Integrand size = 20, antiderivative size = 430 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) x}{e^8}-\frac {\left (c d^2-b d e+a e^2\right )^4}{2 e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^2}{2 e^9}-\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^3}{3 e^9}+\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^4}{2 e^9}-\frac {4 c^3 (2 c d-b e) (d+e x)^5}{5 e^9}+\frac {c^4 (d+e x)^6}{6 e^9}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) \log (d+e x)}{e^9} \]
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Time = 0.45 (sec) , antiderivative size = 430, 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} \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\frac {(d+e x)^2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{2 e^9}+\frac {c^2 (d+e x)^4 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9}-\frac {4 c (d+e x)^3 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac {2 \log (d+e x) \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}-\frac {4 x (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^4}{2 e^9 (d+e x)^2}-\frac {4 c^3 (d+e x)^5 (2 c d-b e)}{5 e^9}+\frac {c^4 (d+e x)^6}{6 e^9} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\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}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^3}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^2}+\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)}+\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 ) (d+e x)}{e^8}+\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 ) (d+e x)^2}{e^8}+\frac {2 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)^3}{e^8}-\frac {4 c^3 (2 c d-b e) (d+e x)^4}{e^8}+\frac {c^4 (d+e x)^5}{e^8}\right ) \, dx \\ & = -\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 ) x}{e^8}-\frac {\left (c d^2-b d e+a e^2\right )^4}{2 e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{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 ) (d+e x)^2}{2 e^9}-\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 ) (d+e x)^3}{3 e^9}+\frac {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)^4}{2 e^9}-\frac {4 c^3 (2 c d-b e) (d+e x)^5}{5 e^9}+\frac {c^4 (d+e x)^6}{6 e^9}+\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 ) \log (d+e x)}{e^9} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\frac {30 e \left (-21 c^4 d^5+20 c^3 d^3 e (3 b d-2 a e)+b^3 e^4 (-3 b d+4 a e)+12 b c e^3 \left (2 b^2 d^2-3 a b d e+a^2 e^2\right )-6 c^2 d e^2 \left (10 b^2 d^2-12 a b d e+3 a^2 e^2\right )\right ) x+15 e^2 \left (15 c^4 d^4+b^4 e^4-8 c^3 d^2 e (5 b d-3 a e)-12 b^2 c e^3 (b d-a e)+6 c^2 e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right ) x^2+20 c e^3 (-c d+b e) \left (5 c^2 d^2+2 b^2 e^2+c e (-7 b d+6 a e)\right ) x^3+15 c^2 e^4 \left (3 c^2 d^2+3 b^2 e^2+2 c e (-3 b d+a e)\right ) x^4+6 c^3 e^5 (-3 c d+4 b e) x^5+5 c^4 e^6 x^6-\frac {15 \left (c d^2+e (-b d+a e)\right )^4}{(d+e x)^2}+\frac {120 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{d+e x}+60 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2 \log (d+e x)}{30 e^9} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(889\) vs. \(2(418)=836\).
Time = 3.21 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.07
method | result | size |
