Integrand size = 20, antiderivative size = 426 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=-\frac {c \left (35 c^3 d^3-4 b^3 e^3+6 b c e^2 (5 b d-2 a e)-20 c^2 d e (3 b d-a e)\right ) x}{e^8}+\frac {c^2 \left (15 c^2 d^2+6 b^2 e^2-4 c e (5 b d-a e)\right ) x^2}{2 e^7}-\frac {c^3 (5 c d-4 b e) x^3}{3 e^6}+\frac {c^4 x^4}{4 e^5}-\frac {\left (c d^2-b d e+a e^2\right )^4}{4 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^9 (d+e x)^3}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{e^9} \]
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Time = 0.43 (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} \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=\frac {\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^9}-\frac {c x \left (-20 c^2 d e (3 b d-a e)+6 b c e^2 (5 b d-2 a e)-4 b^3 e^3+35 c^3 d^3\right )}{e^8}-\frac {\left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )^2}{e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^9 (d+e x)}+\frac {c^2 x^2 \left (-4 c e (5 b d-a e)+6 b^2 e^2+15 c^2 d^2\right )}{2 e^7}-\frac {\left (a e^2-b d e+c d^2\right )^4}{4 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^3}-\frac {c^3 x^3 (5 c d-4 b e)}{3 e^6}+\frac {c^4 x^4}{4 e^5} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c \left (-35 c^3 d^3+4 b^3 e^3-6 b c e^2 (5 b d-2 a e)+20 c^2 d e (3 b d-a e)\right )}{e^8}+\frac {c^2 \left (15 c^2 d^2+6 b^2 e^2-4 c e (5 b d-a e)\right ) x}{e^7}-\frac {c^3 (5 c d-4 b e) x^2}{e^6}+\frac {c^4 x^3}{e^5}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^5}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (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 )}{e^8 (d+e x)^3}+\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)^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^8 (d+e x)}\right ) \, dx \\ & = -\frac {c \left (35 c^3 d^3-4 b^3 e^3+6 b c e^2 (5 b d-2 a e)-20 c^2 d e (3 b d-a e)\right ) x}{e^8}+\frac {c^2 \left (15 c^2 d^2+6 b^2 e^2-4 c e (5 b d-a e)\right ) x^2}{2 e^7}-\frac {c^3 (5 c d-4 b e) x^3}{3 e^6}+\frac {c^4 x^4}{4 e^5}-\frac {\left (c d^2-b d e+a e^2\right )^4}{4 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^9 (d+e x)^3}-\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 )}{e^9 (d+e x)^2}+\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 )}{e^9 (d+e x)}+\frac {\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 ) \log (d+e x)}{e^9} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=\frac {12 c e \left (-35 c^3 d^3+4 b^3 e^3-6 b c e^2 (5 b d-2 a e)+20 c^2 d e (3 b d-a e)\right ) x+6 c^2 e^2 \left (15 c^2 d^2+6 b^2 e^2+4 c e (-5 b d+a e)\right ) x^2+4 c^3 e^3 (-5 c d+4 b e) x^3+3 c^4 e^4 x^4-\frac {3 \left (c d^2+e (-b d+a e)\right )^4}{(d+e x)^4}+\frac {16 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^3}-\frac {12 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2}{(d+e x)^2}+\frac {48 (2 c d-b e) \left (7 c^3 d^4-2 c^2 d^2 e (7 b d-5 a e)+b^2 e^3 (-b d+a e)+c e^2 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right )}{d+e x}+12 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{12 e^9} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(887\) vs. \(2(416)=832\).
