Integrand size = 20, antiderivative size = 443 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^4}{10 e^9 (d+e x)^{10}}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{9 e^9 (d+e x)^9}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{4 e^9 (d+e x)^8}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{7 e^9 (d+e x)^7}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{6 e^9 (d+e x)^6}+\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{5 e^9 (d+e x)^5}-\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{2 e^9 (d+e x)^4}+\frac {4 c^3 (2 c d-b e)}{3 e^9 (d+e x)^3}-\frac {c^4}{2 e^9 (d+e x)^2} \]
[Out]
Time = 0.27 (sec) , antiderivative size = 443, 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} \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=-\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{6 e^9 (d+e x)^6}-\frac {c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9 (d+e x)^4}+\frac {4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^9 (d+e x)^7}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{4 e^9 (d+e x)^8}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{9 e^9 (d+e x)^9}-\frac {\left (a e^2-b d e+c d^2\right )^4}{10 e^9 (d+e x)^{10}}+\frac {4 c^3 (2 c d-b e)}{3 e^9 (d+e x)^3}-\frac {c^4}{2 e^9 (d+e x)^2} \]
[In]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^{11}}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^{10}}+\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)^9}+\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)^8}+\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)^7}+\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)^6}+\frac {2 c^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)^5}-\frac {4 c^3 (2 c d-b e)}{e^8 (d+e x)^4}+\frac {c^4}{e^8 (d+e x)^3}\right ) \, dx \\ & = -\frac {\left (c d^2-b d e+a e^2\right )^4}{10 e^9 (d+e x)^{10}}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{9 e^9 (d+e x)^9}-\frac {\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 )}{4 e^9 (d+e x)^8}+\frac {4 (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 )}{7 e^9 (d+e x)^7}-\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 )}{6 e^9 (d+e x)^6}+\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 )}{5 e^9 (d+e x)^5}-\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{2 e^9 (d+e x)^4}+\frac {4 c^3 (2 c d-b e)}{3 e^9 (d+e x)^3}-\frac {c^4}{2 e^9 (d+e x)^2} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 731, normalized size of antiderivative = 1.65 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=-\frac {14 c^4 \left (d^8+10 d^7 e x+45 d^6 e^2 x^2+120 d^5 e^3 x^3+210 d^4 e^4 x^4+252 d^3 e^5 x^5+210 d^2 e^6 x^6+120 d e^7 x^7+45 e^8 x^8\right )+e^4 \left (126 a^4 e^4+56 a^3 b e^3 (d+10 e x)+21 a^2 b^2 e^2 \left (d^2+10 d e x+45 e^2 x^2\right )+6 a b^3 e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+b^4 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )+2 c e^3 \left (7 a^3 e^3 \left (d^2+10 d e x+45 e^2 x^2\right )+9 a^2 b e^2 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+6 a b^2 e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+2 b^3 \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )+3 c^2 e^2 \left (2 a^2 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+4 a b e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+3 b^2 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )+2 c^3 e \left (3 a e \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )+7 b \left (d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right )\right )}{1260 e^9 (d+e x)^{10}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(879\) vs. \(2(426)=852\).
