Integrand size = 28, antiderivative size = 427 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{5 e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{7/2}}{7 e^8}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{9/2}}{3 e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{11/2}}{11 e^8}-\frac {10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{13/2}}{13 e^8}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{15/2}}{5 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{17/2}}{17 e^8}+\frac {4 c^4 (d+e x)^{19/2}}{19 e^8} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {785} \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{11/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{11 e^8}+\frac {2 c^2 (d+e x)^{15/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac {10 c (d+e x)^{13/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{13 e^8}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac {14 c^3 (d+e x)^{17/2} (2 c d-b e)}{17 e^8}+\frac {4 c^4 (d+e x)^{19/2}}{19 e^8} \]
[In]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}{e^7}+\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 ) (d+e x)^{5/2}}{e^7}+\frac {3 (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 ) (d+e x)^{7/2}}{e^7}+\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)^{9/2}}{e^7}+\frac {5 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)^{11/2}}{e^7}+\frac {3 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)^{13/2}}{e^7}-\frac {7 c^3 (2 c d-b e) (d+e x)^{15/2}}{e^7}+\frac {2 c^4 (d+e x)^{17/2}}{e^7}\right ) \, dx \\ & = -\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{5 e^8}+\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 ) (d+e x)^{7/2}}{7 e^8}-\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 ) (d+e x)^{9/2}}{3 e^8}+\frac {2 \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)^{11/2}}{11 e^8}-\frac {10 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)^{13/2}}{13 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)^{15/2}}{5 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{17/2}}{17 e^8}+\frac {4 c^4 (d+e x)^{19/2}}{19 e^8} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.41 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{5/2} \left (-14 c^4 \left (2048 d^7-5120 d^6 e x+8960 d^5 e^2 x^2-13440 d^4 e^3 x^3+18480 d^3 e^4 x^4-24024 d^2 e^5 x^5+30030 d e^6 x^6-36465 e^7 x^7\right )+4199 b e^4 \left (231 a^3 e^3+99 a^2 b e^2 (-2 d+5 e x)+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )+323 c e^3 \left (858 a^3 e^3 (-2 d+5 e x)+429 a^2 b e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+156 a b^2 e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+5 b^3 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )-969 c^2 e^2 \left (26 a^2 e^2 \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )-5 a b e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+b^2 \left (256 d^5-640 d^4 e x+1120 d^3 e^2 x^2-1680 d^2 e^3 x^3+2310 d e^4 x^4-3003 e^5 x^5\right )\right )+19 c^3 e \left (34 a e \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )+7 b \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )\right )}{4849845 e^8} \]
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[Out]
Time = 0.78 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.17
