Integrand size = 20, antiderivative size = 414 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=\frac {c^2 \left (21 c^2 d^2+6 b^2 e^2-4 c e (6 b d-a e)\right ) x}{e^8}-\frac {c^3 (3 c d-2 b e) x^2}{e^7}+\frac {c^4 x^3}{3 e^6}-\frac {\left (c d^2-b d e+a e^2\right )^4}{5 e^9 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^4}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{3 e^9 (d+e x)^3}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)^2}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{e^9 (d+e x)}-\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \log (d+e x)}{e^9} \]
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Time = 0.39 (sec) , antiderivative size = 414, 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} \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=-\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^9 (d+e x)}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac {4 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac {c^2 x \left (-4 c e (6 b d-a e)+6 b^2 e^2+21 c^2 d^2\right )}{e^8}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^4}{5 e^9 (d+e x)^5}-\frac {c^3 x^2 (3 c d-2 b e)}{e^7}+\frac {c^4 x^3}{3 e^6} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^2 \left (21 c^2 d^2+6 b^2 e^2-4 c e (6 b d-a e)\right )}{e^8}-\frac {2 c^3 (3 c d-2 b e) x}{e^7}+\frac {c^4 x^2}{e^6}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^6}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^5}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^8 (d+e x)^4}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right )}{e^8 (d+e x)^3}+\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{e^8 (d+e x)^2}+\frac {4 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right )}{e^8 (d+e x)}\right ) \, dx \\ & = \frac {c^2 \left (21 c^2 d^2+6 b^2 e^2-4 c e (6 b d-a e)\right ) x}{e^8}-\frac {c^3 (3 c d-2 b e) x^2}{e^7}+\frac {c^4 x^3}{3 e^6}-\frac {\left (c d^2-b d e+a e^2\right )^4}{5 e^9 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^4}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{3 e^9 (d+e x)^3}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)^2}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{e^9 (d+e x)}-\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \log (d+e x)}{e^9} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=\frac {15 c^2 e \left (21 c^2 d^2+6 b^2 e^2+4 c e (-6 b d+a e)\right ) x+15 c^3 e^2 (-3 c d+2 b e) x^2+5 c^4 e^3 x^3-\frac {3 \left (c d^2+e (-b d+a e)\right )^4}{(d+e x)^5}+\frac {15 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^4}-\frac {10 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2}{(d+e x)^3}+\frac {30 (2 c d-b e) \left (7 c^3 d^4-2 c^2 d^2 e (7 b d-5 a e)+b^2 e^3 (-b d+a e)+c e^2 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right )}{(d+e x)^2}-\frac {15 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right )}{d+e x}-60 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2+c e (-7 b d+3 a e)\right ) \log (d+e x)}{15 e^9} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(886\) vs. \(2(408)=816\).
Time = 3.00 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.14
method | result | size |