Time = 9.56 (sec) , antiderivative size = 880, normalized size of antiderivative = 1.99
method | result | size |
risch | \(\frac {-\frac {c^{4} x^{8}}{2 e}-\frac {4 c^{3} \left (b e +c d \right ) x^{7}}{3 e^{2}}-\frac {c^{2} \left (6 a c \,e^{2}+9 b^{2} e^{2}+14 b c d e +14 c^{2} d^{2}\right ) x^{6}}{6 e^{3}}-\frac {c \left (12 a b c \,e^{3}+6 c^{2} a d \,e^{2}+4 b^{3} e^{3}+9 b^{2} d \,e^{2} c +14 b \,c^{2} d^{2} e +14 c^{3} d^{3}\right ) x^{5}}{5 e^{4}}-\frac {\left (6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}+6 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}+4 b^{3} c d \,e^{3}+9 b^{2} c^{2} d^{2} e^{2}+14 b \,c^{3} d^{3} e +14 c^{4} d^{4}\right ) x^{4}}{6 e^{5}}-\frac {2 \left (18 a^{2} b c \,e^{5}+6 d \,e^{4} a^{2} c^{2}+6 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+12 a b \,c^{2} d^{2} e^{3}+6 d^{3} e^{2} c^{3} a +b^{4} d \,e^{4}+4 b^{3} c \,d^{2} e^{3}+9 b^{2} c^{2} d^{3} e^{2}+14 b \,c^{3} d^{4} e +14 c^{4} d^{5}\right ) x^{3}}{21 e^{6}}-\frac {\left (14 e^{6} c \,a^{3}+21 a^{2} b^{2} e^{6}+18 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+6 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+12 a b \,c^{2} d^{3} e^{3}+6 d^{4} e^{2} c^{3} a +b^{4} d^{2} e^{4}+4 b^{3} c \,d^{3} e^{3}+9 b^{2} c^{2} d^{4} e^{2}+14 b \,c^{3} d^{5} e +14 d^{6} c^{4}\right ) x^{2}}{28 e^{7}}-\frac {\left (56 a^{3} b \,e^{7}+14 d \,e^{6} c \,a^{3}+21 a^{2} b^{2} d \,e^{6}+18 a^{2} b c \,d^{2} e^{5}+6 d^{3} e^{4} a^{2} c^{2}+6 a \,b^{3} d^{2} e^{5}+12 a \,b^{2} c \,d^{3} e^{4}+12 a b \,c^{2} d^{4} e^{3}+6 d^{5} e^{2} c^{3} a +b^{4} d^{3} e^{4}+4 b^{3} c \,d^{4} e^{3}+9 b^{2} c^{2} d^{5} e^{2}+14 b \,c^{3} d^{6} e +14 d^{7} c^{4}\right ) x}{126 e^{8}}-\frac {126 a^{4} e^{8}+56 a^{3} b d \,e^{7}+14 a^{3} c \,d^{2} e^{6}+21 a^{2} b^{2} d^{2} e^{6}+18 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+6 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+12 a b \,c^{2} d^{5} e^{3}+6 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+4 b^{3} c \,d^{5} e^{3}+9 b^{2} c^{2} d^{6} e^{2}+14 b \,c^{3} d^{7} e +14 c^{4} d^{8}}{1260 e^{9}}}{\left (e x +d \right )^{10}}\) | \(880\) |
default | \(-\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}}{9 e^{9} \left (e x +d \right )^{9}}-\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}}{10 e^{9} \left (e x +d \right )^{10}}-\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 )}{5 e^{9} \left (e x +d \right )^{5}}-\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}}{8 e^{9} \left (e x +d \right )^{8}}-\frac {4 c^{3} \left (b e -2 c d \right )}{3 e^{9} \left (e x +d \right )^{3}}-\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}}{7 e^{9} \left (e x +d \right )^{7}}-\frac {c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right )}{2 e^{9} \left (e x +d \right )^{4}}-\frac {c^{4}}{2 e^{9} \left (e x +d \right )^{2}}-\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}}{6 e^{9} \left (e x +d \right )^{6}}\) | \(914\) |
