Integrand size = 20, antiderivative size = 426 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=-\frac {c^3 (7 c d-4 b e) x}{e^8}+\frac {c^4 x^2}{2 e^7}-\frac {\left (c d^2-b d e+a e^2\right )^4}{6 e^9 (d+e x)^6}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{5 e^9 (d+e x)^5}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{2 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{3 e^9 (d+e x)^3}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{2 e^9 (d+e x)^2}+\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) \log (d+e x)}{e^9} \]
[Out]
Time = 0.36 (sec) , antiderivative size = 426, 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} \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=-\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{2 e^9 (d+e x)^2}+\frac {4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9 (d+e x)^4}+\frac {2 c^2 \log (d+e x) \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right )^4}{6 e^9 (d+e x)^6}-\frac {c^3 x (7 c d-4 b e)}{e^8}+\frac {c^4 x^2}{2 e^7} \]
[In]
[Out]
Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {c^3 (7 c d-4 b e)}{e^8}+\frac {c^4 x}{e^7}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^7}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^6}+\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)^5}+\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)^4}+\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)^3}+\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)^2}+\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)}\right ) \, dx \\ & = -\frac {c^3 (7 c d-4 b e) x}{e^8}+\frac {c^4 x^2}{2 e^7}-\frac {\left (c d^2-b d e+a e^2\right )^4}{6 e^9 (d+e x)^6}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{5 e^9 (d+e x)^5}-\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 )}{2 e^9 (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+b^2 e^2-c e (7 b d-3 a e)\right )}{3 e^9 (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 )}{2 e^9 (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^9 (d+e x)}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) \log (d+e x)}{e^9} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=\frac {c^4 \left (1023 d^8+5298 d^7 e x+10725 d^6 e^2 x^2+10100 d^5 e^3 x^3+3375 d^4 e^4 x^4-1170 d^3 e^5 x^5-1035 d^2 e^6 x^6-120 d e^7 x^7+15 e^8 x^8\right )-e^4 \left (5 a^4 e^4+4 a^3 b e^3 (d+6 e x)+3 a^2 b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b^3 e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+b^4 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )-2 c e^3 \left (a^3 e^3 \left (d^2+6 d e x+15 e^2 x^2\right )+3 a^2 b e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+6 a b^2 e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+10 b^3 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )-3 c^2 e^2 \left (2 a^2 e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+20 a b e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )-b^2 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )+2 c^3 e \left (a d e \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )-b \left (669 d^7+3594 d^6 e x+7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5-360 d e^6 x^6-60 e^7 x^7\right )\right )+60 c^2 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) (d+e x)^6 \log (d+e x)}{30 e^9 (d+e x)^6} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(888\) vs. \(2(415)=830\).
Time = 5.04 (sec) , antiderivative size = 889, normalized size of antiderivative = 2.09
method | result | size |
