Integrand size = 20, antiderivative size = 256 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^3 x}{e^6}-\frac {\left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^5}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{4 e^7 (d+e x)^4}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{2 e^7 (d+e x)^2}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7} \]
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Time = 0.17 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac {c^3 x}{e^6} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3}{e^6}+\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^6}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^5}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)^4}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^6 (d+e x)^3}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 (d+e x)^2}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)}\right ) \, dx \\ & = \frac {c^3 x}{e^6}-\frac {\left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^5}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{4 e^7 (d+e x)^4}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{2 e^7 (d+e x)^2}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {2 c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )+e^3 \left (4 a^3 e^3+3 a^2 b e^2 (d+5 e x)+2 a b^2 e \left (d^2+5 d e x+10 e^2 x^2\right )+b^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+2 c e^2 \left (a^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a b e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+6 b^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+c^2 e \left (12 a e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-b d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 c^2 (2 c d-b e) (d+e x)^5 \log (d+e x)}{20 e^7 (d+e x)^5} \]
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Time = 3.24 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.73
method | result | size |
norman | \(\frac {\frac {c^{3} x^{6}}{e}-\frac {4 e^{6} a^{3}+3 a^{2} b d \,e^{5}+2 d^{2} e^{4} a^{2} c +2 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+12 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}-137 b \,c^{2} d^{5} e +274 c^{3} d^{6}}{20 e^{7}}-\frac {\left (3 a \,c^{2} e^{2}+3 b^{2} e^{2} c -15 d e b \,c^{2}+30 c^{3} d^{2}\right ) x^{4}}{e^{3}}-\frac {\left (6 a b c \,e^{3}+12 c^{2} a d \,e^{2}+b^{3} e^{3}+12 b^{2} d \,e^{2} c -90 b \,c^{2} d^{2} e +180 c^{3} d^{3}\right ) x^{3}}{2 e^{4}}-\frac {\left (2 e^{4} a^{2} c +2 a \,b^{2} e^{4}+6 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-110 d^{3} e b \,c^{2}+220 d^{4} c^{3}\right ) x^{2}}{2 e^{5}}-\frac {\left (3 a^{2} b \,e^{5}+2 d \,e^{4} a^{2} c +2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+12 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+12 b^{2} c \,d^{3} e^{2}-125 b \,c^{2} d^{4} e +250 d^{5} c^{3}\right ) x}{4 e^{6}}}{\left (e x +d \right )^{5}}+\frac {3 c^{2} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{7}}\) | \(444\) |