norman | \(\frac {-\frac {c^{4} x^{8}}{2 e}-\frac {4 \left (e^{2} c^{3} b +d e \,c^{4}\right ) x^{7}}{3 e^{3}}-\frac {\left (6 a \,c^{3} e^{3}+9 b^{2} c^{2} e^{3}+14 d \,e^{2} c^{3} b +14 d^{2} e \,c^{4}\right ) x^{6}}{6 e^{4}}-\frac {\left (12 a b \,c^{2} e^{4}+6 a \,c^{3} d \,e^{3}+4 b^{3} c \,e^{4}+9 b^{2} c^{2} d \,e^{3}+14 b \,c^{3} d^{2} e^{2}+14 c^{4} d^{3} e \right ) x^{5}}{5 e^{5}}-\frac {\left (6 e^{5} a^{2} c^{2}+12 a \,b^{2} c \,e^{5}+12 a b \,c^{2} d \,e^{4}+6 d^{2} e^{3} c^{3} a +b^{4} e^{5}+4 b^{3} c d \,e^{4}+9 b^{2} c^{2} d^{2} e^{3}+14 b \,c^{3} d^{3} e^{2}+14 d^{4} e \,c^{4}\right ) x^{4}}{6 e^{6}}-\frac {2 \left (18 a^{2} b c \,e^{6}+6 a^{2} c^{2} d \,e^{5}+6 a \,b^{3} e^{6}+12 a \,b^{2} c d \,e^{5}+12 a b \,c^{2} d^{2} e^{4}+6 a \,c^{3} d^{3} e^{3}+b^{4} d \,e^{5}+4 b^{3} c \,d^{2} e^{4}+9 b^{2} c^{2} d^{3} e^{3}+14 b \,c^{3} d^{4} e^{2}+14 c^{4} d^{5} e \right ) x^{3}}{21 e^{7}}-\frac {\left (14 e^{7} c \,a^{3}+21 a^{2} b^{2} e^{7}+18 a^{2} b c d \,e^{6}+6 d^{2} e^{5} a^{2} c^{2}+6 a \,b^{3} d \,e^{6}+12 a \,b^{2} c \,d^{2} e^{5}+12 a b \,c^{2} d^{3} e^{4}+6 d^{4} e^{3} c^{3} a +b^{4} d^{2} e^{5}+4 b^{3} c \,d^{3} e^{4}+9 b^{2} c^{2} d^{4} e^{3}+14 b \,c^{3} d^{5} e^{2}+14 d^{6} e \,c^{4}\right ) x^{2}}{28 e^{8}}-\frac {\left (56 a^{3} b \,e^{8}+14 a^{3} c d \,e^{7}+21 a^{2} b^{2} d \,e^{7}+18 a^{2} b c \,d^{2} e^{6}+6 a^{2} c^{2} d^{3} e^{5}+6 a \,b^{3} d^{2} e^{6}+12 a \,b^{2} c \,d^{3} e^{5}+12 a b \,c^{2} d^{4} e^{4}+6 a \,c^{3} d^{5} e^{3}+b^{4} d^{3} e^{5}+4 b^{3} c \,d^{4} e^{4}+9 b^{2} c^{2} d^{5} e^{3}+14 b \,c^{3} d^{6} e^{2}+14 c^{4} d^{7} e \right ) x}{126 e^{9}}-\frac {126 a^{4} e^{9}+56 a^{3} b d \,e^{8}+14 a^{3} c \,d^{2} e^{7}+21 a^{2} b^{2} d^{2} e^{7}+18 a^{2} b c \,d^{3} e^{6}+6 a^{2} c^{2} d^{4} e^{5}+6 a \,b^{3} d^{3} e^{6}+12 a \,b^{2} c \,d^{4} e^{5}+12 a b \,c^{2} d^{5} e^{4}+6 a \,c^{3} d^{6} e^{3}+b^{4} d^{4} e^{5}+4 b^{3} c \,d^{5} e^{4}+9 b^{2} c^{2} d^{6} e^{3}+14 b \,c^{3} d^{7} e^{2}+14 c^{4} d^{8} e}{1260 e^{10}}}{\left (e x +d \right )^{10}}\) | \(914\) |
gosper | \(\text {Expression too large to display}\) | \(1015\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1021\) |
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (425) = 850\).
