Integrand size = 35, antiderivative size = 173 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}+\frac {c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac {e^3}{\left (c d^2-a e^2\right )^4 (d+e x)}-\frac {4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5} \]
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Time = 0.10 (sec) , antiderivative size = 173, 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} \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac {c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}-\frac {e^3}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac {4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5} \]
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Rule 46
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a e+c d x)^4 (d+e x)^2} \, dx \\ & = \int \left (\frac {c^2 d^2}{\left (c d^2-a e^2\right )^2 (a e+c d x)^4}-\frac {2 c^2 d^2 e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^3}+\frac {3 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac {4 c^2 d^2 e^3}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac {e^4}{\left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {4 c d e^4}{\left (c d^2-a e^2\right )^5 (d+e x)}\right ) \, dx \\ & = -\frac {c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}+\frac {c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac {e^3}{\left (c d^2-a e^2\right )^4 (d+e x)}-\frac {4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {\frac {c d \left (c d^2-a e^2\right )^3}{(a e+c d x)^3}-\frac {3 c d e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {9 c d e^2 \left (c d^2-a e^2\right )}{a e+c d x}+\frac {3 c d^2 e^3-3 a e^5}{d+e x}+12 c d e^3 \log (a e+c d x)-12 c d e^3 \log (d+e x)}{3 \left (-c d^2+a e^2\right )^5} \]
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Time = 2.46 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {c d}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d x +a e \right )^{3}}+\frac {4 c d \,e^{3} \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{5}}-\frac {3 c d \,e^{2}}{\left (e^{2} a -c \,d^{2}\right )^{4} \left (c d x +a e \right )}-\frac {c d e}{\left (e^{2} a -c \,d^{2}\right )^{3} \left (c d x +a e \right )^{2}}-\frac {e^{3}}{\left (e^{2} a -c \,d^{2}\right )^{4} \left (e x +d \right )}-\frac {4 c d \,e^{3} \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{5}}\) | \(173\) |
risch | \(\frac {-\frac {4 c^{3} d^{3} e^{3} x^{3}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {2 c^{2} d^{2} \left (5 e^{2} a +c \,d^{2}\right ) e^{2} x^{2}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {2 \left (11 a^{2} e^{4}+8 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) d e c x}{3 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}-\frac {3 e^{6} a^{3}+13 d^{2} e^{4} a^{2} c -5 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{3 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}}{\left (c d x +a e \right )^{2} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )}-\frac {4 d \,e^{3} c \ln \left (e x +d \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {4 d \,e^{3} c \ln \left (-c d x -a e \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}\) | \(546\) |
parallelrisch | \(-\frac {-12 x^{3} c^{7} d^{8} e^{4}-6 x^{2} c^{7} d^{9} e^{3}+2 x \,c^{7} d^{10} e^{2}+10 a^{3} c^{4} d^{5} e^{7}-18 a^{2} c^{5} d^{7} e^{5}+6 a \,c^{6} d^{9} e^{3}+3 a^{4} c^{3} d^{3} e^{9}+12 x^{3} a \,c^{6} d^{6} e^{6}+30 x^{2} a^{2} c^{5} d^{5} e^{7}-24 x^{2} a \,c^{6} d^{7} e^{5}+22 x \,a^{3} c^{4} d^{4} e^{8}-6 x \,a^{2} c^{5} d^{6} e^{6}-18 x a \,c^{6} d^{8} e^{4}+12 \ln \left (e x +d \right ) x^{4} c^{7} d^{7} e^{5}-12 \ln \left (c d x +a e \right ) x^{4} c^{7} d^{7} e^{5}+12 \ln \left (e x +d \right ) x^{3} c^{7} d^{8} e^{4}-12 \ln \left (c d x +a e \right ) x^{3} c^{7} d^{8} e^{4}+12 \ln \left (e x +d \right ) a^{3} c^{4} d^{5} e^{7}-12 \ln \left (c d x +a e \right ) a^{3} c^{4} d^{5} e^{7}-36 \ln \left (c d x +a e \right ) x^{2} a \,c^{6} d^{7} e^{5}+12 \ln \left (e x +d \right ) x \,a^{3} c^{4} d^{4} e^{8}+36 \ln \left (e x +d \right ) x \,a^{2} c^{5} d^{6} e^{6}-12 \ln \left (c d x +a e \right ) x \,a^{3} c^{4} d^{4} e^{8}-36 \ln \left (c d x +a e \right ) x \,a^{2} c^{5} d^{6} e^{6}+36 \ln \left (e x +d \right ) x^{3} a \,c^{6} d^{6} e^{6}-36 \ln \left (c d x +a e \right ) x^{3} a \,c^{6} d^{6} e^{6}+36 \ln \left (e x +d \right ) x^{2} a^{2} c^{5} d^{5} e^{7}+36 \ln \left (e x +d \right ) x^{2} a \,c^{6} d^{7} e^{5}-36 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{5} d^{5} e^{7}-c^{7} d^{11} e}{3 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right ) \left (c d x +a e \right )^{2} d^{3} e \,c^{3}}\) | \(670\) |
norman | \(\frac {-\frac {4 c^{3} d^{3} e^{5} x^{5}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {\left (-10 a \,c^{4} d^{4} e^{8}-10 d^{6} e^{6} c^{5}\right ) x^{4}}{e^{2} d^{2} c^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {\left (-a^{3} c^{3} d^{2} e^{10}-19 a^{2} c^{4} d^{4} e^{8}-19 a \,c^{5} d^{6} e^{6}-d^{8} e^{4} c^{6}\right ) x^{2}}{e^{2} d^{2} c^{3} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {a \left (-2 a^{2} c^{3} d^{2} e^{8}-16 a \,c^{4} d^{4} e^{6}-2 e^{4} d^{6} c^{5}\right ) x}{e d \,c^{3} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {-3 c^{3} e^{6} d^{2} a^{3}-13 c^{4} d^{4} e^{4} a^{2}+5 c^{5} d^{6} e^{2} a -c^{6} d^{8}}{3 c^{3} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {\left (-22 a^{2} c^{4} d^{4} e^{10}-76 a \,c^{5} d^{6} e^{8}-22 c^{6} d^{8} e^{6}\right ) x^{3}}{3 e^{3} d^{3} c^{3} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}-\frac {4 d \,e^{3} c \ln \left (e x +d \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {4 d \,e^{3} c \ln \left (c d x +a e \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}\) | \(792\) |
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Leaf count of result is larger than twice the leaf count of optimal. 837 vs. \(2 (171) = 342\).
Time = 0.28 (sec) , antiderivative size = 837, normalized size of antiderivative = 4.84 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {c^{4} d^{8} - 6 \, a c^{3} d^{6} e^{2} + 18 \, a^{2} c^{2} d^{4} e^{4} - 10 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8} + 12 \, {\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (c^{4} d^{6} e^{2} + 4 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 2 \, {\left (c^{4} d^{7} e - 9 \, a c^{3} d^{5} e^{3} - 3 \, a^{2} c^{2} d^{3} e^{5} + 11 \, a^{3} c d e^{7}\right )} x + 12 \, {\left (c^{4} d^{4} e^{4} x^{4} + a^{3} c d^{2} e^{6} + {\left (c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right ) - 12 \, {\left (c^{4} d^{4} e^{4} x^{4} + a^{3} c d^{2} e^{6} + {\left (c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (a^{3} c^{5} d^{11} e^{3} - 5 \, a^{4} c^{4} d^{9} e^{5} + 10 \, a^{5} c^{3} d^{7} e^{7} - 10 \, a^{6} c^{2} d^{5} e^{9} + 5 \, a^{7} c d^{3} e^{11} - a^{8} d e^{13} + {\left (c^{8} d^{13} e - 5 \, a c^{7} d^{11} e^{3} + 10 \, a^{2} c^{6} d^{9} e^{5} - 10 \, a^{3} c^{5} d^{7} e^{7} + 5 \, a^{4} c^{4} d^{5} e^{9} - a^{5} c^{3} d^{3} e^{11}\right )} x^{4} + {\left (c^{8} d^{14} - 2 \, a c^{7} d^{12} e^{2} - 5 \, a^{2} c^{6} d^{10} e^{4} + 20 \, a^{3} c^{5} d^{8} e^{6} - 25 \, a^{4} c^{4} d^{6} e^{8} + 14 \, a^{5} c^{3} d^{4} e^{10} - 3 \, a^{6} c^{2} d^{2} e^{12}\right )} x^{3} + 3 \, {\left (a c^{7} d^{13} e - 4 \, a^{2} c^{6} d^{11} e^{3} + 5 \, a^{3} c^{5} d^{9} e^{5} - 5 \, a^{5} c^{3} d^{5} e^{9} + 4 \, a^{6} c^{2} d^{3} e^{11} - a^{7} c d e^{13}\right )} x^{2} + {\left (3 \, a^{2} c^{6} d^{12} e^{2} - 14 \, a^{3} c^{5} d^{10} e^{4} + 25 \, a^{4} c^{4} d^{8} e^{6} - 20 \, a^{5} c^{3} d^{6} e^{8} + 5 \, a^{6} c^{2} d^{4} e^{10} + 2 \, a^{7} c d^{2} e^{12} - a^{8} e^{14}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1006 vs. \(2 (160) = 320\).
