2\,e^3\,x^5} \]
[In]
[Out]