Time = 0.36 (sec) , antiderivative size = 906, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=-\frac {630 \, c^{4} e^{8} x^{8} + 14 \, c^{4} d^{8} + 14 \, b c^{3} d^{7} e + 56 \, a^{3} b d e^{7} + 126 \, a^{4} e^{8} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 7 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 1680 \, {\left (c^{4} d e^{7} + b c^{3} e^{8}\right )} x^{7} + 210 \, {\left (14 \, c^{4} d^{2} e^{6} + 14 \, b c^{3} d e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 252 \, {\left (14 \, c^{4} d^{3} e^{5} + 14 \, b c^{3} d^{2} e^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 210 \, {\left (14 \, c^{4} d^{4} e^{4} + 14 \, b c^{3} d^{3} e^{5} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} + 4 \, {\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} + 120 \, {\left (14 \, c^{4} d^{5} e^{3} + 14 \, b c^{3} d^{4} e^{4} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} + 4 \, {\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} + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 45 \, {\left (14 \, c^{4} d^{6} e^{2} + 14 \, b c^{3} d^{5} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + 7 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 10 \, {\left (14 \, c^{4} d^{7} e + 14 \, b c^{3} d^{6} e^{2} + 56 \, a^{3} b e^{8} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 7 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{1260 \, {\left (e^{19} x^{10} + 10 \, d e^{18} x^{9} + 45 \, d^{2} e^{17} x^{8} + 120 \, d^{3} e^{16} x^{7} + 210 \, d^{4} e^{15} x^{6} + 252 \, d^{5} e^{14} x^{5} + 210 \, d^{6} e^{13} x^{4} + 120 \, d^{7} e^{12} x^{3} + 45 \, d^{8} e^{11} x^{2} + 10 \, d^{9} e^{10} x + d^{10} e^{9}\right )}} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (425) = 850\).
Time = 0.24 (sec) , antiderivative size = 906, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=-\frac {630 \, c^{4} e^{8} x^{8} + 14 \, c^{4} d^{8} + 14 \, b c^{3} d^{7} e + 56 \, a^{3} b d e^{7} + 126 \, a^{4} e^{8} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 7 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 1680 \, {\left (c^{4} d e^{7} + b c^{3} e^{8}\right )} x^{7} + 210 \, {\left (14 \, c^{4} d^{2} e^{6} + 14 \, b c^{3} d e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 252 \, {\left (14 \, c^{4} d^{3} e^{5} + 14 \, b c^{3} d^{2} e^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 210 \, {\left (14 \, c^{4} d^{4} e^{4} + 14 \, b c^{3} d^{3} e^{5} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} + 4 \, {\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} + 120 \, {\left (14 \, c^{4} d^{5} e^{3} + 14 \, b c^{3} d^{4} e^{4} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} + 4 \, {\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} + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 45 \, {\left (14 \, c^{4} d^{6} e^{2} + 14 \, b c^{3} d^{5} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + 7 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 10 \, {\left (14 \, c^{4} d^{7} e + 14 \, b c^{3} d^{6} e^{2} + 56 \, a^{3} b e^{8} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 7 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{1260 \, {\left (e^{19} x^{10} + 10 \, d e^{18} x^{9} + 45 \, d^{2} e^{17} x^{8} + 120 \, d^{3} e^{16} x^{7} + 210 \, d^{4} e^{15} x^{6} + 252 \, d^{5} e^{14} x^{5} + 210 \, d^{6} e^{13} x^{4} + 120 \, d^{7} e^{12} x^{3} + 45 \, d^{8} e^{11} x^{2} + 10 \, d^{9} e^{10} x + d^{10} e^{9}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1014 vs. \(2 (425) = 850\).
