Integrand size = 22, antiderivative size = 286 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{3 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{13 e^7}-\frac {2 c^2 (2 c d-b e) (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \]
2/5*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(5/2)/e^7-6/7*(-b*e+2*c*d)*(a*e^2-b*d*e+ c*d^2)^2*(e*x+d)^(7/2)/e^7+2/3*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e* (-a*e+5*b*d))*(e*x+d)^(9/2)/e^7-2/11*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c* e*(-3*a*e+5*b*d))*(e*x+d)^(11/2)/e^7+6/13*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5 *b*d))*(e*x+d)^(13/2)/e^7-2/5*c^2*(-b*e+2*c*d)*(e*x+d)^(15/2)/e^7+2/17*c^3 *(e*x+d)^(17/2)/e^7
Time = 0.34 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.38 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{5/2} \left (c^3 \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )+221 e^3 \left (231 a^3 e^3+99 a^2 b e^2 (-2 d+5 e x)+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )+17 c e^2 \left (143 a^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 a b e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+3 b^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )-17 c^2 e \left (-3 a e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+b \left (256 d^5-640 d^4 e x+1120 d^3 e^2 x^2-1680 d^2 e^3 x^3+2310 d e^4 x^4-3003 e^5 x^5\right )\right )\right )}{255255 e^7} \]
(2*(d + e*x)^(5/2)*(c^3*(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) + 221*e ^3*(231*a^3*e^3 + 99*a^2*b*e^2*(-2*d + 5*e*x) + 11*a*b^2*e*(8*d^2 - 20*d*e *x + 35*e^2*x^2) + b^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) ) + 17*c*e^2*(143*a^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 78*a*b*e*(-16* d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 3*b^2*(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)) - 17*c^2*e*(-3*a*e* (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) + b*(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))))/(255255*e^7)
Time = 0.43 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1140, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^{9/2} (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6}+\frac {3 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {3 (d+e x)^{5/2} (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6}+\frac {(d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^3}{e^6}-\frac {3 c^2 (d+e x)^{13/2} (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^{15/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^7}-\frac {2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^7}+\frac {2 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac {6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac {2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7}\) |
(2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(5/2))/(5*e^7) - (6*(2*c*d - b*e)*( c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2))/(7*e^7) + (2*(c*d^2 - b*d*e + a* e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(9/2))/(3*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^ (11/2))/(11*e^7) + (6*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x )^(13/2))/(13*e^7) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15/2))/(5*e^7) + (2*c ^3*(d + e*x)^(17/2))/(17*e^7)
3.23.83.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
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 | \(\frac {2 \left (15015 c^{3} e^{8} x^{8}+51051 b \,c^{2} e^{8} x^{7}+18018 c^{3} d \,e^{7} x^{7}+58905 a \,c^{2} e^{8} x^{6}+58905 b^{2} c \,e^{8} x^{6}+62832 b \,c^{2} d \,e^{7} x^{6}+231 c^{3} d^{2} e^{6} x^{6}+139230 a b c \,e^{8} x^{5}+74970 a \,c^{2} d \,e^{7} x^{5}+23205 b^{3} e^{8} x^{5}+74970 b^{2} c d \,e^{7} x^{5}+1071 b \,c^{2} d^{2} e^{6} x^{5}-252 c^{3} d^{3} e^{5} x^{5}+85085 a^{2} c \,e^{8} x^{4}+85085 a \,b^{2} e^{8} x^{4}+185640 a b c d \,e^{7} x^{4}+1785 a \,c^{2} d^{2} e^{6} x^{4}+30940 b^{3} d \,e^{7} x^{4}+1785 b^{2} c \,d^{2} e^{6} x^{4}-1190 b \,c^{2} d^{3} e^{5} x^{4}+280 c^{3} d^{4} e^{4} x^{4}+109395 a^{2} b \,e^{8} x^{3}+121550 a^{2} c d \,e^{7} x^{3}+121550 a \,b^{2} d \,e^{7} x^{3}+6630 a b c \,d^{2} e^{6} x^{3}-2040 a \,c^{2} d^{3} e^{5} x^{3}+1105 b^{3} d^{2} e^{6} x^{3}-2040 b^{2} c \,d^{3} e^{5} x^{3}+1360 b \,c^{2} d^{4} e^{4} x^{3}-320 c^{3} d^{5} e^{3} x^{3}+51051 a^{3} e^{8} x^{2}+175032 a^{2} b d \,e^{7} x^{2}+7293 a^{2} c \,d^{2} e^{6} x^{2}+7293 a \,b^{2} d^{2} e^{6} x^{2}-7956 a b c \,d^{3} e^{5} x^{2}+2448 a \,c^{2} d^{4} e^{4} x^{2}-1326 