norman | \(\frac {-\frac {5 a^{4} e^{8}+4 a^{3} b d \,e^{7}+2 a^{3} c \,d^{2} e^{6}+3 a^{2} b^{2} d^{2} e^{6}+6 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+60 a b \,c^{2} d^{5} e^{3}-294 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+20 b^{3} c \,d^{5} e^{3}-441 b^{2} c^{2} d^{6} e^{2}+2058 b \,c^{3} d^{7} e -2058 c^{4} d^{8}}{30 e^{9}}+\frac {c^{4} x^{8}}{2 e}-\frac {2 \left (6 a b \,c^{2} e^{3}-12 a \,c^{3} d \,e^{2}+2 b^{3} c \,e^{3}-18 b^{2} c^{2} d \,e^{2}+84 b \,c^{3} d^{2} e -84 d^{3} c^{4}\right ) x^{5}}{e^{4}}-\frac {\left (6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}+60 a b \,c^{2} d \,e^{3}-180 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}+20 b^{3} c d \,e^{3}-270 b^{2} c^{2} d^{2} e^{2}+1260 b \,c^{3} d^{3} e -1260 c^{4} d^{4}\right ) x^{4}}{2 e^{5}}-\frac {2 \left (6 a^{2} b c \,e^{5}+6 d \,e^{4} a^{2} c^{2}+2 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}-220 d^{3} e^{2} c^{3} a +b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}-330 b^{2} c^{2} d^{3} e^{2}+1540 b \,c^{3} d^{4} e -1540 c^{4} d^{5}\right ) x^{3}}{3 e^{6}}-\frac {\left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+6 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+2 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+60 a b \,c^{2} d^{3} e^{3}-250 d^{4} e^{2} c^{3} a +b^{4} d^{2} e^{4}+20 b^{3} c \,d^{3} e^{3}-375 b^{2} c^{2} d^{4} e^{2}+1750 b \,c^{3} d^{5} e -1750 d^{6} c^{4}\right ) x^{2}}{2 e^{7}}-\frac {\left (4 a^{3} b \,e^{7}+2 d \,e^{6} c \,a^{3}+3 a^{2} b^{2} d \,e^{6}+6 a^{2} b c \,d^{2} e^{5}+6 d^{3} e^{4} a^{2} c^{2}+2 a \,b^{3} d^{2} e^{5}+12 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}-274 d^{5} e^{2} c^{3} a +b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}-411 b^{2} c^{2} d^{5} e^{2}+1918 b \,c^{3} d^{6} e -1918 d^{7} c^{4}\right ) x}{5 e^{8}}+\frac {4 c^{3} \left (b e -c d \right ) x^{7}}{e^{2}}}{\left (e x +d \right )^{6}}+\frac {2 c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(889\) |
default | \(\frac {c^{3} \left (\frac {1}{2} c e \,x^{2}+4 b e x -7 c d x \right )}{e^{8}}-\frac {4 a^{3} b \,e^{7}-8 d \,e^{6} c \,a^{3}-12 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}-24 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}-48 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}-24 d^{5} e^{2} c^{3} a -4 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}-36 b^{2} c^{2} d^{5} e^{2}+28 b \,c^{3} d^{6} e -8 d^{7} c^{4}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {4 c \left (3 a b c \,e^{3}-6 c^{2} a d \,e^{2}+b^{3} e^{3}-9 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right )}{e^{9} \left (e x +d \right )}-\frac {12 a^{2} b c \,e^{5}-24 d \,e^{4} a^{2} c^{2}+4 a \,b^{3} e^{5}-48 a \,b^{2} c d \,e^{4}+120 a b \,c^{2} d^{2} e^{3}-80 d^{3} e^{2} c^{3} a -4 b^{4} d \,e^{4}+40 b^{3} c \,d^{2} e^{3}-120 b^{2} c^{2} d^{3} e^{2}+140 b \,c^{3} d^{4} e -56 c^{4} d^{5}}{3 e^{9} \left (e x +d \right )^{3}}-\frac {4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}-36 a^{2} b c d \,e^{5}+36 d^{2} e^{4} a^{2} c^{2}-12 a \,b^{3} d \,e^{5}+72 a \,b^{2} c \,d^{2} e^{4}-120 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} c^{3} a +6 b^{4} d^{2} e^{4}-40 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}-84 b \,c^{3} d^{5} e +28 d^{6} c^{4}}{4 e^{9} \left (e x +d \right )^{4}}-\frac {6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+60 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}-20 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}-140 b \,c^{3} d^{3} e +70 c^{4} d^{4}}{2 e^{9} \left (e x +d \right )^{2}}+\frac {2 c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{9}}-\frac {a^{4} e^{8}-4 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-12 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}-12 a b \,c^{2} d^{5} e^{3}+4 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}-4 b^{3} c \,d^{5} e^{3}+6 b^{2} c^{2} d^{6} e^{2}-4 b \,c^{3} d^{7} e +c^{4} d^{8}}{6 e^{9} \left (e x +d \right )^{6}}\) | \(900\) |