Time = 0.28 (sec) , antiderivative size = 1014, normalized size of antiderivative = 2.29 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=-\frac {630 \, c^{4} e^{8} x^{8} + 1680 \, c^{4} d e^{7} x^{7} + 1680 \, b c^{3} e^{8} x^{7} + 2940 \, c^{4} d^{2} e^{6} x^{6} + 2940 \, b c^{3} d e^{7} x^{6} + 1890 \, b^{2} c^{2} e^{8} x^{6} + 1260 \, a c^{3} e^{8} x^{6} + 3528 \, c^{4} d^{3} e^{5} x^{5} + 3528 \, b c^{3} d^{2} e^{6} x^{5} + 2268 \, b^{2} c^{2} d e^{7} x^{5} + 1512 \, a c^{3} d e^{7} x^{5} + 1008 \, b^{3} c e^{8} x^{5} + 3024 \, a b c^{2} e^{8} x^{5} + 2940 \, c^{4} d^{4} e^{4} x^{4} + 2940 \, b c^{3} d^{3} e^{5} x^{4} + 1890 \, b^{2} c^{2} d^{2} e^{6} x^{4} + 1260 \, a c^{3} d^{2} e^{6} x^{4} + 840 \, b^{3} c d e^{7} x^{4} + 2520 \, a b c^{2} d e^{7} x^{4} + 210 \, b^{4} e^{8} x^{4} + 2520 \, a b^{2} c e^{8} x^{4} + 1260 \, a^{2} c^{2} e^{8} x^{4} + 1680 \, c^{4} d^{5} e^{3} x^{3} + 1680 \, b c^{3} d^{4} e^{4} x^{3} + 1080 \, b^{2} c^{2} d^{3} e^{5} x^{3} + 720 \, a c^{3} d^{3} e^{5} x^{3} + 480 \, b^{3} c d^{2} e^{6} x^{3} + 1440 \, a b c^{2} d^{2} e^{6} x^{3} + 120 \, b^{4} d e^{7} x^{3} + 1440 \, a b^{2} c d e^{7} x^{3} + 720 \, a^{2} c^{2} d e^{7} x^{3} + 720 \, a b^{3} e^{8} x^{3} + 2160 \, a^{2} b c e^{8} x^{3} + 630 \, c^{4} d^{6} e^{2} x^{2} + 630 \, b c^{3} d^{5} e^{3} x^{2} + 405 \, b^{2} c^{2} d^{4} e^{4} x^{2} + 270 \, a c^{3} d^{4} e^{4} x^{2} + 180 \, b^{3} c d^{3} e^{5} x^{2} + 540 \, a b c^{2} d^{3} e^{5} x^{2} + 45 \, b^{4} d^{2} e^{6} x^{2} + 540 \, a b^{2} c d^{2} e^{6} x^{2} + 270 \, a^{2} c^{2} d^{2} e^{6} x^{2} + 270 \, a b^{3} d e^{7} x^{2} + 810 \, a^{2} b c d e^{7} x^{2} + 945 \, a^{2} b^{2} e^{8} x^{2} + 630 \, a^{3} c e^{8} x^{2} + 140 \, c^{4} d^{7} e x + 140 \, b c^{3} d^{6} e^{2} x + 90 \, b^{2} c^{2} d^{5} e^{3} x + 60 \, a c^{3} d^{5} e^{3} x + 40 \, b^{3} c d^{4} e^{4} x + 120 \, a b c^{2} d^{4} e^{4} x + 10 \, b^{4} d^{3} e^{5} x + 120 \, a b^{2} c d^{3} e^{5} x + 60 \, a^{2} c^{2} d^{3} e^{5} x + 60 \, a b^{3} d^{2} e^{6} x + 180 \, a^{2} b c d^{2} e^{6} x + 210 \, a^{2} b^{2} d e^{7} x + 140 \, a^{3} c d e^{7} x + 560 \, a^{3} b e^{8} x + 14 \, c^{4} d^{8} + 14 \, b c^{3} d^{7} e + 9 \, b^{2} c^{2} d^{6} e^{2} + 6 \, a c^{3} d^{6} e^{2} + 4 \, b^{3} c d^{5} e^{3} + 12 \, a b c^{2} d^{5} e^{3} + b^{4} d^{4} e^{4} + 12 \, a b^{2} c d^{4} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{4} + 6 \, a b^{3} d^{3} e^{5} + 18 \, a^{2} b c d^{3} e^{5} + 21 \, a^{2} b^{2} d^{2} e^{6} + 14 \, a^{3} c d^{2} e^{6} + 56 \, a^{3} b d e^{7} + 126 \, a^{4} e^{8}}{1260 \, {\left (e x + d\right )}^{10} e^{9}} \]
