Integrand size = 22, antiderivative size = 286 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{3 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{13 e^7}-\frac {2 c^2 (2 c d-b e) (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \]
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Time = 0.09 (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} \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {6 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^7}-\frac {2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^7}+\frac {2 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac {6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac {2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \]
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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)^{3/2}}{e^6}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{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)^{7/2}}{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)^{9/2}}{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)^{11/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{13/2}}{e^6}+\frac {c^3 (d+e x)^{15/2}}{e^6}\right ) \, dx \\ & = \frac {2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac {2 \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)^{9/2}}{3 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)^{11/2}}{11 e^7}+\frac {6 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{13 e^7}-\frac {2 c^2 (2 c d-b e) (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.38 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{5/2} \left (c^3 \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )+221 e^3 \left (231 a^3 e^3+99 a^2 b e^2 (-2 d+5 e x)+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )+17 c e^2 \left (143 a^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 a b e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+3 b^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )-17 c^2 e \left (-3 a e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+b \left (256 d^5-640 d^4 e x+1120 d^3 e^2 x^2-1680 d^2 e^3 x^3+2310 d e^4 x^4-3003 e^5 x^5\right )\right )\right )}{255255 e^7} \]
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Time = 0.29 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.12
method | result | size |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (\left (\frac {5 c^{3} x^{6}}{17}+\left (b \,x^{5}+\frac {15}{13} a \,x^{4}\right ) c^{2}+\frac {5 \left (\frac {9}{13} b^{2} x^{2}+\frac {18}{11} a b x +a^{2}\right ) x^{2} c}{3}+a^{3}+\frac {5 a \,b^{2} x^{2}}{3}+\frac {15 a^{2} b x}{7}+\frac {5 b^{3} x^{3}}{11}\right ) e^{6}-\frac {6 \left (\frac {14 c^{3} x^{5}}{51}+\frac {140 x^{3} \left (\frac {11 b x}{12}+a \right ) c^{2}}{143}+\left (\frac {140}{143} b^{2} x^{3}+\frac {70}{33} a b \,x^{2}+\frac {10}{9} a^{2} x \right ) c +b \left (\frac {35}{99} b^{2} x^{2}+\frac {10}{9} a b x +a^{2}\right )\right ) d \,e^{5}}{7}+\frac {8 \left (\frac {105 c^{3} x^{4}}{221}+\frac {210 x^{2} \left (b x +a \right ) c^{2}}{143}+\left (\frac {210}{143} b^{2} x^{2}+\frac {30}{11} a b x +a^{2}\right ) c +b^{2} \left (\frac {5 b x}{11}+a \right )\right ) d^{2} e^{4}}{21}-\frac {32 \left (\frac {70 c^{3} x^{3}}{221}+\frac {10 x \left (\frac {7 b x}{6}+a \right ) c^{2}}{13}+b \left (\frac {10 b x}{13}+a \right ) c +\frac {b^{3}}{6}\right ) d^{3} e^{3}}{77}+\frac {128 \left (\frac {35 c^{2} x^{2}}{51}+\left (\frac {5 b x}{3}+a \right ) c +b^{2}\right ) c \,d^{4} e^{2}}{1001}-\frac {256 c^{2} \left (\frac {10 c x}{17}+b \right ) d^{5} e}{3003}+\frac {1024 c^{3} d^{6}}{51051}\right )}{5 e^{7}}\) | \(319\) |
derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {15}{2}}}{5}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\) | \(357\) |
default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {15}{2}}}{5}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\) | \(357\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (15015 c^{3} x^{6} e^{6}+51051 b \,c^{2} e^{6} x^{5}-12012 c^{3} d \,e^{5} x^{5}+58905 a \,c^{2} e^{6} x^{4}+58905 b^{2} c \,e^{6} x^{4}-39270 b \,c^{2} d \,e^{5} x^{4}+9240 c^{3} d^{2} e^{4} x^{4}+139230 a b c \,e^{6} x^{3}-42840 a \,c^{2} d \,e^{5} x^{3}+23205 b^{3} e^{6} x^{3}-42840 b^{2} c d \,e^{5} x^{3}+28560 b \,c^{2} d^{2} e^{4} x^{3}-6720 c^{3} d^{3} e^{3} x^{3}+85085 a^{2} c \,e^{6} x^{2}+85085 a \,b^{2} e^{6} x^{2}-92820 a b c d \,e^{5} x^{2}+28560 a \,c^{2} d^{2} e^{4} x^{2}-15470 b^{3} d \,e^{5} x^{2}+28560 b^{2} c \,d^{2} e^{4} x^{2}-19040 b \,c^{2} d^{3} e^{3} x^{2}+4480 c^{3} d^{4} e^{2} x^{2}+109395 a^{2} b \,e^{6} x -48620 a^{2} c d \,e^{5} x -48620 a \,b^{2} d \,e^{5} x +53040 a b c \,d^{2} e^{4} x -16320 a \,c^{2} d^{3} e^{3} x +8840 b^{3} d^{2} e^{4} x -16320 b^{2} c \,d^{3} e^{3} x +10880 b \,c^{2} d^{4} e^{2} x -2560 c^{3} d^{5} e x +51051 a^{3} e^{6}-43758 a^{2} b d \,e^{5}+19448 a^{2} c \,d^{2} e^{4}+19448 a \,b^{2} d^{2} e^{4}-21216 a b c \,d^{3} e^{3}+6528 a \,c^{2} d^{4} e^{2}-3536 b^{3} d^{3} e^{3}+6528 b^{2} c \,d^{4} e^{2}-4352 b \,c^{2} d^{5} e +1024 c^{3} d^{6}\right )}{255255 e^{7}}\) | \(495\) |
trager | \(\frac {2 \left (15015 c^{3} e^{8} x^{8}+51051 b \,c^{2} e^{8} x^{7}+18018 c^{3} d \,e^{7} x^{7}+58905 a \,c^{2} e^{8} x^{6}+58905 b^{2} c \,e^{8} x^{6}+62832 b \,c^{2} d \,e^{7} x^{6}+231 c^{3} d^{2} e^{6} x^{6}+139230 a b c \,e^{8} x^{5}+74970 a \,c^{2} d \,e^{7} x^{5}+23205 b^{3} e^{8} x^{5}+74970 b^{2} c d \,e^{7} x^{5}+1071 b \,c^{2} d^{2} e^{6} x^{5}-252 c^{3} d^{3} e^{5} x^{5}+85085 a^{2} c \,e^{8} x^{4}+85085 a \,b^{2} e^{8} x^{4}+185640 a b c d \,e^{7} x^{4}+1785 a \,c^{2} d^{2} e^{6} x^{4}+30940 b^{3} d \,e^{7} x^{4}+1785 b^{2} c \,d^{2} e^{6} x^{4}-1190 b \,c^{2} d^{3} e^{5} x^{4}+280 c^{3} d^{4} e^{4} x^{4}+109395 a^{2} b \,e^{8} x^{3}+121550 a^{2} c d \,e^{7} x^{3}+121550 a \,b^{2} d \,e^{7} x^{3}+6630 a b c \,d^{2} e^{6} x^{3}-2040 a \,c^{2} d^{3} e^{5} x^{3}+1105 