risch | \(\frac {c^{3} x}{e^{6}}+\frac {\left (-3 a \,c^{2} e^{5}-3 b^{2} c \,e^{5}+15 b \,c^{2} d \,e^{4}-15 c^{3} d^{2} e^{3}\right ) x^{4}-\frac {e^{2} \left (6 a b c \,e^{3}+12 c^{2} a d \,e^{2}+b^{3} e^{3}+12 b^{2} d \,e^{2} c -90 b \,c^{2} d^{2} e +100 c^{3} d^{3}\right ) x^{3}}{2}-\frac {e \left (2 e^{4} a^{2} c +2 a \,b^{2} e^{4}+6 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-110 d^{3} e b \,c^{2}+130 d^{4} c^{3}\right ) x^{2}}{2}+\left (-\frac {3}{4} a^{2} b \,e^{5}-\frac {1}{2} d \,e^{4} a^{2} c -\frac {1}{2} a \,b^{2} d \,e^{4}-\frac {3}{2} a b c \,d^{2} e^{3}-3 d^{3} e^{2} c^{2} a -\frac {1}{4} b^{3} d^{2} e^{3}-3 b^{2} c \,d^{3} e^{2}+\frac {125}{4} b \,c^{2} d^{4} e -\frac {77}{2} d^{5} c^{3}\right ) x -\frac {4 e^{6} a^{3}+3 a^{2} b d \,e^{5}+2 d^{2} e^{4} a^{2} c +2 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+12 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}-137 b \,c^{2} d^{5} e +174 c^{3} d^{6}}{20 e}}{e^{6} \left (e x +d \right )^{5}}+\frac {3 c^{2} \ln \left (e x +d \right ) b}{e^{6}}-\frac {6 c^{3} d \ln \left (e x +d \right )}{e^{7}}\) | \(449\) |
default | \(\frac {c^{3} x}{e^{6}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +c^{3} d^{6}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{e^{7} \left (e x +d \right )}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{4 e^{7} \left (e x +d \right )^{4}}-\frac {6 a b c \,e^{3}-12 c^{2} a d \,e^{2}+b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{2 e^{7} \left (e x +d \right )^{2}}+\frac {3 c^{2} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{7}}\) | \(453\) |
parallelrisch | \(\frac {-60 a \,c^{2} e^{6} x^{4}-600 c^{3} d^{2} e^{4} x^{4}-6 a b c \,d^{3} e^{3}-b^{3} d^{3} e^{3}-12 d^{4} e^{2} c^{2} a -2 d^{2} e^{4} a^{2} c -1800 x^{3} c^{3} d^{3} e^{3}-2200 x^{2} c^{3} d^{4} e^{2}-274 c^{3} d^{6}-60 x^{2} a b c d \,e^{5}-30 x a b c \,d^{2} e^{4}-10 x \,a^{2} c d \,e^{5}-60 x a \,c^{2} d^{3} e^{3}+20 x^{6} c^{3} e^{6}-12 b^{2} c \,d^{4} e^{2}-3 a^{2} b d \,e^{5}-2 a \,b^{2} d^{2} e^{4}+600 \ln \left (e x +d \right ) x^{2} b \,c^{2} d^{3} e^{3}-4 e^{6} a^{3}-120 \ln \left (e x +d \right ) c^{3} d^{6}-120 x^{3} a \,c^{2} d \,e^{5}-120 x^{2} a \,c^{2} d^{2} e^{4}-20 x^{2} a^{2} c \,e^{6}+137 b \,c^{2} d^{5} e -1250 x \,c^{3} d^{5} e -60 x^{4} b^{2} c \,e^{6}-20 x^{2} a \,b^{2} e^{6}-10 x^{2} b^{3} d \,e^{5}-15 x \,a^{2} b \,e^{6}-5 x \,b^{3} d^{2} e^{4}-10 x^{3} b^{3} e^{6}+60 \ln \left (e x +d \right ) x^{5} b \,c^{2} e^{6}-120 \ln \left (e x +d \right ) x^{5} c^{3} d \,e^{5}-600 \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{4}+60 \ln \left (e x +d \right ) b \,c^{2} d^{5} e -120 x^{2} b^{2} c \,d^{2} e^{4}+1100 x^{2} b \,c^{2} d^{3} e^{3}-10 x a \,b^{2} d \,e^{5}-60 x \,b^{2} c \,d^{3} e^{3}+625 x b \,c^{2} d^{4} e^{2}+300 x^{4} b \,c^{2} d \,e^{5}-60 x^{3} a b c \,e^{6}-120 x^{3} b^{2} c d \,e^{5}+900 x^{3} b \,c^{2} d^{2} e^{4}+600 \ln \left (e x +d \right ) x^{3} b \,c^{2} d^{2} e^{4}-1200 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}-1200 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}-600 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +300 \ln \left (e x +d \right ) x b \,c^{2} d^{4} e^{2}+300 \ln \left (e x +d \right ) x^{4} b \,c^{2} d \,e^{5}}{20 e^{7} \left (e x +d \right )^{5}}\) | \(693\) |
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Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (250) = 500\).