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Time = 10.10 (sec) , antiderivative size = 979, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=-\frac {\frac {126\,a^4\,e^8+56\,a^3\,b\,d\,e^7+14\,a^3\,c\,d^2\,e^6+21\,a^2\,b^2\,d^2\,e^6+18\,a^2\,b\,c\,d^3\,e^5+6\,a^2\,c^2\,d^4\,e^4+6\,a\,b^3\,d^3\,e^5+12\,a\,b^2\,c\,d^4\,e^4+12\,a\,b\,c^2\,d^5\,e^3+6\,a\,c^3\,d^6\,e^2+b^4\,d^4\,e^4+4\,b^3\,c\,d^5\,e^3+9\,b^2\,c^2\,d^6\,e^2+14\,b\,c^3\,d^7\,e+14\,c^4\,d^8}{1260\,e^9}+\frac {2\,x^3\,\left (18\,a^2\,b\,c\,e^5+6\,a^2\,c^2\,d\,e^4+6\,a\,b^3\,e^5+12\,a\,b^2\,c\,d\,e^4+12\,a\,b\,c^2\,d^2\,e^3+6\,a\,c^3\,d^3\,e^2+b^4\,d\,e^4+4\,b^3\,c\,d^2\,e^3+9\,b^2\,c^2\,d^3\,e^2+14\,b\,c^3\,d^4\,e+14\,c^4\,d^5\right )}{21\,e^6}+\frac {x^4\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3+6\,a\,c^3\,d^2\,e^2+b^4\,e^4+4\,b^3\,c\,d\,e^3+9\,b^2\,c^2\,d^2\,e^2+14\,b\,c^3\,d^3\,e+14\,c^4\,d^4\right )}{6\,e^5}+\frac {x\,\left (56\,a^3\,b\,e^7+14\,a^3\,c\,d\,e^6+21\,a^2\,b^2\,d\,e^6+18\,a^2\,b\,c\,d^2\,e^5+6\,a^2\,c^2\,d^3\,e^4+6\,a\,b^3\,d^2\,e^5+12\,a\,b^2\,c\,d^3\,e^4+12\,a\,b\,c^2\,d^4\,e^3+6\,a\,c^3\,d^5\,e^2+b^4\,d^3\,e^4+4\,b^3\,c\,d^4\,e^3+9\,b^2\,c^2\,d^5\,e^2+14\,b\,c^3\,d^6\,e+14\,c^4\,d^7\right )}{126\,e^8}+\frac {c^4\,x^8}{2\,e}+\frac {x^2\,\left (14\,a^3\,c\,e^6+21\,a^2\,b^2\,e^6+18\,a^2\,b\,c\,d\,e^5+6\,a^2\,c^2\,d^2\,e^4+6\,a\,b^3\,d\,e^5+12\,a\,b^2\,c\,d^2\,e^4+12\,a\,b\,c^2\,d^3\,e^3+6\,a\,c^3\,d^4\,e^2+b^4\,d^2\,e^4+4\,b^3\,c\,d^3\,e^3+9\,b^2\,c^2\,d^4\,e^2+14\,b\,c^3\,d^5\,e+14\,c^4\,d^6\right )}{28\,e^7}+\frac {4\,c^3\,x^7\,\left (b\,e+c\,d\right )}{3\,e^2}+\frac {c^2\,x^6\,\left (9\,b^2\,e^2+14\,b\,c\,d\,e+14\,c^2\,d^2+6\,a\,c\,e^2\right )}{6\,e^3}+\frac {c\,x^5\,\left (4\,b^3\,e^3+9\,b^2\,c\,d\,e^2+14\,b\,c^2\,d^2\,e+12\,a\,b\,c\,e^3+14\,c^3\,d^3+6\,a\,c^2\,d\,e^2\right )}{5\,e^4}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \]
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