norman | \(\frac {-\frac {a^{4} e^{8}+4 a^{3} b d \,e^{7}-12 a^{3} c \,d^{2} e^{6}-18 a^{2} b^{2} d^{2} e^{6}+108 a^{2} b c \,d^{3} e^{5}-108 a^{2} c^{2} d^{4} e^{4}+36 a \,b^{3} d^{3} e^{5}-216 a \,b^{2} c \,d^{4} e^{4}+360 a b \,c^{2} d^{5} e^{3}-180 a \,c^{3} d^{6} e^{2}-18 b^{4} d^{4} e^{4}+120 b^{3} c \,d^{5} e^{3}-270 b^{2} c^{2} d^{6} e^{2}+252 b \,c^{3} d^{7} e -84 c^{4} d^{8}}{2 e^{9}}+\frac {c^{4} x^{8}}{6 e}+\frac {\left (18 c^{2} a^{2} e^{4}+36 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+30 c^{3} a \,d^{2} e^{2}+3 b^{4} e^{4}-20 b^{3} c d \,e^{3}+45 b^{2} c^{2} d^{2} e^{2}-42 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}-18 d \,e^{4} a^{2} c^{2}+6 a \,b^{3} e^{5}-36 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}-30 d^{3} e^{2} c^{3} a -3 b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}-45 b^{2} c^{2} d^{3} e^{2}+42 b \,c^{3} d^{4} e -14 c^{4} d^{5}\right ) x^{3}}{3 e^{6}}-\frac {2 \left (2 a^{3} b \,e^{7}-4 d \,e^{6} c \,a^{3}-6 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}-36 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}-72 a \,b^{2} c \,d^{3} e^{4}+120 a b \,c^{2} d^{4} e^{3}-60 d^{5} e^{2} c^{3} a -6 b^{4} d^{3} e^{4}+40 b^{3} c \,d^{4} e^{3}-90 b^{2} c^{2} d^{5} e^{2}+84 b \,c^{3} d^{6} e -28 d^{7} c^{4}\right ) x}{e^{8}}+\frac {c \left (60 a b c \,e^{3}-30 c^{2} a d \,e^{2}+20 b^{3} e^{3}-45 b^{2} d \,e^{2} c +42 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) x^{5}}{15 e^{4}}+\frac {c^{2} \left (30 a c \,e^{2}+45 b^{2} e^{2}-42 b c d e +14 c^{2} d^{2}\right ) x^{6}}{30 e^{3}}+\frac {4 c^{3} \left (3 b e -c d \right ) x^{7}}{15 e^{2}}}{\left (e x +d \right )^{2}}+\frac {2 \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}-18 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}-6 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}-60 a b \,c^{2} d^{3} e^{3}+30 d^{4} e^{2} c^{3} a +3 b^{4} d^{2} e^{4}-20 b^{3} c \,d^{3} e^{3}+45 b^{2} c^{2} d^{4} e^{2}-42 b \,c^{3} d^{5} e +14 d^{6} c^{4}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(890\) |
default | \(\frac {-21 c^{4} d^{5} x +a \,c^{3} e^{5} x^{4}+\frac {1}{2} b^{4} e^{5} x^{2}-6 b^{2} c^{2} d \,e^{4} x^{3}+8 b \,c^{3} d^{2} e^{3} x^{3}-18 d \,e^{4} a^{2} c^{2} x -40 d^{3} e^{2} c^{3} a x -36 a \,b^{2} c d \,e^{4} x +72 a b \,c^{2} d^{2} e^{3} x +6 a \,b^{2} c \,e^{5} x^{2}+12 a \,c^{3} d^{2} e^{3} x^{2}-6 b^{3} c d \,e^{4} x^{2}+\frac {1}{6} c^{4} x^{6} e^{5}-3 b \,c^{3} d \,e^{4} x^{4}+18 b^{2} c^{2} d^{2} e^{3} x^{2}-20 b \,c^{3} d^{3} e^{2} x^{2}+12 a^{2} b c \,e^{5} x +\frac {4}{5} b \,c^{3} e^{5} x^{5}-\frac {3}{5} c^{4} d \,e^{4} x^{5}+\frac {3}{2} b^{2} c^{2} e^{5} x^{4}+\frac {3}{2} c^{4} d^{2} e^{3} x^{4}+\frac {4}{3} b^{3} c \,e^{5} x^{3}-\frac {10}{3} c^{4} d^{3} e^{2} x^{3}+3 a^{2} c^{2} e^{5} x^{2}+\frac {15}{2} c^{4} d^{4} e \,x^{2}+4 a \,b^{3} e^{5} x -3 b^{4} d \,e^{4} x -18 a b \,c^{2} d \,e^{4} x^{2}+24 b^{3} c \,d^{2} e^{3} x -60 b^{2} c^{2} d^{3} e^{2} x +60 b \,c^{3} d^{4} e x +4 a b \,c^{2} e^{5} x^{3}-4 a \,c^{3} d \,e^{4} x^{3}}{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}}{e^{9} \left (e x +d \right )}-\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}}{2 e^{9} \left (e x +d \right )^{2}}+\frac {\left (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}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(952\) |
risch | \(\text {Expression too large to display}\) | \(1023\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1578\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1218 vs. \(2 (418) = 836\).
Time = 0.66 (sec) , antiderivative size = 1218, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (422) = 844\).