b^{3} d^{2} e^{6} x^{3}-2040 b^{2} c \,d^{3} e^{5} x^{3}+1360 b \,c^{2} d^{4} e^{4} x^{3}-320 c^{3} d^{5} e^{3} x^{3}+51051 a^{3} e^{8} x^{2}+175032 a^{2} b d \,e^{7} x^{2}+7293 a^{2} c \,d^{2} e^{6} x^{2}+7293 a \,b^{2} d^{2} e^{6} x^{2}-7956 a b c \,d^{3} e^{5} x^{2}+2448 a \,c^{2} d^{4} e^{4} x^{2}-1326 b^{3} d^{3} e^{5} x^{2}+2448 b^{2} c \,d^{4} e^{4} x^{2}-1632 b \,c^{2} d^{5} e^{3} x^{2}+384 c^{3} d^{6} e^{2} x^{2}+102102 a^{3} d \,e^{7} x +21879 a^{2} b \,d^{2} e^{6} x -9724 a^{2} c \,d^{3} e^{5} x -9724 a \,b^{2} d^{3} e^{5} x +10608 a b c \,d^{4} e^{4} x -3264 a \,c^{2} d^{5} e^{3} x +1768 b^{3} d^{4} e^{4} x -3264 b^{2} c \,d^{5} e^{3} x +2176 b \,c^{2} d^{6} e^{2} x -512 c^{3} d^{7} e x +51051 a^{3} d^{2} e^{6}-43758 a^{2} b \,d^{3} e^{5}+19448 a^{2} c \,d^{4} e^{4}+19448 a \,b^{2} d^{4} e^{4}-21216 a b c \,d^{5} e^{3}+6528 a \,c^{2} d^{6} e^{2}-3536 b^{3} d^{5} e^{3}+6528 b^{2} c \,d^{6} e^{2}-4352 b \,c^{2} d^{7} e +1024 c^{3} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\) | \(783\) |
risch | \(\frac {2 \left (15015 c^{3} e^{8} x^{8}+51051 b \,c^{2} e^{8} x^{7}+18018 c^{3} d \,e^{7} x^{7}+58905 a \,c^{2} e^{8} x^{6}+58905 b^{2} c \,e^{8} x^{6}+62832 b \,c^{2} d \,e^{7} x^{6}+231 c^{3} d^{2} e^{6} x^{6}+139230 a b c \,e^{8} x^{5}+74970 a \,c^{2} d \,e^{7} x^{5}+23205 b^{3} e^{8} x^{5}+74970 b^{2} c d \,e^{7} x^{5}+1071 b \,c^{2} d^{2} e^{6} x^{5}-252 c^{3} d^{3} e^{5} x^{5}+85085 a^{2} c \,e^{8} x^{4}+85085 a \,b^{2} e^{8} x^{4}+185640 a b c d \,e^{7} x^{4}+1785 a \,c^{2} d^{2} e^{6} x^{4}+30940 b^{3} d \,e^{7} x^{4}+1785 b^{2} c \,d^{2} e^{6} x^{4}-1190 b \,c^{2} d^{3} e^{5} x^{4}+280 c^{3} d^{4} e^{4} x^{4}+109395 a^{2} b \,e^{8} x^{3}+121550 a^{2} c d \,e^{7} x^{3}+121550 a \,b^{2} d \,e^{7} x^{3}+6630 a b c \,d^{2} e^{6} x^{3}-2040 a \,c^{2} d^{3} e^{5} x^{3}+1105 b^{3} d^{2} e^{6} x^{3}-2040 b^{2} c \,d^{3} e^{5} x^{3}+1360 b \,c^{2} d^{4} e^{4} x^{3}-320 c^{3} d^{5} e^{3} x^{3}+51051 a^{3} e^{8} x^{2}+175032 a^{2} b d \,e^{7} x^{2}+7293 a^{2} c \,d^{2} e^{6} x^{2}+7293 a \,b^{2} d^{2} e^{6} x^{2}-7956 a b c \,d^{3} e^{5} x^{2}+2448 a \,c^{2} d^{4} e^{4} x^{2}-1326 b^{3} d^{3} e^{5} x^{2}+2448 b^{2} c \,d^{4} e^{4} x^{2}-1632 b \,c^{2} d^{5} e^{3} x^{2}+384 c^{3} d^{6} e^{2} x^{2}+102102 a^{3} d \,e^{7} x +21879 a^{2} b \,d^{2} e^{6} x -9724 a^{2} c \,d^{3} e^{5} x -9724 a \,b^{2} d^{3} e^{5} x +10608 a b c \,d^{4} e^{4} x -3264 a \,c^{2} d^{5} e^{3} x +1768 b^{3} d^{4} e^{4} x -3264 b^{2} c \,d^{5} e^{3} x +2176 b \,c^{2} d^{6} e^{2} x -512 c^{3} d^{7} e x +51051 a^{3} d^{2} e^{6}-43758 a^{2} b \,d^{3} e^{5}+19448 a^{2} c \,d^{4} e^{4}+19448 a \,b^{2} d^{4} e^{4}-21216 a b c \,d^{5} e^{3}+6528 a \,c^{2} d^{6} e^{2}-3536 b^{3} d^{5} e^{3}+6528 b^{2} c \,d^{6} e^{2}-4352 b \,c^{2} d^{7} e +1024 c^{3} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\) | \(783\) |
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Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (258) = 516\).