norman | \(\frac {-\frac {3 a^{4} e^{8}+3 a^{3} b d \,e^{7}+2 a^{3} c \,d^{2} e^{6}+3 a^{2} b^{2} d^{2} e^{6}+9 a^{2} b c \,d^{3} e^{5}+18 a^{2} c^{2} d^{4} e^{4}+3 a \,b^{3} d^{3} e^{5}+36 a \,b^{2} c \,d^{4} e^{4}-411 a b \,c^{2} d^{5} e^{3}+822 a \,c^{3} d^{6} e^{2}+3 b^{4} d^{4} e^{4}-137 b^{3} c \,d^{5} e^{3}+1233 b^{2} c^{2} d^{6} e^{2}-2877 b \,c^{3} d^{7} e +1918 c^{4} d^{8}}{15 e^{9}}+\frac {c^{4} x^{8}}{3 e}-\frac {\left (6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+120 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}-20 b^{3} c d \,e^{3}+180 b^{2} c^{2} d^{2} e^{2}-420 b \,c^{3} d^{3} e +280 c^{4} d^{4}\right ) x^{4}}{e^{5}}-\frac {2 \left (3 a^{2} b c \,e^{5}+6 d \,e^{4} a^{2} c^{2}+a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}-90 a b \,c^{2} d^{2} e^{3}+180 d^{3} e^{2} c^{3} a +b^{4} d \,e^{4}-30 b^{3} c \,d^{2} e^{3}+270 b^{2} c^{2} d^{3} e^{2}-630 b \,c^{3} d^{4} e +420 c^{4} d^{5}\right ) x^{3}}{e^{6}}-\frac {2 \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+9 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}+3 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}-330 a b \,c^{2} d^{3} e^{3}+660 d^{4} e^{2} c^{3} a +3 b^{4} d^{2} e^{4}-110 b^{3} c \,d^{3} e^{3}+990 b^{2} c^{2} d^{4} e^{2}-2310 b \,c^{3} d^{5} e +1540 d^{6} c^{4}\right ) x^{2}}{3 e^{7}}-\frac {\left (3 a^{3} b \,e^{7}+2 d \,e^{6} c \,a^{3}+3 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}+18 d^{3} e^{4} a^{2} c^{2}+3 a \,b^{3} d^{2} e^{5}+36 a \,b^{2} c \,d^{3} e^{4}-375 a b \,c^{2} d^{4} e^{3}+750 d^{5} e^{2} c^{3} a +3 b^{4} d^{3} e^{4}-125 b^{3} c \,d^{4} e^{3}+1125 b^{2} c^{2} d^{5} e^{2}-2625 b \,c^{3} d^{6} e +1750 d^{7} c^{4}\right ) x}{3 e^{8}}+\frac {2 c^{2} \left (6 a c \,e^{2}+9 b^{2} e^{2}-21 b c d e +14 c^{2} d^{2}\right ) x^{6}}{3 e^{3}}+\frac {2 c^{3} \left (3 b e -2 c d \right ) x^{7}}{3 e^{2}}}{\left (e x +d \right )^{5}}+\frac {4 c \left (3 a b c \,e^{3}-6 c^{2} a d \,e^{2}+b^{3} e^{3}-9 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(887\) |
default | \(\frac {c^{2} \left (\frac {1}{3} c^{2} e^{2} x^{3}+2 b c \,e^{2} x^{2}-3 c^{2} d e \,x^{2}+4 a c \,e^{2} x +6 b^{2} e^{2} x -24 b c d e x +21 c^{2} d^{2} x \right )}{e^{8}}-\frac {a^{4} e^{8}-4 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-12 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}-12 a b \,c^{2} d^{5} e^{3}+4 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}-4 b^{3} c \,d^{5} e^{3}+6 b^{2} c^{2} d^{6} e^{2}-4 b \,c^{3} d^{7} e +c^{4} d^{8}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+60 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}-20 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}-140 b \,c^{3} d^{3} e +70 c^{4} d^{4}}{e^{9} \left (e x +d \right )}-\frac {4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}-36 a^{2} b c d \,e^{5}+36 d^{2} e^{4} a^{2} c^{2}-12 a \,b^{3} d \,e^{5}+72 a \,b^{2} c \,d^{2} e^{4}-120 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} c^{3} a +6 b^{4} d^{2} e^{4}-40 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}-84 b \,c^{3} d^{5} e +28 d^{6} c^{4}}{3 e^{9} \left (e x +d \right )^{3}}-\frac {4 a^{3} b \,e^{7}-8 d \,e^{6} c \,a^{3}-12 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}-24 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}-48 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}-24 d^{5} e^{2} c^{3} a -4 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}-36 b^{2} c^{2} d^{5} e^{2}+28 b \,c^{3} d^{6} e -8 d^{7} c^{4}}{4 e^{9} \left (e x +d \right )^{4}}-\frac {12 a^{2} b c \,e^{5}-24 d \,e^{4} a^{2} c^{2}+4 a \,b^{3} e^{5}-48 a \,b^{2} c d \,e^{4}+120 a b \,c^{2} d^{2} e^{3}-80 d^{3} e^{2} c^{3} a -4 b^{4} d \,e^{4}+40 b^{3} c \,d^{2} e^{3}-120 b^{2} c^{2} d^{3} e^{2}+140 b \,c^{3} d^{4} e -56 c^{4} d^{5}}{2 e^{9} \left (e x +d \right )^{2}}+\frac {4 c \left (3 a b c \,e^{3}-6 c^{2} a d \,e^{2}+b^{3} e^{3}-9 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(902\) |