Time = 9.79 (sec) , antiderivative size = 906, normalized size of antiderivative = 2.11 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\frac {c^{4} x^{6}}{6 e^{3}} + x^{5} \cdot \left (\frac {4 b c^{3}}{5 e^{3}} - \frac {3 c^{4} d}{5 e^{4}}\right ) + x^{4} \left (\frac {a c^{3}}{e^{3}} + \frac {3 b^{2} c^{2}}{2 e^{3}} - \frac {3 b c^{3} d}{e^{4}} + \frac {3 c^{4} d^{2}}{2 e^{5}}\right ) + x^{3} \cdot \left (\frac {4 a b c^{2}}{e^{3}} - \frac {4 a c^{3} d}{e^{4}} + \frac {4 b^{3} c}{3 e^{3}} - \frac {6 b^{2} c^{2} d}{e^{4}} + \frac {8 b c^{3} d^{2}}{e^{5}} - \frac {10 c^{4} d^{3}}{3 e^{6}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c^{2}}{e^{3}} + \frac {6 a b^{2} c}{e^{3}} - \frac {18 a b c^{2} d}{e^{4}} + \frac {12 a c^{3} d^{2}}{e^{5}} + \frac {b^{4}}{2 e^{3}} - \frac {6 b^{3} c d}{e^{4}} + \frac {18 b^{2} c^{2} d^{2}}{e^{5}} - \frac {20 b c^{3} d^{3}}{e^{6}} + \frac {15 c^{4} d^{4}}{2 e^{7}}\right ) + x \left (\frac {12 a^{2} b c}{e^{3}} - \frac {18 a^{2} c^{2} d}{e^{4}} + \frac {4 a b^{3}}{e^{3}} - \frac {36 a b^{2} c d}{e^{4}} + \frac {72 a b c^{2} d^{2}}{e^{5}} - \frac {40 a c^{3} d^{3}}{e^{6}} - \frac {3 b^{4} d}{e^{4}} + \frac {24 b^{3} c d^{2}}{e^{5}} - \frac {60 b^{2} c^{2} d^{3}}{e^{6}} + \frac {60 b c^{3} d^{4}}{e^{7}} - \frac {21 c^{4} d^{5}}{e^{8}}\right ) + \frac {- a^{4} e^{8} - 4 a^{3} b d e^{7} + 12 a^{3} c d^{2} e^{6} + 18 a^{2} b^{2} d^{2} e^{6} - 60 a^{2} b c d^{3} e^{5} + 42 a^{2} c^{2} d^{4} e^{4} - 20 a b^{3} d^{3} e^{5} + 84 a b^{2} c d^{4} e^{4} - 108 a b c^{2} d^{5} e^{3} + 44 a c^{3} d^{6} e^{2} + 7 b^{4} d^{4} e^{4} - 36 b^{3} c d^{5} e^{3} + 66 b^{2} c^{2} d^{6} e^{2} - 52 b c^{3} d^{7} e + 15 c^{4} d^{8} + x \left (- 8 a^{3} b e^{8} + 16 a^{3} c d e^{7} + 24 a^{2} b^{2} d e^{7} - 72 a^{2} b c d^{2} e^{6} + 48 a^{2} c^{2} d^{3} e^{5} - 24 a b^{3} d^{2} e^{6} + 96 a b^{2} c d^{3} e^{5} - 120 a b c^{2} d^{4} e^{4} + 48 a c^{3} d^{5} e^{3} + 8 b^{4} d^{3} e^{5} - 40 b^{3} c d^{4} e^{4} + 72 b^{2} c^{2} d^{5} e^{3} - 56 b c^{3} d^{6} e^{2} + 16 c^{4} d^{7} e\right )}{2 d^{2} e^{9} + 4 d e^{10} x + 2 e^{11} x^{2}} + \frac {2 \left (a e^{2} - b d e + c d^{2}\right )^{2} \cdot \left (2 a c e^{2} + 3 b^{2} e^{2} - 14 b c d e + 14 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{9}} \]
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none
Time = 0.23 (sec) , antiderivative size = 819, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\frac {15 \, c^{4} d^{8} - 52 \, b c^{3} d^{7} e - 4 \, a^{3} b d e^{7} - a^{4} e^{8} + 22 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 36 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + 7 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 20 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 8 \, {\left (2 \, c^{4} d^{7} e - 7 \, b c^{3} d^{6} e^{2} - a^{3} b e^{8} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 5 \, {\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} - 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}{2 \, {\left (e^{11} x^{2} + 2 \, d e^{10} x + d^{2} e^{9}\right )}} + \frac {5 \, c^{4} e^{5} x^{6} - 6 \, {\left (3 \, c^{4} d e^{4} - 4 \, b c^{3} e^{5}\right )} x^{5} + 15 \, {\left (3 \, c^{4} d^{2} e^{3} - 6 \, b c^{3} d e^{4} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{5}\right )} x^{4} - 20 \, {\left (5 \, c^{4} d^{3} e^{2} - 12 \, b c^{3} d^{2} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{4} - 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{5}\right )} x^{3} + 15 \, {\left (15 \, c^{4} d^{4} e - 40 \, b c^{3} d^{3} e^{2} + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{3} - 12 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{5}\right )} x^{2} - 30 \, {\left (21 \, c^{4} d^{5} - 60 \, b c^{3} d^{4} e + 20 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 24 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} x}{30 \, e^{8}} + \frac {2 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{4} - 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{6}\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. 958 vs. \(2 (418) = 836\).