Time = 2.79 (sec) , antiderivative size = 1006, normalized size of antiderivative = 5.82 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=- \frac {4 c d e^{3} \log {\left (x + \frac {- \frac {4 a^{6} c d e^{15}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {24 a^{5} c^{2} d^{3} e^{13}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {60 a^{4} c^{3} d^{5} e^{11}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {80 a^{3} c^{4} d^{7} e^{9}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {60 a^{2} c^{5} d^{9} e^{7}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {24 a c^{6} d^{11} e^{5}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 a c d e^{5} - \frac {4 c^{7} d^{13} e^{3}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 c^{2} d^{3} e^{3}}{8 c^{2} d^{2} e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {4 c d e^{3} \log {\left (x + \frac {\frac {4 a^{6} c d e^{15}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {24 a^{5} c^{2} d^{3} e^{13}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {60 a^{4} c^{3} d^{5} e^{11}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {80 a^{3} c^{4} d^{7} e^{9}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {60 a^{2} c^{5} d^{9} e^{7}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {24 a c^{6} d^{11} e^{5}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 a c d e^{5} + \frac {4 c^{7} d^{13} e^{3}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 c^{2} d^{3} e^{3}}{8 c^{2} d^{2} e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {- 3 a^{3} e^{6} - 13 a^{2} c d^{2} e^{4} + 5 a c^{2} d^{4} e^{2} - c^{3} d^{6} - 12 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 30 a c^{2} d^{2} e^{4} - 6 c^{3} d^{4} e^{2}\right ) + x \left (- 22 a^{2} c d e^{5} - 16 a c^{2} d^{3} e^{3} + 2 c^{3} d^{5} e\right )}{3 a^{7} d e^{11} - 12 a^{6} c d^{3} e^{9} + 18 a^{5} c^{2} d^{5} e^{7} - 12 a^{4} c^{3} d^{7} e^{5} + 3 a^{3} c^{4} d^{9} e^{3} + x^{4} \cdot \left (3 a^{4} c^{3} d^{3} e^{9} - 12 a^{3} c^{4} d^{5} e^{7} + 18 a^{2} c^{5} d^{7} e^{5} - 12 a c^{6} d^{9} e^{3} + 3 c^{7} d^{11} e\right ) + x^{3} \cdot \left (9 a^{5} c^{2} d^{2} e^{10} - 33 a^{4} c^{3} d^{4} e^{8} + 42 a^{3} c^{4} d^{6} e^{6} - 18 a^{2} c^{5} d^{8} e^{4} - 3 a c^{6} d^{10} e^{2} + 3 c^{7} d^{12}\right ) + x^{2} \cdot \left (9 a^{6} c d e^{11} - 27 a^{5} c^{2} d^{3} e^{9} + 18 a^{4} c^{3} d^{5} e^{7} + 18 a^{3} c^{4} d^{7} e^{5} - 27 a^{2} c^{5} d^{9} e^{3} + 9 a c^{6} d^{11} e\right ) + x \left (3 a^{7} e^{12} - 3 a^{6} c d^{2} e^{10} - 18 a^{5} c^{2} d^{4} e^{8} + 42 a^{4} c^{3} d^{6} e^{6} - 33 a^{3} c^{4} d^{8} e^{4} + 9 a^{2} c^{5} d^{10} e^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (171) = 342\).
