Integrand size = 22, antiderivative size = 286 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{7/2}}{7 e^7}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{9/2}}{3 e^7}+\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{11 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{13/2}}{13 e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{15/2}}{5 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{17/2}}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7} \]
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Time = 0.13 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {712} \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 c (d+e x)^{15/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{13/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{13 e^7}+\frac {6 (d+e x)^{11/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^7}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^3}{7 e^7}-\frac {6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7} \]
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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 )^3 (d+e x)^{5/2}}{e^6}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{e^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{9/2}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{11/2}}{e^6}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{15/2}}{e^6}+\frac {c^3 (d+e x)^{17/2}}{e^6}\right ) \, dx \\ & = \frac {2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{7/2}}{7 e^7}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{9/2}}{3 e^7}+\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{11 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{13/2}}{13 e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{15/2}}{5 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{17/2}}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.39 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{7/2} \left (5 c^3 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )+1615 e^3 \left (429 a^3 e^3+143 a^2 b e^2 (-2 d+7 e x)+13 a b^2 e \left (8 d^2-28 d e x+63 e^2 x^2\right )+b^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )\right )+323 c e^2 \left (65 a^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+30 a b e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )-19 c^2 e \left (-17 a e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+5 b \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )\right )}{4849845 e^7} \]
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Time = 0.31 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.11
method | result | size |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (\left (\frac {7 c^{3} x^{6}}{19}+\frac {7 x^{4} \left (\frac {15 b x}{17}+a \right ) c^{2}}{5}+\left (\frac {7}{5} b^{2} x^{4}+\frac {21}{11} a^{2} x^{2}+\frac {42}{13} a b \,x^{3}\right ) c +a^{3}+\frac {21 a \,b^{2} x^{2}}{11}+\frac {7 a^{2} b x}{3}+\frac {7 b^{3} x^{3}}{13}\right ) e^{6}-\frac {2 \left (\frac {126 c^{3} x^{5}}{323}+\frac {84 \left (\frac {65 b x}{68}+a \right ) x^{3} c^{2}}{65}+\frac {14 \left (\frac {66}{65} b^{2} x^{2}+\frac {27}{13} a b x +a^{2}\right ) x c}{11}+b \left (\frac {63}{143} b^{2} x^{2}+\frac {14}{11} a b x +a^{2}\right )\right ) d \,e^{5}}{3}+\frac {8 \left (\frac {231 c^{3} x^{4}}{323}+\frac {126 x^{2} \left (\frac {55 b x}{51}+a \right ) c^{2}}{65}+\left (\frac {126}{65} b^{2} x^{2}+\frac {42}{13} a b x +a^{2}\right ) c +b^{2} \left (\frac {7 b x}{13}+a \right )\right ) d^{2} e^{4}}{33}-\frac {32 d^{3} \left (\frac {154 c^{3} x^{3}}{323}+\frac {14 \left (\frac {45 b x}{34}+a \right ) x \,c^{2}}{15}+b \left (\frac {14 b x}{15}+a \right ) c +\frac {b^{3}}{6}\right ) e^{3}}{143}+\frac {128 \left (\frac {315 c^{2} x^{2}}{323}+\left (\frac {35 b x}{17}+a \right ) c +b^{2}\right ) c \,d^{4} e^{2}}{2145}-\frac {256 c^{2} \left (\frac {14 c x}{19}+b \right ) d^{5} e}{7293}+\frac {1024 c^{3} d^{6}}{138567}\right )}{7 e^{7}}\) | \(318\) |
derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {6 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{7}}\) | \(357\) |
default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {6 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{7}}\) | \(357\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (255255 c^{3} x^{6} e^{6}+855855 b \,c^{2} e^{6} x^{5}-180180 c^{3} d \,e^{5} x^{5}+969969 a \,c^{2} e^{6} x^{4}+969969 b^{2} c \,e^{6} x^{4}-570570 b \,c^{2} d \,e^{5} x^{4}+120120 c^{3} d^{2} e^{4} x^{4}+2238390 a b c \,e^{6} x^{3}-596904 a \,c^{2} d \,e^{5} x^{3}+373065 b^{3} e^{6} x^{3}-596904 b^{2} c d \,e^{5} x^{3}+351120 b \,c^{2} d^{2} e^{4} x^{3}-73920 c^{3} d^{3} e^{3} x^{3}+1322685 a^{2} c \,e^{6} x^{2}+1322685 a \,b^{2} e^{6} x^{2}-1220940 a b c d \,e^{5} x^{2}+325584 a \,c^{2} d^{2} e^{4} x^{2}-203490 b^{3} d \,e^{5} x^{2}+325584 b^{2} c \,d^{2} e^{4} x^{2}-191520 b \,c^{2} d^{3} e^{3} x^{2}+40320 c^{3} d^{4} e^{2} x^{2}+1616615 a^{2} b \,e^{6} x -587860 a^{2} c d \,e^{5} x -587860 a \,b^{2} d \,e^{5} x +542640 a b c \,d^{2} e^{4} x -144704 a \,c^{2} d^{3} e^{3} x +90440 b^{3} d^{2} e^{4} x -144704 b^{2} c \,d^{3} e^{3} x +85120 b \,c^{2} d^{4} e^{2} x -17920 c^{3} d^{5} e x +692835 a^{3} e^{6}-461890 a^{2} b d \,e^{5}+167960 a^{2} c \,d^{2} e^{4}+167960 a \,b^{2} d^{2} e^{4}-155040 a b c \,d^{3} e^{3}+41344 a \,c^{2} d^{4} e^{2}-25840 b^{3} d^{3} e^{3}+41344 b^{2} c \,d^{4} e^{2}-24320 b \,c^{2} d^{5} e +5120 c^{3} d^{6}\right )}{4849845 e^{7}}\) | \(495\) |
trager | \(\frac {2 \left (255255 c^{3} e^{9} x^{9}+855855 b \,c^{2} e^{9} x^{8}+585585 c^{3} d \,e^{8} x^{8}+969969 a \,c^{2} e^{9} x^{7}+969969 b^{2} c \,e^{9} x^{7}+1996995 b \,c^{2} d \,e^{8} x^{7}+345345 c^{3} d^{2} e^{7} x^{7}+2238390 a b c \,e^{9} x^{6}+2313003 a \,c^{2} d \,e^{8} x^{6}+373065 b^{3} e^{9} x^{6}+2313003 b^{2} c d \,e^{8} x^{6}+1206975 b \,c^{2} d^{2} e^{7} x^{6}+1155 c^{3} d^{3} e^{6} x^{6}+1322685 a^{2} c \,e^{9} x^{5}+1322685 a \,b^{2} e^{9} x^{5}+5494230 a b c d \,e^{8} x^{5}+1444779 a \,c^{2} d^{2} e^{7} x^{5}+915705 b^{3} d \,e^{8} x^{5}+1444779 b^{2} c \,d^{2} e^{7} x^{5}+5985 b \,c^{2} d^{3} e^{6} x^{5}-1260 c^{3} d^{4} e^{5} x^{5}+1616615 a^{2} b \,e^{9} x^{4}+3380195 a^{2} c d \,e^{8} x^{4}+3380195 a \,b^{2} d \,e^{8} x^{4}+3594990 a b c \,d^{2} e^{7} x^{4}+11305 a \,c^{2} d^{3} e^{6} x^{4}+599165 b^{3} d^{2} e^{7} x^{4}+11305 b^{2} c \,d^{3} e^{6} x^{4}-6650 b \,c^{2} d^{4} e^{5} x^{4}+1400 c^{3} d^{5} e^{4} x^{4}+692835 a^{3} e^{9} x^{3}+4387955 a^{2} b d \,e^{8} x^{3}+2372435 a^{2} c \,d^{2} e^{7} x^{3}+2372435 a \,b^{2} d^{2} e^{7} x^{3}+48450 a b c \,d^{3} e^{6} x^{3}-12920 a \,c^{2} d^{4} e^{5} x^{3}+8075 b^{3} d^{3} e^{6} x^{3}-12920 b^{2} c \,d^{4} e^{5} x^{3}+7600 b \,c^{2} d^{5} e^{4} x^{3}-1600 c^{3} d^{6} e^{3} x^{3}+2078505 a^{3} d \,e^{8} x^{2}+3464175 a^{2} b \,d^{2} e^{7} x^{2}+62985 a^{2} c \,d^{3} e^{6} x^{2}+62985 a \,b^{2} d^{3} e^{6} x^{2}-58140 a b c \,d^{4} e^{5} x^{2}+15504 a \,c^{2} d^{5} e^{4} x^{2}-9690 b^{3} d^{4} e^{5} x^{2}+15504 b^{2} c \,d^{5} e^{4} x^{2}-9120 b \,c^{2} d^{6} e^{3} x^{2}+1920 c^{3} d^{7} e^{2} x^{2}+2078505 a^{3} d^{2} e^{7} x +230945 a^{2} b \,d^{3} e^{6} x -83980 a^{2} c \,d^{4} e^{5} x -83980 a \,b^{2} d^{4} e^{5} x +77520 a b c \,d^{5} e^{4} x -20672 a \,c^{2} d^{6} e^{3} x +12920 b^{3} d^{5} e^{4} x -20672 b^{2} c \,d^{6} e^{3} x +12160 b \,c^{2} d^{7} e^{2} x -2560 c^{3} d^{8} e x +692835 a^{3} d^{3} e^{6}-461890 a^{2} b \,d^{4} e^{5}+167960 a^{2} c \,d^{5} e^{4}+167960 a \,b^{2} d^{5} e^{4}-155040 a b c \,d^{6} e^{3}+41344 a \,c^{2} d^{7} e^{2}-25840 b^{3} d^{6} e^{3}+41344 b^{2} c \,d^{7} e^{2}-24320 b \,c^{2} d^{8} e +5120 c^{3} d^{9}\right ) \sqrt {e x +d}}{4849845 e^{7}}\) | \(929\) |
risch | \(\frac {2 \left (255255 c^{3} e^{9} x^{9}+855855 b \,c^{2} e^{9} x^{8}+585585 c^{3} d \,e^{8} x^{8}+969969 a \,c^{2} e^{9} x^{7}+969969 b^{2} c \,e^{9} x^{7}+1996995 b \,c^{2} d \,e^{8} x^{7}+345345 c^{3} d^{2} e^{7} x^{7}+2238390 a b c \,e^{9} x^{6}+2313003 a \,c^{2} d \,e^{8} x^{6}+373065 b^{3} e^{9} x^{6}+2313003 b^{2} c d \,e^{8} x^{6}+1206975 b \,c^{2} d^{2} e^{7} x^{6}+1155 c^{3} d^{3} e^{6} x^{6}+1322685 a^{2} c \,e^{9} x^{5}+1322685 a \,b^{2} e^{9} x^{5}+5494230 a b c d \,e^{8} x^{5}+1444779 a \,c^{2} d^{2} e^{7} x^{5}+915705 b^{3} d \,e^{8} x^{5}+1444779 b^{2} c \,d^{2} e^{7} x^{5}+5985 b \,c^{2} d^{3} e^{6} x^{5}-1260 c^{3} d^{4} e^{5} x^{5}+1616615 a^{2} b \,e^{9} x^{4}+3380195 a^{2} c d \,e^{8} x^{4}+3380195 a \,b^{2} d \,e^{8} x^{4}+3594990 a b c \,d^{2} e^{7} x^{4}+11305 a \,c^{2} d^{3} e^{6} x^{4}+599165 b^{3} d^{2} e^{7} x^{4}+11305 b^{2} c \,d^{3} e^{6} x^{4}-6650 b \,c^{2} d^{4} e^{5} x^{4}+1400 c^{3} d^{5} e^{4} x^{4}+692835 a^{3} e^{9} x^{3}+4387955 a^{2} b d \,e^{8} x^{3}+2372435 a^{2} c \,d^{2} e^{7} x^{3}+2372435 a \,b^{2} d^{2} e^{7} x^{3}+48450 a b c \,d^{3} e^{6} x^{3}-12920 a \,c^{2} d^{4} e^{5} x^{3}+8075 b^{3} d^{3} e^{6} x^{3}-12920 b^{2} c \,d^{4} e^{5} x^{3}+7600 b \,c^{2} d^{5} e^{4} x^{3}-1600 c^{3} d^{6} e^{3} x^{3}+2078505 a^{3} d \,e^{8} x^{2}+3464175 a^{2} b \,d^{2} e^{7} x^{2}+62985 a^{2} c \,d^{3} e^{6} x^{2}+62985 a \,b^{2} d^{3} e^{6} x^{2}-58140 a b c \,d^{4} e^{5} x^{2}+15504 a \,c^{2} d^{5} e^{4} x^{2}-9690 b^{3} d^{4} e^{5} x^{2}+15504 b^{2} c \,d^{5} e^{4} x^{2}-9120 b \,c^{2} d^{6} e^{3} x^{2}+1920 c^{3} d^{7} e^{2} x^{2}+2078505 a^{3} d^{2} e^{7} x +230945 a^{2} b \,d^{3} e^{6} x -83980 a^{2} c \,d^{4} e^{5} x -83980 a \,b^{2} d^{4} e^{5} x +77520 a b c \,d^{5} e^{4} x -20672 a \,c^{2} d^{6} e^{3} x +12920 b^{3} d^{5} e^{4} x -20672 b^{2} c \,d^{6} e^{3} x +12160 b \,c^{2} d^{7} e^{2} x -2560 c^{3} d^{8} e x +692835 a^{3} d^{3} e^{6}-461890 a^{2} b \,d^{4} e^{5}+167960 a^{2} c \,d^{5} e^{4}+167960 a \,b^{2} d^{5} e^{4}-155040 a b c \,d^{6} e^{3}+41344 a \,c^{2} d^{7} e^{2}-25840 b^{3} d^{6} e^{3}+41344 b^{2} c \,d^{7} e^{2}-24320 b \,c^{2} d^{8} e +5120 c^{3} d^{9}\right ) \sqrt {e x +d}}{4849845 e^{7}}\) | \(929\) |
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Leaf count of result is larger than twice the leaf count of optimal. 727 vs. \(2 (258) = 516\).