method | result | size |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (\left (\frac {10 c^{4} x^{7}}{19}+\left (\frac {35}{17} b \,x^{6}+2 a \,x^{5}\right ) c^{3}+3 \left (b^{2} x^{5}+\frac {10}{11} a^{2} x^{3}+\frac {25}{13} a b \,x^{4}\right ) c^{2}+5 \left (\frac {5}{13} b^{3} x^{4}+\frac {2}{7} a^{3} x +\frac {12}{11} a \,b^{2} x^{3}+a^{2} b \,x^{2}\right ) c +b \left (\frac {5}{3} a \,b^{2} x^{2}+\frac {15}{7} b \,a^{2} x +\frac {5}{11} x^{3} b^{3}+a^{3}\right )\right ) e^{7}-\frac {4 d \left (\frac {245 c^{4} x^{6}}{323}+\frac {35 \left (\frac {91 b x}{85}+a \right ) x^{4} c^{3}}{13}+\frac {35 \left (\frac {33}{26} b^{2} x^{2}+\frac {30}{13} a b x +a^{2}\right ) x^{2} c^{2}}{11}+\left (\frac {350}{143} x^{3} b^{3}+\frac {70}{11} a \,b^{2} x^{2}+5 b \,a^{2} x +a^{3}\right ) c +\frac {35 b^{4} x^{2}}{66}+\frac {5 a \,b^{3} x}{3}+\frac {3 b^{2} a^{2}}{2}\right ) e^{6}}{7}+\frac {8 d^{2} \left (\frac {98 c^{4} x^{5}}{323}+\frac {140 \left (\frac {77 b x}{68}+a \right ) x^{3} c^{3}}{143}+\frac {10 \left (\frac {21}{13} b^{2} x^{2}+\frac {35}{13} a b x +a^{2}\right ) x \,c^{2}}{11}+b \left (\frac {350}{429} b^{2} x^{2}+\frac {20}{11} a b x +a^{2}\right ) c +\frac {b^{3} \left (\frac {5 b x}{11}+a \right )}{3}\right ) e^{5}}{7}-\frac {32 d^{3} \left (\frac {2695 c^{4} x^{4}}{4199}+\frac {70 \left (\frac {21 b x}{17}+a \right ) x^{2} c^{3}}{39}+\left (\frac {35}{13} b^{2} x^{2}+\frac {50}{13} a b x +a^{2}\right ) c^{2}+2 \left (\frac {25}{39} b^{3} x +b^{2} a \right ) c +\frac {b^{4}}{6}\right ) e^{4}}{77}+\frac {640 d^{4} \left (\frac {98 c^{3} x^{3}}{323}+\frac {2 \left (\frac {49 b x}{34}+a \right ) x \,c^{2}}{3}+b \left (b x +a \right ) c +\frac {b^{3}}{3}\right ) c \,e^{3}}{1001}-\frac {512 d^{5} \left (\frac {245 c^{2} x^{2}}{323}+\left (\frac {35 b x}{17}+a \right ) c +\frac {3 b^{2}}{2}\right ) c^{2} e^{2}}{3003}+\frac {1024 d^{6} \left (\frac {10 c x}{19}+b \right ) c^{3} e}{7293}-\frac {4096 c^{4} d^{7}}{138567}\right )}{5 e^{8}}\) | \(500\) |
derivativedivides | \(\frac {\frac {4 c^{4} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {14 \left (b e -2 c d \right ) c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} c^{2}+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )+2 c \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )+6 c \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{8}}\) | \(713\) |
default | \(\frac {\frac {4 c^{4} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {14 \left (b e -2 c d \right ) c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} c^{2}+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )+2 c \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )+6 c \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{8}}\) | \(713\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (510510 x^{7} c^{4} e^{7}+1996995 x^{6} b \,c^{3} e^{7}-420420 x^{6} c^{4} d \,e^{6}+1939938 x^{5} a \,c^{3} e^{7}+2909907 x^{5} b^{2} c^{2} e^{7}-1597596 x^{5} b \,c^{3} d \,e^{6}+336336 x^{5} c^{4} d^{2} e^{5}+5595975 x^{4} a b \,c^{2} e^{7}-1492260 x^{4} a \,c^{3} d \,e^{6}+1865325 x^{4} b^{3} c \,e^{7}-2238390 x^{4} b^{2} c^{2} d \,e^{6}+1228920 x^{4} b \,c^{3} d^{2} e^{5}-258720 x^{4} c^{4} d^{3} e^{4}+2645370 x^{3} a^{2} c^{2} e^{7}+5290740 x^{3} a \,b^{2} c \,e^{7}-4069800 x^{3} a b \,c^{2} d \,e^{6}+1085280 x^{3} a \,c^{3} d^{2} e^{5}+440895 x^{3} b^{4} e^{7}-1356600 x^{3} b^{3} c d \,e^{6}+1627920 x^{3} b^{2} c^{2} d^{2} e^{5}-893760 x^{3} b \,c^{3} d^{3} e^{4}+188160 x^{3} c^{4} d^{4} e^{3}+4849845 x^{2} a^{2} b c \,e^{7}-1763580 x^{2} a^{2} c^{2} d \,e^{6}+1616615 x^{2} a \,b^{3} e^{7}-3527160 x^{2} a \,b^{2} c d \,e^{6}+2713200 x^{2} a b \,c^{2} d^{2} e^{5}-723520 x^{2} a \,c^{3} d^{3} e^{4}-293930 x^{2} b^{4} d \,e^{6}+904400 x^{2} b^{3} c \,d^{2} e^{5}-1085280 x^{2} b^{2} c^{2} d^{3} e^{4}+595840 x^{2} b \,c^{3} d^{4} e^{3}-125440 x^{2} c^{4} d^{5} e^{2}+1385670 x \,a^{3} c \,e^{7}+2078505 x \,a^{2} b^{2} e^{7}-2771340 x \,a^{2} b c d \,e^{6}+1007760 x \,a^{2} c^{2} d^{2} e^{5}-923780 x a \,b^{3} d \,e^{6}+2015520 x a \,b^{2} c \,d^{2} e^{5}-1550400 x a b \,c^{2} d^{3} e^{4}+413440 x a \,c^{3} d^{4} e^{3}+167960 x \,b^{4} d^{2} e^{5}-516800 x \,b^{3} c \,d^{3} e^{4}+620160 x \,b^{2} c^{2} d^{4} e^{3}-340480 x b \,c^{3} d^{5} e^{2}+71680 x \,c^{4} d^{6} e +969969 a^{3} b \,e^{7}-554268 a^{3} c d \,e^{6}-831402 a^{2} b^{2} d \,e^{6}+1108536 a^{2} b c \,d^{2} e^{5}-403104 a^{2} c^{2} d^{3} e^{4}+369512 a \,b^{3} d^{2} e^{5}-806208 a \,b^{2} c \,d^{3} e^{4}+620160 a b \,c^{2} d^{4} e^{3}-165376 a \,c^{3} d^{5} e^{2}-67184 b^{4} d^{3} e^{4}+206720 b^{3} c \,d^{4} e^{3}-248064 b^{2} c^{2} d^{5} e^{2}+136192 b \,c^{3} d^{6} e -28672 c^{4} d^{7}\right )}{4849845 e^{8}}\) | \(795\) |