Time = 0.27 (sec) , antiderivative size = 958, normalized size of antiderivative = 2.23 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\frac {2 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 45 \, b^{2} c^{2} d^{4} e^{2} + 30 \, a c^{3} d^{4} e^{2} - 20 \, b^{3} c d^{3} e^{3} - 60 \, a b c^{2} d^{3} e^{3} + 3 \, b^{4} d^{2} e^{4} + 36 \, a b^{2} c d^{2} e^{4} + 18 \, a^{2} c^{2} d^{2} e^{4} - 6 \, a b^{3} d e^{5} - 18 \, a^{2} b c d e^{5} + 3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{9}} + \frac {15 \, c^{4} d^{8} - 52 \, b c^{3} d^{7} e + 66 \, b^{2} c^{2} d^{6} e^{2} + 44 \, a c^{3} d^{6} e^{2} - 36 \, b^{3} c d^{5} e^{3} - 108 \, a b c^{2} d^{5} e^{3} + 7 \, b^{4} d^{4} e^{4} + 84 \, a b^{2} c d^{4} e^{4} + 42 \, a^{2} c^{2} d^{4} e^{4} - 20 \, a b^{3} d^{3} e^{5} - 60 \, a^{2} b c d^{3} e^{5} + 18 \, a^{2} b^{2} d^{2} e^{6} + 12 \, a^{3} c d^{2} e^{6} - 4 \, a^{3} b d e^{7} - a^{4} e^{8} + 8 \, {\left (2 \, c^{4} d^{7} e - 7 \, b c^{3} d^{6} e^{2} + 9 \, b^{2} c^{2} d^{5} e^{3} + 6 \, a c^{3} d^{5} e^{3} - 5 \, b^{3} c d^{4} e^{4} - 15 \, 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} - 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} - a^{3} b e^{8}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{9}} + \frac {5 \, c^{4} e^{15} x^{6} - 18 \, c^{4} d e^{14} x^{5} + 24 \, b c^{3} e^{15} x^{5} + 45 \, c^{4} d^{2} e^{13} x^{4} - 90 \, b c^{3} d e^{14} x^{4} + 45 \, b^{2} c^{2} e^{15} x^{4} + 30 \, a c^{3} e^{15} x^{4} - 100 \, c^{4} d^{3} e^{12} x^{3} + 240 \, b c^{3} d^{2} e^{13} x^{3} - 180 \, b^{2} c^{2} d e^{14} x^{3} - 120 \, a c^{3} d e^{14} x^{3} + 40 \, b^{3} c e^{15} x^{3} + 120 \, a b c^{2} e^{15} x^{3} + 225 \, c^{4} d^{4} e^{11} x^{2} - 600 \, b c^{3} d^{3} e^{12} x^{2} + 540 \, b^{2} c^{2} d^{2} e^{13} x^{2} + 360 \, a c^{3} d^{2} e^{13} x^{2} - 180 \, b^{3} c d e^{14} x^{2} - 540 \, a b c^{2} d e^{14} x^{2} + 15 \, b^{4} e^{15} x^{2} + 180 \, a b^{2} c e^{15} x^{2} + 90 \, a^{2} c^{2} e^{15} x^{2} - 630 \, c^{4} d^{5} e^{10} x + 1800 \, b c^{3} d^{4} e^{11} x - 1800 \, b^{2} c^{2} d^{3} e^{12} x - 1200 \, a c^{3} d^{3} e^{12} x + 720 \, b^{3} c d^{2} e^{13} x + 2160 \, a b c^{2} d^{2} e^{13} x - 90 \, b^{4} d e^{14} x - 1080 \, a b^{2} c d e^{14} x - 540 \, a^{2} c^{2} d e^{14} x + 120 \, a b^{3} e^{15} x + 360 \, a^{2} b c e^{15} x}{30 \, e^{18}} \]
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Time = 9.93 (sec) , antiderivative size = 1444, normalized size of antiderivative = 3.36 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\text {Too large to display} \]
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