risch | \(\frac {c^{4} x^{2}}{2 e^{7}}+\frac {4 c^{3} b x}{e^{7}}-\frac {7 c^{4} d x}{e^{8}}+\frac {\left (-12 e^{7} c^{2} a b +24 d \,e^{6} c^{3} a -4 b^{3} c \,e^{7}+36 b^{2} c^{2} d \,e^{6}-84 b \,c^{3} d^{2} e^{5}+56 d^{3} e^{4} c^{4}\right ) x^{5}-\frac {e^{3} \left (6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}+60 a b \,c^{2} d \,e^{3}-180 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}+20 b^{3} c d \,e^{3}-270 b^{2} c^{2} d^{2} e^{2}+700 b \,c^{3} d^{3} e -490 c^{4} d^{4}\right ) x^{4}}{2}-\frac {2 e^{2} \left (6 a^{2} b c \,e^{5}+6 d \,e^{4} a^{2} c^{2}+2 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}-220 d^{3} e^{2} c^{3} a +b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}-330 b^{2} c^{2} d^{3} e^{2}+910 b \,c^{3} d^{4} e -658 c^{4} d^{5}\right ) x^{3}}{3}-\frac {e \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+6 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+2 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+60 a b \,c^{2} d^{3} e^{3}-250 d^{4} e^{2} c^{3} a +b^{4} d^{2} e^{4}+20 b^{3} c \,d^{3} e^{3}-375 b^{2} c^{2} d^{4} e^{2}+1078 b \,c^{3} d^{5} e -798 d^{6} c^{4}\right ) x^{2}}{2}+\left (-\frac {4}{5} a^{3} b \,e^{7}-\frac {2}{5} d \,e^{6} c \,a^{3}-\frac {3}{5} a^{2} b^{2} d \,e^{6}-\frac {6}{5} a^{2} b c \,d^{2} e^{5}-\frac {6}{5} d^{3} e^{4} a^{2} c^{2}-\frac {2}{5} a \,b^{3} d^{2} e^{5}-\frac {12}{5} a \,b^{2} c \,d^{3} e^{4}-12 a b \,c^{2} d^{4} e^{3}+\frac {274}{5} d^{5} e^{2} c^{3} a -\frac {1}{5} b^{4} d^{3} e^{4}-4 b^{3} c \,d^{4} e^{3}+\frac {411}{5} b^{2} c^{2} d^{5} e^{2}-\frac {1218}{5} b \,c^{3} d^{6} e +\frac {918}{5} d^{7} c^{4}\right ) x -\frac {5 a^{4} e^{8}+4 a^{3} b d \,e^{7}+2 a^{3} c \,d^{2} e^{6}+3 a^{2} b^{2} d^{2} e^{6}+6 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+60 a b \,c^{2} d^{5} e^{3}-294 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+20 b^{3} c \,d^{5} e^{3}-441 b^{2} c^{2} d^{6} e^{2}+1338 b \,c^{3} d^{7} e -1023 c^{4} d^{8}}{30 e}}{e^{8} \left (e x +d \right )^{6}}+\frac {4 c^{3} \ln \left (e x +d \right ) a}{e^{7}}+\frac {6 c^{2} \ln \left (e x +d \right ) b^{2}}{e^{7}}-\frac {28 c^{3} \ln \left (e x +d \right ) b d}{e^{8}}+\frac {28 c^{4} \ln \left (e x +d \right ) d^{2}}{e^{9}}\) | \(910\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1519\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1191 vs. \(2 (414) = 828\).
Time = 0.33 (sec) , antiderivative size = 1191, normalized size of antiderivative = 2.80 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 866 vs. \(2 (414) = 828\).
Time = 0.23 (sec) , antiderivative size = 866, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=\frac {1023 \, c^{4} d^{8} - 1338 \, b c^{3} d^{7} e - 4 \, a^{3} b d e^{7} - 5 \, a^{4} e^{8} + 147 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 20 \, {\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} - 2 \, {\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} + 120 \, {\left (14 \, c^{4} d^{3} e^{5} - 21 \, b c^{3} d^{2} e^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 15 \, {\left (490 \, c^{4} d^{4} e^{4} - 700 \, b c^{3} d^{3} e^{5} + 90 \, {\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} + 20 \, {\left (658 \, c^{4} d^{5} e^{3} - 910 \, b c^{3} d^{4} e^{4} + 110 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} - 20 \, {\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} - 2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 15 \, {\left (798 \, c^{4} d^{6} e^{2} - 1078 \, b c^{3} d^{5} e^{3} + 125 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 20 \, {\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} - 2 \, {\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} + 6 \, {\left (918 \, c^{4} d^{7} e - 1218 \, b c^{3} d^{6} e^{2} - 4 \, a^{3} b e^{8} + 137 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 20 \, {\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} - 2 \, {\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}{30 \, {\left (e^{15} x^{6} + 6 \, d e^{14} x^{5} + 15 \, d^{2} e^{13} x^{4} + 20 \, d^{3} e^{12} x^{3} + 15 \, d^{4} e^{11} x^{2} + 6 \, d^{5} e^{10} x + d^{6} e^{9}\right )}} + \frac {c^{4} e x^{2} - 2 \, {\left (7 \, c^{4} d - 4 \, b c^{3} e\right )} x}{2 \, e^{8}} + \frac {2 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{9}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (414) = 828\).