risch | \(\frac {c^{4} x^{3}}{3 e^{6}}+\frac {2 c^{3} b \,x^{2}}{e^{6}}-\frac {3 c^{4} d \,x^{2}}{e^{7}}+\frac {4 c^{3} a x}{e^{6}}+\frac {6 c^{2} b^{2} x}{e^{6}}-\frac {24 c^{3} b d x}{e^{7}}+\frac {21 c^{4} d^{2} x}{e^{8}}+\frac {\left (-6 e^{7} a^{2} c^{2}-12 a \,b^{2} c \,e^{7}+60 a b \,c^{2} d \,e^{6}-60 d^{2} e^{5} c^{3} a -b^{4} e^{7}+20 b^{3} c d \,e^{6}-90 b^{2} c^{2} d^{2} e^{5}+140 c^{3} b \,d^{3} e^{4}-70 d^{4} e^{3} c^{4}\right ) x^{4}-2 e^{2} \left (3 a^{2} b c \,e^{5}+6 d \,e^{4} a^{2} c^{2}+a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}-90 a b \,c^{2} d^{2} e^{3}+100 d^{3} e^{2} c^{3} a +b^{4} d \,e^{4}-30 b^{3} c \,d^{2} e^{3}+150 b^{2} c^{2} d^{3} e^{2}-245 b \,c^{3} d^{4} e +126 c^{4} d^{5}\right ) x^{3}-\frac {2 e \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+9 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}+3 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}-330 a b \,c^{2} d^{3} e^{3}+390 d^{4} e^{2} c^{3} a +3 b^{4} d^{2} e^{4}-110 b^{3} c \,d^{3} e^{3}+585 b^{2} c^{2} d^{4} e^{2}-987 b \,c^{3} d^{5} e +518 d^{6} c^{4}\right ) x^{2}}{3}+\left (-a^{3} b \,e^{7}-\frac {2}{3} d \,e^{6} c \,a^{3}-a^{2} b^{2} d \,e^{6}-3 a^{2} b c \,d^{2} e^{5}-6 d^{3} e^{4} a^{2} c^{2}-a \,b^{3} d^{2} e^{5}-12 a \,b^{2} c \,d^{3} e^{4}+125 a b \,c^{2} d^{4} e^{3}-154 d^{5} e^{2} c^{3} a -b^{4} d^{3} e^{4}+\frac {125}{3} b^{3} c \,d^{4} e^{3}-231 b^{2} c^{2} d^{5} e^{2}+399 b \,c^{3} d^{6} e -\frac {638}{3} d^{7} c^{4}\right ) x -\frac {3 a^{4} e^{8}+3 a^{3} b d \,e^{7}+2 a^{3} c \,d^{2} e^{6}+3 a^{2} b^{2} d^{2} e^{6}+9 a^{2} b c \,d^{3} e^{5}+18 a^{2} c^{2} d^{4} e^{4}+3 a \,b^{3} d^{3} e^{5}+36 a \,b^{2} c \,d^{4} e^{4}-411 a b \,c^{2} d^{5} e^{3}+522 a \,c^{3} d^{6} e^{2}+3 b^{4} d^{4} e^{4}-137 b^{3} c \,d^{5} e^{3}+783 b^{2} c^{2} d^{6} e^{2}-1377 b \,c^{3} d^{7} e +743 c^{4} d^{8}}{15 e}}{e^{8} \left (e x +d \right )^{5}}+\frac {12 c^{2} \ln \left (e x +d \right ) a b}{e^{6}}-\frac {24 c^{3} \ln \left (e x +d \right ) a d}{e^{7}}+\frac {4 c \ln \left (e x +d \right ) b^{3}}{e^{6}}-\frac {36 c^{2} \ln \left (e x +d \right ) b^{2} d}{e^{7}}+\frac {84 c^{3} \ln \left (e x +d \right ) b \,d^{2}}{e^{8}}-\frac {56 c^{4} \ln \left (e x +d \right ) d^{3}}{e^{9}}\) | \(928\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1651\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1262 vs. \(2 (408) = 816\).
Time = 0.28 (sec) , antiderivative size = 1262, normalized size of antiderivative = 3.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 850 vs. \(2 (408) = 816\).
Time = 0.22 (sec) , antiderivative size = 850, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=-\frac {743 \, c^{4} d^{8} - 1377 \, b c^{3} d^{7} e + 3 \, a^{3} b d e^{7} + 3 \, a^{4} e^{8} + 261 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 137 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 15 \, {\left (70 \, c^{4} d^{4} e^{4} - 140 \, b c^{3} d^{3} e^{5} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 30 \, {\left (126 \, c^{4} d^{5} e^{3} - 245 \, b c^{3} d^{4} e^{4} + 50 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} - 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} + {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 10 \, {\left (518 \, c^{4} d^{6} e^{2} - 987 \, b c^{3} d^{5} e^{3} + 195 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 110 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 5 \, {\left (638 \, c^{4} d^{7} e - 1197 \, b c^{3} d^{6} e^{2} + 3 \, a^{3} b e^{8} + 231 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 125 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{15 \, {\left (e^{14} x^{5} + 5 \, d e^{13} x^{4} + 10 \, d^{2} e^{12} x^{3} + 10 \, d^{3} e^{11} x^{2} + 5 \, d^{4} e^{10} x + d^{5} e^{9}\right )}} + \frac {c^{4} e^{2} x^{3} - 3 \, {\left (3 \, c^{4} d e - 2 \, b c^{3} e^{2}\right )} x^{2} + 3 \, {\left (21 \, c^{4} d^{2} - 24 \, b c^{3} d e + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} x}{3 \, e^{8}} - \frac {4 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 909 vs. \(2 (408) = 816\).