trager | \(\text {Expression too large to display}\) | \(1225\) |
risch | \(\text {Expression too large to display}\) | \(1225\) |
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 958 vs. \(2 (395) = 790\).
Time = 0.26 (sec) , antiderivative size = 958, normalized size of antiderivative = 2.24 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (510510 \, c^{4} e^{9} x^{9} - 28672 \, c^{4} d^{9} + 136192 \, b c^{3} d^{8} e + 969969 \, a^{3} b d^{2} e^{7} - 82688 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{7} e^{2} + 206720 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{6} e^{3} - 67184 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{5} e^{4} + 369512 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{4} e^{5} - 277134 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3} e^{6} + 15015 \, {\left (40 \, c^{4} d e^{8} + 133 \, b c^{3} e^{9}\right )} x^{8} + 3003 \, {\left (2 \, c^{4} d^{2} e^{7} + 798 \, b c^{3} d e^{8} + 323 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{9}\right )} x^{7} - 231 \, {\left (28 \, c^{4} d^{3} e^{6} - 133 \, b c^{3} d^{2} e^{7} - 5168 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{8} - 8075 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{9}\right )} x^{6} + 21 \, {\left (336 \, c^{4} d^{4} e^{5} - 1596 \, b c^{3} d^{3} e^{6} + 969 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{7} + 113050 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{8} + 20995 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{9}\right )} x^{5} - 35 \, {\left (224 \, c^{4} d^{5} e^{4} - 1064 \, b c^{3} d^{4} e^{5} + 646 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{6} - 1615 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{7} - 16796 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{8} - 46189 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{9}\right )} x^{4} + 5 \, {\left (1792 \, c^{4} d^{6} e^{3} - 8512 \, b c^{3} d^{5} e^{4} + 5168 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{5} - 12920 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{6} + 4199 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{7} + 461890 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{8} + 138567 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{9}\right )} x^{3} - 3 \, {\left (3584 \, c^{4} d^{7} e^{2} - 17024 \, b c^{3} d^{6} e^{3} - 323323 \, a^{3} b e^{9} + 10336 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{4} - 25840 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{5} + 8398 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{6} - 46189 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{7} - 369512 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{8}\right )} x^{2} + {\left (14336 \, c^{4} d^{8} e - 68096 \, b c^{3} d^{7} e^{2} + 1939938 \, a^{3} b d e^{8} + 41344 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{3} - 103360 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{4} + 33592 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{5} - 184756 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{6} + 138567 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{7}\right )} x\right )} \sqrt {e x + d}}{4849845 \, e^{8}} \]