Time = 3.23 (sec) , antiderivative size = 888, normalized size of antiderivative = 2.08
method | result | size |
norman | \(\frac {-\frac {3 a^{4} e^{8}+4 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}+36 a^{2} b c \,d^{3} e^{5}-150 a^{2} c^{2} d^{4} e^{4}+12 a \,b^{3} d^{3} e^{5}-300 a \,b^{2} c \,d^{4} e^{4}+1500 a b \,c^{2} d^{5} e^{3}-1500 a \,c^{3} d^{6} e^{2}-25 b^{4} d^{4} e^{4}+500 b^{3} c \,d^{5} e^{3}-2250 b^{2} c^{2} d^{6} e^{2}+3500 b \,c^{3} d^{7} e -1750 c^{4} d^{8}}{12 e^{9}}+\frac {c^{4} x^{8}}{4 e}-\frac {4 \left (3 a^{2} b c \,e^{5}-6 d \,e^{4} a^{2} c^{2}+a \,b^{3} e^{5}-12 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}-60 d^{3} e^{2} c^{3} a -b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}-90 b^{2} c^{2} d^{3} e^{2}+140 b \,c^{3} d^{4} e -70 c^{4} d^{5}\right ) x^{3}}{e^{6}}-\frac {\left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+18 a^{2} b c d \,e^{5}-54 d^{2} e^{4} a^{2} c^{2}+6 a \,b^{3} d \,e^{5}-108 a \,b^{2} c \,d^{2} e^{4}+540 a b \,c^{2} d^{3} e^{3}-540 d^{4} e^{2} c^{3} a -9 b^{4} d^{2} e^{4}+180 b^{3} c \,d^{3} e^{3}-810 b^{2} c^{2} d^{4} e^{2}+1260 b \,c^{3} d^{5} e -630 d^{6} c^{4}\right ) x^{2}}{e^{7}}-\frac {2 \left (2 a^{3} b \,e^{7}+2 d \,e^{6} c \,a^{3}+3 a^{2} b^{2} d \,e^{6}+18 a^{2} b c \,d^{2} e^{5}-66 d^{3} e^{4} a^{2} c^{2}+6 a \,b^{3} d^{2} e^{5}-132 a \,b^{2} c \,d^{3} e^{4}+660 a b \,c^{2} d^{4} e^{3}-660 d^{5} e^{2} c^{3} a -11 b^{4} d^{3} e^{4}+220 b^{3} c \,d^{4} e^{3}-990 b^{2} c^{2} d^{5} e^{2}+1540 b \,c^{3} d^{6} e -770 d^{7} c^{4}\right ) x}{3 e^{8}}+\frac {2 c \left (6 a b c \,e^{3}-6 c^{2} a d \,e^{2}+2 b^{3} e^{3}-9 b^{2} d \,e^{2} c +14 b \,c^{2} d^{2} e -7 c^{3} d^{3}\right ) x^{5}}{e^{4}}+\frac {c^{2} \left (6 a c \,e^{2}+9 b^{2} e^{2}-14 b c d e +7 c^{2} d^{2}\right ) x^{6}}{3 e^{3}}+\frac {2 c^{3} \left (2 b e -c d \right ) x^{7}}{3 e^{2}}}{\left (e x +d \right )^{4}}+\frac {\left (6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+60 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}-20 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}-140 b \,c^{3} d^{3} e +70 c^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(888\) |
default | \(\frac {c \left (\frac {1}{4} c^{3} x^{4} e^{3}+\frac {4}{3} b \,c^{2} e^{3} x^{3}-\frac {5}{3} c^{3} d \,e^{2} x^{3}+2 a \,c^{2} e^{3} x^{2}+3 b^{2} c \,e^{3} x^{2}-10 b \,c^{2} d \,e^{2} x^{2}+\frac {15}{2} c^{3} d^{2} e \,x^{2}+12 a b c \,e^{3} x -20 a \,c^{2} d \,e^{2} x +4 b^{3} e^{3} x -30 b^{2} d \,e^{2} c x +60 b \,c^{2} d^{2} e x -35 c^{3} d^{3} x \right )}{e^{8}}-\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}}{e^{9} \left (e x +d \right )}-\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}}{3 e^{9} \left (e x +d \right )^{3}}-\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}}{4 e^{9} \left (e x +d \right )^{4}}-\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}}{2 e^{9} \left (e x +d \right )^{2}}+\frac {\left (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}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(912\) |
risch | \(\frac {c^{4} x^{4}}{4 e^{5}}+\frac {4 c^{3} b \,x^{3}}{3 e^{5}}-\frac {5 c^{4} d \,x^{3}}{3 e^{6}}+\frac {2 c^{3} a \,x^{2}}{e^{5}}+\frac {3 c^{2} b^{2} x^{2}}{e^{5}}-\frac {10 c^{3} b d \,x^{2}}{e^{6}}+\frac {15 c^{4} d^{2} x^{2}}{2 e^{7}}+\frac {12 c^{2} a b x}{e^{5}}-\frac {20 c^{3} a d x}{e^{6}}+\frac {4 c \,b^{3} x}{e^{5}}-\frac {30 c^{2} b^{2} d x}{e^{6}}+\frac {60 c^{3} b \,d^{2} x}{e^{7}}-\frac {35 c^{4} d^{3} x}{e^{8}}+\frac {\left (-12 a^{2} b c \,e^{7}+24 d \,e^{6} a^{2} c^{2}-4 a \,b^{3} e^{7}+48 a \,b^{2} c d \,e^{6}-120 a b \,c^{2} d^{2} e^{5}+80 d^{3} e^{4} c^{3} a +4 b^{4} d \,e^{6}-40 b^{3} c \,d^{2} e^{5}+120 b^{2} c^{2} d^{3} e^{4}-140 d^{4} c^{3} b \,e^{3}+56 d^{5} e^{2} c^{4}\right ) x^{3}-e \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+18 a^{2} b c d \,e^{5}-54 d^{2} e^{4} a^{2} c^{2}+6 a \,b^{3} d \,e^{5}-108 a \,b^{2} c \,d^{2} e^{4}+300 a b \,c^{2} d^{3} e^{3}-210 d^{4} e^{2} c^{3} a -9 b^{4} d^{2} e^{4}+100 b^{3} c \,d^{3} e^{3}-315 b^{2} c^{2} d^{4} e^{2}+378 b \,c^{3} d^{5} e -154 d^{6} c^{4}\right ) x^{2}+\left (-\frac {4}{3} a^{3} b \,e^{7}-\frac {4}{3} d \,e^{6} c \,a^{3}-2 a^{2} b^{2} d \,e^{6}-12 a^{2} b c \,d^{2} e^{5}+44 d^{3} e^{4} a^{2} c^{2}-4 a \,b^{3} d^{2} e^{5}+88 a \,b^{2} c \,d^{3} e^{4}-260 a b \,c^{2} d^{4} e^{3}+188 d^{5} e^{2} c^{3} a +\frac {22}{3} b^{4} d^{3} e^{4}-\frac {260}{3} b^{3} c \,d^{4} e^{3}+282 b^{2} c^{2} d^{5} e^{2}-\frac {1036}{3} b \,c^{3} d^{6} e +\frac {428}{3} d^{7} c^{4}\right ) x -\frac {3 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}+36 a^{2} b c \,d^{3} e^{5}-150 a^{2} c^{2} d^{4} e^{4}+12 a \,b^{3} d^{3} e^{5}-300 a \,b^{2} c \,d^{4} e^{4}+924 a b \,c^{2} d^{5} e^{3}-684 a \,c^{3} d^{6} e^{2}-25 b^{4} d^{4} e^{4}+308 b^{3} c \,d^{5} e^{3}-1026 b^{2} c^{2} d^{6} e^{2}+1276 b \,c^{3} d^{7} e -533 c^{4} d^{8}}{12 e}}{e^{8} \left (e x +d \right )^{4}}+\frac {6 \ln \left (e x +d \right ) c^{2} a^{2}}{e^{5}}+\frac {12 \ln \left (e x +d \right ) a \,b^{2} c}{e^{5}}-\frac {60 \ln \left (e x +d \right ) a b \,c^{2} d}{e^{6}}+\frac {60 \ln \left (e x +d \right ) c^{3} a \,d^{2}}{e^{7}}+\frac {b^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {20 \ln \left (e x +d \right ) b^{3} c d}{e^{6}}+\frac {90 \ln \left (e x +d \right ) b^{2} c^{2} d^{2}}{e^{7}}-\frac {140 \ln \left (e x +d \right ) b \,c^{3} d^{3}}{e^{8}}+\frac {70 \ln \left (e x +d \right ) c^{4} d^{4}}{e^{9}}\) | \(957\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1784\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1303 vs. \(2 (416) = 832\).
Time = 0.30 (sec) , antiderivative size = 1303, normalized size of antiderivative = 3.06 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 843 vs. \(2 (416) = 832\).
Time = 0.22 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=\frac {533 \, c^{4} d^{8} - 1276 \, b c^{3} d^{7} e - 4 \, a^{3} b d e^{7} - 3 \, a^{4} e^{8} + 342 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 308 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + 25 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} - 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 48 \, {\left (14 \, c^{4} d^{5} e^{3} - 35 \, b c^{3} d^{4} e^{4} + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} - 10 \, {\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} + 12 \, {\left (154 \, c^{4} d^{6} e^{2} - 378 \, b c^{3} d^{5} e^{3} + 105 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 100 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + 9 \, {\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} - {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 8 \, {\left (214 \, c^{4} d^{7} e - 518 \, b c^{3} d^{6} e^{2} - 2 \, a^{3} b e^{8} + 141 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 130 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + 11 \, {\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} - {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{12 \, {\left (e^{13} x^{4} + 4 \, d e^{12} x^{3} + 6 \, d^{2} e^{11} x^{2} + 4 \, d^{3} e^{10} x + d^{4} e^{9}\right )}} + \frac {3 \, c^{4} e^{3} x^{4} - 4 \, {\left (5 \, c^{4} d e^{2} - 4 \, b c^{3} e^{3}\right )} x^{3} + 6 \, {\left (15 \, c^{4} d^{2} e - 20 \, b c^{3} d e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{3}\right )} x^{2} - 12 \, {\left (35 \, c^{4} d^{3} - 60 \, b c^{3} d^{2} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} x}{12 \, e^{8}} + \frac {{\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\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. 1301 vs. \(2 (416) = 832\).