Time = 0.26 (sec) , antiderivative size = 910, normalized size of antiderivative = 2.14 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=\frac {2 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + 3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{9}} + \frac {c^{4} e^{7} x^{2} - 14 \, c^{4} d e^{6} x + 8 \, b c^{3} e^{7} x}{2 \, e^{14}} + \frac {1023 \, c^{4} d^{8} - 1338 \, b c^{3} d^{7} e + 441 \, b^{2} c^{2} d^{6} e^{2} + 294 \, a c^{3} d^{6} e^{2} - 20 \, b^{3} c d^{5} e^{3} - 60 \, 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} - 2 \, a b^{3} d^{3} e^{5} - 6 \, 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} - 4 \, a^{3} b d e^{7} - 5 \, a^{4} e^{8} + 120 \, {\left (14 \, c^{4} d^{3} e^{5} - 21 \, b c^{3} d^{2} e^{6} + 9 \, b^{2} c^{2} d e^{7} + 6 \, a c^{3} d e^{7} - b^{3} c e^{8} - 3 \, a b c^{2} e^{8}\right )} x^{5} + 15 \, {\left (490 \, c^{4} d^{4} e^{4} - 700 \, b c^{3} d^{3} e^{5} + 270 \, b^{2} c^{2} d^{2} e^{6} + 180 \, 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} + 20 \, {\left (658 \, c^{4} d^{5} e^{3} - 910 \, b c^{3} d^{4} e^{4} + 330 \, b^{2} c^{2} d^{3} e^{5} + 220 \, a c^{3} d^{3} e^{5} - 20 \, b^{3} c d^{2} e^{6} - 60 \, 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} - 2 \, a b^{3} e^{8} - 6 \, a^{2} b c e^{8}\right )} x^{3} + 15 \, {\left (798 \, c^{4} d^{6} e^{2} - 1078 \, b c^{3} d^{5} e^{3} + 375 \, b^{2} c^{2} d^{4} e^{4} + 250 \, a c^{3} d^{4} e^{4} - 20 \, b^{3} c d^{3} e^{5} - 60 \, a b c^{2} d^{3} e^{5} - b^{4} d^{2} e^{6} - 12 \, a b^{2} c d^{2} e^{6} - 6 \, a^{2} c^{2} d^{2} e^{6} - 2 \, a b^{3} d e^{7} - 6 \, a^{2} b c d e^{7} - 3 \, a^{2} b^{2} e^{8} - 2 \, a^{3} c e^{8}\right )} x^{2} + 6 \, {\left (918 \, c^{4} d^{7} e - 1218 \, b c^{3} d^{6} e^{2} + 411 \, b^{2} c^{2} d^{5} e^{3} + 274 \, a c^{3} d^{5} e^{3} - 20 \, b^{3} c d^{4} e^{4} - 60 \, a b c^{2} d^{4} e^{4} - b^{4} d^{3} e^{5} - 12 \, a b^{2} c d^{3} e^{5} - 6 \, a^{2} c^{2} d^{3} e^{5} - 2 \, a b^{3} d^{2} e^{6} - 6 \, a^{2} b c d^{2} e^{6} - 3 \, a^{2} b^{2} d e^{7} - 2 \, a^{3} c d e^{7} - 4 \, a^{3} b e^{8}\right )} x}{30 \, {\left (e x + d\right )}^{6} e^{9}} \]