Time = 0.42 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.34 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {20 \, c^{3} e^{6} x^{6} + 100 \, c^{3} d e^{5} x^{5} - 174 \, c^{3} d^{6} + 137 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - 4 \, a^{3} e^{6} - 12 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 2 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 20 \, {\left (5 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 10 \, {\left (80 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 10 \, {\left (120 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 5 \, {\left (150 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 60 \, {\left (2 \, c^{3} d^{6} - b c^{2} d^{5} e + {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (2 \, c^{3} d^{2} e^{4} - b c^{2} d e^{5}\right )} x^{4} + 10 \, {\left (2 \, c^{3} d^{3} e^{3} - b c^{2} d^{2} e^{4}\right )} x^{3} + 10 \, {\left (2 \, c^{3} d^{4} e^{2} - b c^{2} d^{3} e^{3}\right )} x^{2} + 5 \, {\left (2 \, c^{3} d^{5} e - b c^{2} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 3 \, a^{2} b d e^{5} + 4 \, a^{3} e^{6} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 60 \, {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 10 \, {\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 10 \, {\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 5 \, {\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{20 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} + \frac {c^{3} x}{e^{6}} - \frac {3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \log \left (e x + d\right )}{e^{7}} \]
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Time = 0.27 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^{3} x}{e^{6}} - \frac {3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} - \frac {174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 12 \, a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 2 \, a b^{2} d^{2} e^{4} + 2 \, a^{2} c d^{2} e^{4} + 3 \, a^{2} b d e^{5} + 4 \, a^{3} e^{6} + 60 \, {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6} + a c^{2} e^{6}\right )} x^{4} + 10 \, {\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 12 \, a c^{2} d e^{5} + b^{3} e^{6} + 6 \, a b c e^{6}\right )} x^{3} + 10 \, {\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 12 \, a c^{2} d^{2} e^{4} + b^{3} d e^{5} + 6 \, a b c d e^{5} + 2 \, a b^{2} e^{6} + 2 \, a^{2} c e^{6}\right )} x^{2} + 5 \, {\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 12 \, a c^{2} d^{3} e^{3} + b^{3} d^{2} e^{4} + 6 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} + 3 \, a^{2} b e^{6}\right )} x}{20 \, {\left (e x + d\right )}^{5} e^{7}} \]
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Time = 9.94 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^3\,x}{e^6}-\frac {x\,\left (\frac {3\,a^2\,b\,e^5}{4}+\frac {a^2\,c\,d\,e^4}{2}+\frac {a\,b^2\,d\,e^4}{2}+\frac {3\,a\,b\,c\,d^2\,e^3}{2}+3\,a\,c^2\,d^3\,e^2+\frac {b^3\,d^2\,e^3}{4}+3\,b^2\,c\,d^3\,e^2-\frac {125\,b\,c^2\,d^4\,e}{4}+\frac {77\,c^3\,d^5}{2}\right )+x^4\,\left (3\,b^2\,c\,e^5-15\,b\,c^2\,d\,e^4+15\,c^3\,d^2\,e^3+3\,a\,c^2\,e^5\right )+\frac {4\,a^3\,e^6+3\,a^2\,b\,d\,e^5+2\,a^2\,c\,d^2\,e^4+2\,a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3+12\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3+12\,b^2\,c\,d^4\,e^2-137\,b\,c^2\,d^5\,e+174\,c^3\,d^6}{20\,e}+x^2\,\left (a^2\,c\,e^5+a\,b^2\,e^5+3\,a\,b\,c\,d\,e^4+6\,a\,c^2\,d^2\,e^3+\frac {b^3\,d\,e^4}{2}+6\,b^2\,c\,d^2\,e^3-55\,b\,c^2\,d^3\,e^2+65\,c^3\,d^4\,e\right )+x^3\,\left (\frac {b^3\,e^5}{2}+6\,b^2\,c\,d\,e^4-45\,b\,c^2\,d^2\,e^3+3\,a\,b\,c\,e^5+50\,c^3\,d^3\,e^2+6\,a\,c^2\,d\,e^4\right )}{d^5\,e^6+5\,d^4\,e^7\,x+10\,d^3\,e^8\,x^2+10\,d^2\,e^9\,x^3+5\,d\,e^{10}\,x^4+e^{11}\,x^5}-\frac {\ln \left (d+e\,x\right )\,\left (6\,c^3\,d-3\,b\,c^2\,e\right )}{e^7} \]
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