Time = 0.27 (sec) , antiderivative size = 1301, normalized size of antiderivative = 3.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=\text {Too large to display} \]
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Time = 9.94 (sec) , antiderivative size = 1005, normalized size of antiderivative = 2.36 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=x^3\,\left (\frac {4\,b\,c^3}{3\,e^5}-\frac {5\,c^4\,d}{3\,e^6}\right )-\frac {x\,\left (\frac {4\,a^3\,b\,e^7}{3}+\frac {4\,a^3\,c\,d\,e^6}{3}+2\,a^2\,b^2\,d\,e^6+12\,a^2\,b\,c\,d^2\,e^5-44\,a^2\,c^2\,d^3\,e^4+4\,a\,b^3\,d^2\,e^5-88\,a\,b^2\,c\,d^3\,e^4+260\,a\,b\,c^2\,d^4\,e^3-188\,a\,c^3\,d^5\,e^2-\frac {22\,b^4\,d^3\,e^4}{3}+\frac {260\,b^3\,c\,d^4\,e^3}{3}-282\,b^2\,c^2\,d^5\,e^2+\frac {1036\,b\,c^3\,d^6\,e}{3}-\frac {428\,c^4\,d^7}{3}\right )-x^3\,\left (-12\,a^2\,b\,c\,e^7+24\,a^2\,c^2\,d\,e^6-4\,a\,b^3\,e^7+48\,a\,b^2\,c\,d\,e^6-120\,a\,b\,c^2\,d^2\,e^5+80\,a\,c^3\,d^3\,e^4+4\,b^4\,d\,e^6-40\,b^3\,c\,d^2\,e^5+120\,b^2\,c^2\,d^3\,e^4-140\,b\,c^3\,d^4\,e^3+56\,c^4\,d^5\,e^2\right )+\frac {3\,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+36\,a^2\,b\,c\,d^3\,e^5-150\,a^2\,c^2\,d^4\,e^4+12\,a\,b^3\,d^3\,e^5-300\,a\,b^2\,c\,d^4\,e^4+924\,a\,b\,c^2\,d^5\,e^3-684\,a\,c^3\,d^6\,e^2-25\,b^4\,d^4\,e^4+308\,b^3\,c\,d^5\,e^3-1026\,b^2\,c^2\,d^6\,e^2+1276\,b\,c^3\,d^7\,e-533\,c^4\,d^8}{12\,e}+x^2\,\left (2\,a^3\,c\,e^7+3\,a^2\,b^2\,e^7+18\,a^2\,b\,c\,d\,e^6-54\,a^2\,c^2\,d^2\,e^5+6\,a\,b^3\,d\,e^6-108\,a\,b^2\,c\,d^2\,e^5+300\,a\,b\,c^2\,d^3\,e^4-210\,a\,c^3\,d^4\,e^3-9\,b^4\,d^2\,e^5+100\,b^3\,c\,d^3\,e^4-315\,b^2\,c^2\,d^4\,e^3+378\,b\,c^3\,d^5\,e^2-154\,c^4\,d^6\,e\right )}{d^4\,e^8+4\,d^3\,e^9\,x+6\,d^2\,e^{10}\,x^2+4\,d\,e^{11}\,x^3+e^{12}\,x^4}-x^2\,\left (\frac {5\,d\,\left (\frac {4\,b\,c^3}{e^5}-\frac {5\,c^4\,d}{e^6}\right )}{2\,e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{2\,e^5}+\frac {5\,c^4\,d^2}{e^7}\right )-x\,\left (\frac {10\,c^4\,d^3}{e^8}+\frac {10\,d^2\,\left (\frac {4\,b\,c^3}{e^5}-\frac {5\,c^4\,d}{e^6}\right )}{e^2}-\frac {5\,d\,\left (\frac {5\,d\,\left (\frac {4\,b\,c^3}{e^5}-\frac {5\,c^4\,d}{e^6}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^5}+\frac {10\,c^4\,d^2}{e^7}\right )}{e}-\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^5}\right )+\frac {c^4\,x^4}{4\,e^5}+\frac {\ln \left (d+e\,x\right )\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4-60\,a\,b\,c^2\,d\,e^3+60\,a\,c^3\,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\right )}{e^9} \]
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