[In]
[Out]
Time = 9.92 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=x\,\left (\frac {4\,b\,c^3}{e^7}-\frac {7\,c^4\,d}{e^8}\right )-\frac {x^3\,\left (4\,a^2\,b\,c\,e^7+4\,a^2\,c^2\,d\,e^6+\frac {4\,a\,b^3\,e^7}{3}+8\,a\,b^2\,c\,d\,e^6+40\,a\,b\,c^2\,d^2\,e^5-\frac {440\,a\,c^3\,d^3\,e^4}{3}+\frac {2\,b^4\,d\,e^6}{3}+\frac {40\,b^3\,c\,d^2\,e^5}{3}-220\,b^2\,c^2\,d^3\,e^4+\frac {1820\,b\,c^3\,d^4\,e^3}{3}-\frac {1316\,c^4\,d^5\,e^2}{3}\right )+x\,\left (\frac {4\,a^3\,b\,e^7}{5}+\frac {2\,a^3\,c\,d\,e^6}{5}+\frac {3\,a^2\,b^2\,d\,e^6}{5}+\frac {6\,a^2\,b\,c\,d^2\,e^5}{5}+\frac {6\,a^2\,c^2\,d^3\,e^4}{5}+\frac {2\,a\,b^3\,d^2\,e^5}{5}+\frac {12\,a\,b^2\,c\,d^3\,e^4}{5}+12\,a\,b\,c^2\,d^4\,e^3-\frac {274\,a\,c^3\,d^5\,e^2}{5}+\frac {b^4\,d^3\,e^4}{5}+4\,b^3\,c\,d^4\,e^3-\frac {411\,b^2\,c^2\,d^5\,e^2}{5}+\frac {1218\,b\,c^3\,d^6\,e}{5}-\frac {918\,c^4\,d^7}{5}\right )+x^4\,\left (3\,a^2\,c^2\,e^7+6\,a\,b^2\,c\,e^7+30\,a\,b\,c^2\,d\,e^6-90\,a\,c^3\,d^2\,e^5+\frac {b^4\,e^7}{2}+10\,b^3\,c\,d\,e^6-135\,b^2\,c^2\,d^2\,e^5+350\,b\,c^3\,d^3\,e^4-245\,c^4\,d^4\,e^3\right )+x^5\,\left (4\,b^3\,c\,e^7-36\,b^2\,c^2\,d\,e^6+84\,b\,c^3\,d^2\,e^5+12\,a\,b\,c^2\,e^7-56\,c^4\,d^3\,e^4-24\,a\,c^3\,d\,e^6\right )+\frac {5\,a^4\,e^8+4\,a^3\,b\,d\,e^7+2\,a^3\,c\,d^2\,e^6+3\,a^2\,b^2\,d^2\,e^6+6\,a^2\,b\,c\,d^3\,e^5+6\,a^2\,c^2\,d^4\,e^4+2\,a\,b^3\,d^3\,e^5+12\,a\,b^2\,c\,d^4\,e^4+60\,a\,b\,c^2\,d^5\,e^3-294\,a\,c^3\,d^6\,e^2+b^4\,d^4\,e^4+20\,b^3\,c\,d^5\,e^3-441\,b^2\,c^2\,d^6\,e^2+1338\,b\,c^3\,d^7\,e-1023\,c^4\,d^8}{30\,e}+x^2\,\left (a^3\,c\,e^7+\frac {3\,a^2\,b^2\,e^7}{2}+3\,a^2\,b\,c\,d\,e^6+3\,a^2\,c^2\,d^2\,e^5+a\,b^3\,d\,e^6+6\,a\,b^2\,c\,d^2\,e^5+30\,a\,b\,c^2\,d^3\,e^4-125\,a\,c^3\,d^4\,e^3+\frac {b^4\,d^2\,e^5}{2}+10\,b^3\,c\,d^3\,e^4-\frac {375\,b^2\,c^2\,d^4\,e^3}{2}+539\,b\,c^3\,d^5\,e^2-399\,c^4\,d^6\,e\right )}{d^6\,e^8+6\,d^5\,e^9\,x+15\,d^4\,e^{10}\,x^2+20\,d^3\,e^{11}\,x^3+15\,d^2\,e^{12}\,x^4+6\,d\,e^{13}\,x^5+e^{14}\,x^6}+\frac {\ln \left (d+e\,x\right )\,\left (6\,b^2\,c^2\,e^2-28\,b\,c^3\,d\,e+28\,c^4\,d^2+4\,a\,c^3\,e^2\right )}{e^9}+\frac {c^4\,x^2}{2\,e^7} \]
[In]
[Out]