Time = 0.27 (sec) , antiderivative size = 727, normalized size of antiderivative = 2.54 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (255255 \, c^{3} e^{9} x^{9} + 5120 \, c^{3} d^{9} - 24320 \, b c^{2} d^{8} e - 461890 \, a^{2} b d^{4} e^{5} + 692835 \, a^{3} d^{3} e^{6} + 41344 \, {\left (b^{2} c + a c^{2}\right )} d^{7} e^{2} - 25840 \, {\left (b^{3} + 6 \, a b c\right )} d^{6} e^{3} + 167960 \, {\left (a b^{2} + a^{2} c\right )} d^{5} e^{4} + 45045 \, {\left (13 \, c^{3} d e^{8} + 19 \, b c^{2} e^{9}\right )} x^{8} + 3003 \, {\left (115 \, c^{3} d^{2} e^{7} + 665 \, b c^{2} d e^{8} + 323 \, {\left (b^{2} c + a c^{2}\right )} e^{9}\right )} x^{7} + 231 \, {\left (5 \, c^{3} d^{3} e^{6} + 5225 \, b c^{2} d^{2} e^{7} + 10013 \, {\left (b^{2} c + a c^{2}\right )} d e^{8} + 1615 \, {\left (b^{3} + 6 \, a b c\right )} e^{9}\right )} x^{6} - 63 \, {\left (20 \, c^{3} d^{4} e^{5} - 95 \, b c^{2} d^{3} e^{6} - 22933 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{7} - 14535 \, {\left (b^{3} + 6 \, a b c\right )} d e^{8} - 20995 \, {\left (a b^{2} + a^{2} c\right )} e^{9}\right )} x^{5} + 35 \, {\left (40 \, c^{3} d^{5} e^{4} - 190 \, b c^{2} d^{4} e^{5} + 46189 \, a^{2} b e^{9} + 323 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{6} + 17119 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{7} + 96577 \, {\left (a b^{2} + a^{2} c\right )} d e^{8}\right )} x^{4} - 5 \, {\left (320 \, c^{3} d^{6} e^{3} - 1520 \, b c^{2} d^{5} e^{4} - 877591 \, a^{2} b d e^{8} - 138567 \, a^{3} e^{9} + 2584 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{5} - 1615 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{6} - 474487 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{7}\right )} x^{3} + 3 \, {\left (640 \, c^{3} d^{7} e^{2} - 3040 \, b c^{2} d^{6} e^{3} + 1154725 \, a^{2} b d^{2} e^{7} + 692835 \, a^{3} d e^{8} + 5168 \, {\left (b^{2} c + a c^{2}\right )} d^{5} e^{4} - 3230 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} e^{5} + 20995 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e^{6}\right )} x^{2} - {\left (2560 \, c^{3} d^{8} e - 12160 \, b c^{2} d^{7} e^{2} - 230945 \, a^{2} b d^{3} e^{6} - 2078505 \, a^{3} d^{2} e^{7} + 20672 \, {\left (b^{2} c + a c^{2}\right )} d^{6} e^{3} - 12920 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} e^{4} + 83980 \, {\left (a b^{2} + a^{2} c\right )} d^{4} e^{5}\right )} x\right )} \sqrt {e x + d}}{4849845 \, e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (284) = 568\).
Time = 1.44 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.21 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {19}{2}}}{19 e^{6}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{2} - 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 a^{2} b e^{5} - 6 a^{2} c d e^{4} - 6 a b^{2} d e^{4} + 18 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{2} + 3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{7 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.42 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (255255 \, {\left (e x + d\right )}^{\frac {19}{2}} c^{3} - 855855 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 969969 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 373065 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 1322685 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 1616615 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 692835 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{4849845 \, e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2883 vs. \(2 (258) = 516\).
Time = 0.30 (sec) , antiderivative size = 2883, normalized size of antiderivative = 10.08 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
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Time = 9.81 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.04 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{11/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{11\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {{\left (d+e\,x\right )}^{15/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{15\,e^7}+\frac {2\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{7\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{13\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{3\,e^7} \]
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