Time = 0.20 (sec) , antiderivative size = 656, normalized size of antiderivative = 3.79 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {4 \, c d e^{3} \log \left (c d x + a e\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} + \frac {4 \, c d e^{3} \log \left (e x + d\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} - \frac {12 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} - 5 \, a c^{2} d^{4} e^{2} + 13 \, a^{2} c d^{2} e^{4} + 3 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} + 5 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (c^{3} d^{5} e - 8 \, a c^{2} d^{3} e^{3} - 11 \, a^{2} c d e^{5}\right )} x}{3 \, {\left (a^{3} c^{4} d^{9} e^{3} - 4 \, a^{4} c^{3} d^{7} e^{5} + 6 \, a^{5} c^{2} d^{5} e^{7} - 4 \, a^{6} c d^{3} e^{9} + a^{7} d e^{11} + {\left (c^{7} d^{11} e - 4 \, a c^{6} d^{9} e^{3} + 6 \, a^{2} c^{5} d^{7} e^{5} - 4 \, a^{3} c^{4} d^{5} e^{7} + a^{4} c^{3} d^{3} e^{9}\right )} x^{4} + {\left (c^{7} d^{12} - a c^{6} d^{10} e^{2} - 6 \, a^{2} c^{5} d^{8} e^{4} + 14 \, a^{3} c^{4} d^{6} e^{6} - 11 \, a^{4} c^{3} d^{4} e^{8} + 3 \, a^{5} c^{2} d^{2} e^{10}\right )} x^{3} + 3 \, {\left (a c^{6} d^{11} e - 3 \, a^{2} c^{5} d^{9} e^{3} + 2 \, a^{3} c^{4} d^{7} e^{5} + 2 \, a^{4} c^{3} d^{5} e^{7} - 3 \, a^{5} c^{2} d^{3} e^{9} + a^{6} c d e^{11}\right )} x^{2} + {\left (3 \, a^{2} c^{5} d^{10} e^{2} - 11 \, a^{3} c^{4} d^{8} e^{4} + 14 \, a^{4} c^{3} d^{6} e^{6} - 6 \, a^{5} c^{2} d^{4} e^{8} - a^{6} c d^{2} e^{10} + a^{7} e^{12}\right )} x\right )}} \]
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (171) = 342\).
Time = 0.28 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.23 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {4 \, c^{2} d^{2} e^{3} \log \left ({\left | c d x + a e \right |}\right )}{c^{6} d^{11} - 5 \, a c^{5} d^{9} e^{2} + 10 \, a^{2} c^{4} d^{7} e^{4} - 10 \, a^{3} c^{3} d^{5} e^{6} + 5 \, a^{4} c^{2} d^{3} e^{8} - a^{5} c d e^{10}} + \frac {4 \, c d e^{4} \log \left ({\left | e x + d \right |}\right )}{c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}} - \frac {c^{4} d^{8} - 6 \, a c^{3} d^{6} e^{2} + 18 \, a^{2} c^{2} d^{4} e^{4} - 10 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8} + 12 \, {\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (c^{4} d^{6} e^{2} + 4 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 2 \, {\left (c^{4} d^{7} e - 9 \, a c^{3} d^{5} e^{3} - 3 \, a^{2} c^{2} d^{3} e^{5} + 11 \, a^{3} c d e^{7}\right )} x}{3 \, {\left (c d^{2} - a e^{2}\right )}^{5} {\left (c d x + a e\right )}^{3} {\left (e x + d\right )}} \]
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Time = 10.26 (sec) , antiderivative size = 617, normalized size of antiderivative = 3.57 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {8\,c\,d\,e^3\,\mathrm {atanh}\left (\frac {a^5\,e^{10}-3\,a^4\,c\,d^2\,e^8+2\,a^3\,c^2\,d^4\,e^6+2\,a^2\,c^3\,d^6\,e^4-3\,a\,c^4\,d^8\,e^2+c^5\,d^{10}}{{\left (a\,e^2-c\,d^2\right )}^5}+\frac {2\,c\,d\,e\,x\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{{\left (a\,e^2-c\,d^2\right )}^5}\right )}{{\left (a\,e^2-c\,d^2\right )}^5}-\frac {\frac {3\,a^3\,e^6+13\,a^2\,c\,d^2\,e^4-5\,a\,c^2\,d^4\,e^2+c^3\,d^6}{3\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {2\,e\,x\,\left (11\,a^2\,c\,d\,e^4+8\,a\,c^2\,d^3\,e^2-c^3\,d^5\right )}{3\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {2\,e^2\,x^2\,\left (c^3\,d^4+5\,a\,c^2\,d^2\,e^2\right )}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}+\frac {4\,c^3\,d^3\,e^3\,x^3}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}}{x\,\left (a^3\,e^4+3\,c\,a^2\,d^2\,e^2\right )+x^3\,\left (c^3\,d^4+3\,a\,c^2\,d^2\,e^2\right )+x^2\,\left (3\,a^2\,c\,d\,e^3+3\,a\,c^2\,d^3\,e\right )+a^3\,d\,e^3+c^3\,d^3\,e\,x^4} \]
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