Time = 0.27 (sec) , antiderivative size = 909, normalized size of antiderivative = 2.20 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=-\frac {4 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 9 \, b^{2} c^{2} d e^{2} + 6 \, a c^{3} d e^{2} - b^{3} c e^{3} - 3 \, a b c^{2} e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{9}} - \frac {743 \, c^{4} d^{8} - 1377 \, b c^{3} d^{7} e + 783 \, b^{2} c^{2} d^{6} e^{2} + 522 \, a c^{3} d^{6} e^{2} - 137 \, b^{3} c d^{5} e^{3} - 411 \, a b c^{2} d^{5} e^{3} + 3 \, b^{4} d^{4} e^{4} + 36 \, a b^{2} c d^{4} e^{4} + 18 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a b^{3} d^{3} e^{5} + 9 \, a^{2} b c d^{3} e^{5} + 3 \, a^{2} b^{2} d^{2} e^{6} + 2 \, a^{3} c d^{2} e^{6} + 3 \, a^{3} b d e^{7} + 3 \, a^{4} e^{8} + 15 \, {\left (70 \, c^{4} d^{4} e^{4} - 140 \, b c^{3} d^{3} e^{5} + 90 \, b^{2} c^{2} d^{2} e^{6} + 60 \, a c^{3} d^{2} e^{6} - 20 \, b^{3} c d e^{7} - 60 \, a b c^{2} d e^{7} + b^{4} e^{8} + 12 \, a b^{2} c e^{8} + 6 \, a^{2} c^{2} e^{8}\right )} x^{4} + 30 \, {\left (126 \, c^{4} d^{5} e^{3} - 245 \, b c^{3} d^{4} e^{4} + 150 \, b^{2} c^{2} d^{3} e^{5} + 100 \, a c^{3} d^{3} e^{5} - 30 \, b^{3} c d^{2} e^{6} - 90 \, a b c^{2} d^{2} e^{6} + b^{4} d e^{7} + 12 \, a b^{2} c d e^{7} + 6 \, a^{2} c^{2} d e^{7} + a b^{3} e^{8} + 3 \, a^{2} b c e^{8}\right )} x^{3} + 10 \, {\left (518 \, c^{4} d^{6} e^{2} - 987 \, b c^{3} d^{5} e^{3} + 585 \, b^{2} c^{2} d^{4} e^{4} + 390 \, a c^{3} d^{4} e^{4} - 110 \, b^{3} c d^{3} e^{5} - 330 \, a b c^{2} d^{3} e^{5} + 3 \, b^{4} d^{2} e^{6} + 36 \, a b^{2} c d^{2} e^{6} + 18 \, a^{2} c^{2} d^{2} e^{6} + 3 \, a b^{3} d e^{7} + 9 \, a^{2} b c d e^{7} + 3 \, a^{2} b^{2} e^{8} + 2 \, a^{3} c e^{8}\right )} x^{2} + 5 \, {\left (638 \, c^{4} d^{7} e - 1197 \, b c^{3} d^{6} e^{2} + 693 \, b^{2} c^{2} d^{5} e^{3} + 462 \, a c^{3} d^{5} e^{3} - 125 \, b^{3} c d^{4} e^{4} - 375 \, a b c^{2} d^{4} e^{4} + 3 \, b^{4} d^{3} e^{5} + 36 \, a b^{2} c d^{3} e^{5} + 18 \, a^{2} c^{2} d^{3} e^{5} + 3 \, a b^{3} d^{2} e^{6} + 9 \, a^{2} b c d^{2} e^{6} + 3 \, a^{2} b^{2} d e^{7} + 2 \, a^{3} c d e^{7} + 3 \, a^{3} b e^{8}\right )} x}{15 \, {\left (e x + d\right )}^{5} e^{9}} + \frac {c^{4} e^{12} x^{3} - 9 \, c^{4} d e^{11} x^{2} + 6 \, b c^{3} e^{12} x^{2} + 63 \, c^{4} d^{2} e^{10} x - 72 \, b c^{3} d e^{11} x + 18 \, b^{2} c^{2} e^{12} x + 12 \, a c^{3} e^{12} x}{3 \, e^{18}} \]