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Time = 1.89 (sec) , antiderivative size = 862, normalized size of antiderivative = 2.02 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 c^{4} \left (d + e x\right )^{\frac {19}{2}}}{19 e^{7}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (7 b c^{3} e - 14 c^{4} d\right )}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (6 a c^{3} e^{2} + 9 b^{2} c^{2} e^{2} - 42 b c^{3} d e + 42 c^{4} d^{2}\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (15 a b c^{2} e^{3} - 30 a c^{3} d e^{2} + 5 b^{3} c e^{3} - 45 b^{2} c^{2} d e^{2} + 105 b c^{3} d^{2} e - 70 c^{4} d^{3}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \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 )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (9 a^{2} b c e^{5} - 18 a^{2} c^{2} d e^{4} + 3 a b^{3} e^{5} - 36 a b^{2} c d e^{4} + 90 a b c^{2} d^{2} e^{3} - 60 a c^{3} d^{3} e^{2} - 3 b^{4} d e^{4} + 30 b^{3} c d^{2} e^{3} - 90 b^{2} c^{2} d^{3} e^{2} + 105 b c^{3} d^{4} e - 42 c^{4} d^{5}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 a^{3} c e^{6} + 3 a^{2} b^{2} e^{6} - 18 a^{2} b c d e^{5} + 18 a^{2} c^{2} d^{2} e^{4} - 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 a c^{3} d^{4} e^{2} + 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 c^{4} d^{6}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{3} b e^{7} - 2 a^{3} c d e^{6} - 3 a^{2} b^{2} d e^{6} + 9 a^{2} b c d^{2} e^{5} - 6 a^{2} c^{2} d^{3} e^{4} + 3 a b^{3} d^{2} e^{5} - 12 a b^{2} c d^{3} e^{4} + 15 a b c^{2} d^{4} e^{3} - 6 a c^{3} d^{5} e^{2} - b^{4} d^{3} e^{4} + 5 b^{3} c d^{4} e^{3} - 9 b^{2} c^{2} d^{5} e^{2} + 7 b c^{3} d^{6} e - 2 c^{4} d^{7}\right )}{5 e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {d^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{4}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.51 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (510510 \, {\left (e x + d\right )}^{\frac {19}{2}} c^{4} - 1996995 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 969969 \, {\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 )} {\left (e x + d\right )}^{\frac {15}{2}} - 1865325 \, {\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 )} {\left (e x + d\right )}^{\frac {13}{2}} + 440895 \, {\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 )} {\left (e x + d\right )}^{\frac {11}{2}} - 1616615 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 692835 \, {\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 )} {\left (e x + d\right )}^{\frac {7}{2}} - 969969 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{4849845 \, e^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2941 vs. \(2 (395) = 790\).
Time = 0.31 (sec) , antiderivative size = 2941, normalized size of antiderivative = 6.89 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
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Time = 0.13 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.04 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{15/2}\,\left (18\,b^2\,c^2\,e^2-84\,b\,c^3\,d\,e+84\,c^4\,d^2+12\,a\,c^3\,e^2\right )}{15\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{19/2}}{19\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (12\,a^2\,c^2\,e^4+24\,a\,b^2\,c\,e^4-120\,a\,b\,c^2\,d\,e^3+120\,a\,c^3\,d^2\,e^2+2\,b^4\,e^4-40\,b^3\,c\,d\,e^3+180\,b^2\,c^2\,d^2\,e^2-280\,b\,c^3\,d^3\,e+140\,c^4\,d^4\right )}{11\,e^8}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{5\,e^8}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (3\,a^2\,c\,e^4+a\,b^2\,e^4-10\,a\,b\,c\,d\,e^3+10\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+8\,b^2\,c\,d^2\,e^2-14\,b\,c^2\,d^3\,e+7\,c^3\,d^4\right )}{3\,e^8}+\frac {2\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{7\,e^8}+\frac {10\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{13\,e^8} \]
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