Time = 0.28 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.16 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, c^{3} e^{8} x^{8} + 1024 \, c^{3} d^{8} - 4352 \, b c^{2} d^{7} e - 43758 \, a^{2} b d^{3} e^{5} + 51051 \, a^{3} d^{2} e^{6} + 6528 \, {\left (b^{2} c + a c^{2}\right )} d^{6} e^{2} - 3536 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} e^{3} + 19448 \, {\left (a b^{2} + a^{2} c\right )} d^{4} e^{4} + 3003 \, {\left (6 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \, {\left (c^{3} d^{2} e^{6} + 272 \, b c^{2} d e^{7} + 255 \, {\left (b^{2} c + a c^{2}\right )} e^{8}\right )} x^{6} - 21 \, {\left (12 \, c^{3} d^{3} e^{5} - 51 \, b c^{2} d^{2} e^{6} - 3570 \, {\left (b^{2} c + a c^{2}\right )} d e^{7} - 1105 \, {\left (b^{3} + 6 \, a b c\right )} e^{8}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{4} e^{4} - 34 \, b c^{2} d^{3} e^{5} + 51 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{6} + 884 \, {\left (b^{3} + 6 \, a b c\right )} d e^{7} + 2431 \, {\left (a b^{2} + a^{2} c\right )} e^{8}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{5} e^{3} - 272 \, b c^{2} d^{4} e^{4} - 21879 \, a^{2} b e^{8} + 408 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{5} - 221 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{6} - 24310 \, {\left (a b^{2} + a^{2} c\right )} d e^{7}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{6} e^{2} - 544 \, b c^{2} d^{5} e^{3} + 58344 \, a^{2} b d e^{7} + 17017 \, a^{3} e^{8} + 816 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{4} - 442 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{5} + 2431 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{6}\right )} x^{2} - {\left (512 \, c^{3} d^{7} e - 2176 \, b c^{2} d^{6} e^{2} - 21879 \, a^{2} b d^{2} e^{6} - 102102 \, a^{3} d e^{7} + 3264 \, {\left (b^{2} c + a c^{2}\right )} d^{5} e^{3} - 1768 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} e^{4} + 9724 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e^{5}\right )} x\right )} \sqrt {e x + d}}{255255 \, e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (284) = 568\).
Time = 1.29 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.21 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{2} - 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 a^{2} b e^{5} - 6 a^{2} c d e^{4} - 6 a b^{2} d e^{4} + 18 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{2} + 3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{5 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.42 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{3} - 51051 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 58905 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 23205 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 109395 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 51051 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{255255 \, e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1970 vs. \(2 (258) = 516\).
Time = 0.28 (sec) , antiderivative size = 1970, normalized size of antiderivative = 6.89 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
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Time = 0.07 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.04 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{9\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{13\,e^7}+\frac {2\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{5\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{11\,e^7}+\frac {6\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{7\,e^7} \]
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