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Time = 0.22 (sec) , antiderivative size = 959, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=x^2\,\left (\frac {2\,b\,c^3}{e^6}-\frac {3\,c^4\,d}{e^7}\right )-x\,\left (\frac {6\,d\,\left (\frac {4\,b\,c^3}{e^6}-\frac {6\,c^4\,d}{e^7}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^6}+\frac {15\,c^4\,d^2}{e^8}\right )-\frac {x^3\,\left (6\,a^2\,b\,c\,e^7+12\,a^2\,c^2\,d\,e^6+2\,a\,b^3\,e^7+24\,a\,b^2\,c\,d\,e^6-180\,a\,b\,c^2\,d^2\,e^5+200\,a\,c^3\,d^3\,e^4+2\,b^4\,d\,e^6-60\,b^3\,c\,d^2\,e^5+300\,b^2\,c^2\,d^3\,e^4-490\,b\,c^3\,d^4\,e^3+252\,c^4\,d^5\,e^2\right )+x\,\left (a^3\,b\,e^7+\frac {2\,a^3\,c\,d\,e^6}{3}+a^2\,b^2\,d\,e^6+3\,a^2\,b\,c\,d^2\,e^5+6\,a^2\,c^2\,d^3\,e^4+a\,b^3\,d^2\,e^5+12\,a\,b^2\,c\,d^3\,e^4-125\,a\,b\,c^2\,d^4\,e^3+154\,a\,c^3\,d^5\,e^2+b^4\,d^3\,e^4-\frac {125\,b^3\,c\,d^4\,e^3}{3}+231\,b^2\,c^2\,d^5\,e^2-399\,b\,c^3\,d^6\,e+\frac {638\,c^4\,d^7}{3}\right )+x^4\,\left (6\,a^2\,c^2\,e^7+12\,a\,b^2\,c\,e^7-60\,a\,b\,c^2\,d\,e^6+60\,a\,c^3\,d^2\,e^5+b^4\,e^7-20\,b^3\,c\,d\,e^6+90\,b^2\,c^2\,d^2\,e^5-140\,b\,c^3\,d^3\,e^4+70\,c^4\,d^4\,e^3\right )+\frac {3\,a^4\,e^8+3\,a^3\,b\,d\,e^7+2\,a^3\,c\,d^2\,e^6+3\,a^2\,b^2\,d^2\,e^6+9\,a^2\,b\,c\,d^3\,e^5+18\,a^2\,c^2\,d^4\,e^4+3\,a\,b^3\,d^3\,e^5+36\,a\,b^2\,c\,d^4\,e^4-411\,a\,b\,c^2\,d^5\,e^3+522\,a\,c^3\,d^6\,e^2+3\,b^4\,d^4\,e^4-137\,b^3\,c\,d^5\,e^3+783\,b^2\,c^2\,d^6\,e^2-1377\,b\,c^3\,d^7\,e+743\,c^4\,d^8}{15\,e}+x^2\,\left (\frac {4\,a^3\,c\,e^7}{3}+2\,a^2\,b^2\,e^7+6\,a^2\,b\,c\,d\,e^6+12\,a^2\,c^2\,d^2\,e^5+2\,a\,b^3\,d\,e^6+24\,a\,b^2\,c\,d^2\,e^5-220\,a\,b\,c^2\,d^3\,e^4+260\,a\,c^3\,d^4\,e^3+2\,b^4\,d^2\,e^5-\frac {220\,b^3\,c\,d^3\,e^4}{3}+390\,b^2\,c^2\,d^4\,e^3-658\,b\,c^3\,d^5\,e^2+\frac {1036\,c^4\,d^6\,e}{3}\right )}{d^5\,e^8+5\,d^4\,e^9\,x+10\,d^3\,e^{10}\,x^2+10\,d^2\,e^{11}\,x^3+5\,d\,e^{12}\,x^4+e^{13}\,x^5}-\frac {\ln \left (d+e\,x\right )\,\left (-4\,b^3\,c\,e^3+36\,b^2\,c^2\,d\,e^2-84\,b\,c^3\,d^2\,e-12\,a\,b\,c^2\,e^3+56\,c^4\,d^3+24\,a\,c^3\,d\,e^2\right )}{e^9}+\frac {c^4\,x^3}{3\,e^6} \]
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