b^{3} d^{3} e^{5} x^{2}+2448 b^{2} c \,d^{4} e^{4} x^{2}-1632 b \,c^{2} d^{5} e^{3} x^{2}+384 c^{3} d^{6} e^{2} x^{2}+102102 a^{3} d \,e^{7} x +21879 a^{2} b \,d^{2} e^{6} x -9724 a^{2} c \,d^{3} e^{5} x -9724 a \,b^{2} d^{3} e^{5} x +10608 a b c \,d^{4} e^{4} x -3264 a \,c^{2} d^{5} e^{3} x +1768 b^{3} d^{4} e^{4} x -3264 b^{2} c \,d^{5} e^{3} x +2176 b \,c^{2} d^{6} e^{2} x -512 c^{3} d^{7} e x +51051 a^{3} d^{2} e^{6}-43758 a^{2} b \,d^{3} e^{5}+19448 a^{2} c \,d^{4} e^{4}+19448 a \,b^{2} d^{4} e^{4}-21216 a b c \,d^{5} e^{3}+6528 a \,c^{2} d^{6} e^{2}-3536 b^{3} d^{5} e^{3}+6528 b^{2} c \,d^{6} e^{2}-4352 b \,c^{2} d^{7} e +1024 c^{3} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\) | \(783\) |
2/5*(e*x+d)^(5/2)*((5/17*c^3*x^6+(b*x^5+15/13*a*x^4)*c^2+5/3*(9/13*b^2*x^2 +18/11*a*b*x+a^2)*x^2*c+a^3+5/3*a*b^2*x^2+15/7*a^2*b*x+5/11*b^3*x^3)*e^6-6 /7*(14/51*c^3*x^5+140/143*x^3*(11/12*b*x+a)*c^2+(140/143*b^2*x^3+70/33*a*b *x^2+10/9*a^2*x)*c+b*(35/99*b^2*x^2+10/9*a*b*x+a^2))*d*e^5+8/21*(105/221*c ^3*x^4+210/143*x^2*(b*x+a)*c^2+(210/143*b^2*x^2+30/11*a*b*x+a^2)*c+b^2*(5/ 11*b*x+a))*d^2*e^4-32/77*(70/221*c^3*x^3+10/13*x*(7/6*b*x+a)*c^2+b*(10/13* b*x+a)*c+1/6*b^3)*d^3*e^3+128/1001*(35/51*c^2*x^2+(5/3*b*x+a)*c+b^2)*c*d^4 *e^2-256/3003*c^2*(10/17*c*x+b)*d^5*e+1024/51051*c^3*d^6)/e^7
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 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, c^{3} e^{8} x^{8} + 1024 \, c^{3} d^{8} - 4352 \, b c^{2} d^{7} e - 43758 \, a^{2} b d^{3} e^{5} + 51051 \, a^{3} d^{2} e^{6} + 6528 \, {\left (b^{2} c + a c^{2}\right )} d^{6} e^{2} - 3536 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} e^{3} + 19448 \, {\left (a b^{2} + a^{2} c\right )} d^{4} e^{4} + 3003 \, {\left (6 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \, {\left (c^{3} d^{2} e^{6} + 272 \, b c^{2} d e^{7} + 255 \, {\left (b^{2} c + a c^{2}\right )} e^{8}\right )} x^{6} - 21 \, {\left (12 \, c^{3} d^{3} e^{5} - 51 \, b c^{2} d^{2} e^{6} - 3570 \, {\left (b^{2} c + a c^{2}\right )} d e^{7} - 1105 \, {\left (b^{3} + 6 \, a b c\right )} e^{8}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{4} e^{4} - 34 \, b c^{2} d^{3} e^{5} + 51 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{6} + 884 \, {\left (b^{3} + 6 \, a b c\right )} d e^{7} + 2431 \, {\left (a b^{2} + a^{2} c\right )} e^{8}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{5} e^{3} - 272 \, b c^{2} d^{4} e^{4} - 21879 \, a^{2} b e^{8} + 408 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{5} - 221 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{6} - 24310 \, {\left (a b^{2} + a^{2} c\right )} d e^{7}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{6} e^{2} - 544 \, b c^{2} d^{5} e^{3} + 58344 \, a^{2} b d e^{7} + 17017 \, a^{3} e^{8} + 816 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{4} - 442 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{5} + 2431 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{6}\right )} x^{2} - {\left (512 \, c^{3} d^{7} e - 2176 \, b c^{2} d^{6} e^{2} - 21879 \, a^{2} b d^{2} e^{6} - 102102 \, a^{3} d e^{7} + 3264 \, {\left (b^{2} c + a c^{2}\right )} d^{5} e^{3} - 1768 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} e^{4} + 9724 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e^{5}\right )} x\right )} \sqrt {e x + d}}{255255 \, e^{7}} \]
2/255255*(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*(b^2*c + a*c^2)*d^6*e^2 - 3536*(b^3 + 6*a*b*c)*d^5*e^3 + 19448*(a*b^2 + a^2*c)*d^4*e^4 + 3003*(6*c^3*d*e^7 + 17 *b*c^2*e^8)*x^7 + 231*(c^3*d^2*e^6 + 272*b*c^2*d*e^7 + 255*(b^2*c + a*c^2) *e^8)*x^6 - 21*(12*c^3*d^3*e^5 - 51*b*c^2*d^2*e^6 - 3570*(b^2*c + a*c^2)*d *e^7 - 1105*(b^3 + 6*a*b*c)*e^8)*x^5 + 35*(8*c^3*d^4*e^4 - 34*b*c^2*d^3*e^ 5 + 51*(b^2*c + a*c^2)*d^2*e^6 + 884*(b^3 + 6*a*b*c)*d*e^7 + 2431*(a*b^2 + a^2*c)*e^8)*x^4 - 5*(64*c^3*d^5*e^3 - 272*b*c^2*d^4*e^4 - 21879*a^2*b*e^8 + 408*(b^2*c + a*c^2)*d^3*e^5 - 221*(b^3 + 6*a*b*c)*d^2*e^6 - 24310*(a*b^ 2 + a^2*c)*d*e^7)*x^3 + 3*(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*(b^2*c + a*c^2)*d^4*e^4 - 442*(b^3 + 6*a*b* c)*d^3*e^5 + 2431*(a*b^2 + a^2*c)*d^2*e^6)*x^2 - (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*(b^2*c + a*c^2) *d^5*e^3 - 1768*(b^3 + 6*a*b*c)*d^4*e^4 + 9724*(a*b^2 + a^2*c)*d^3*e^5)*x) *sqrt(e*x + d)/e^7
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 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{2} - 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 a^{2} b e^{5} - 6 a^{2} c d e^{4} - 6 a b^{2} d e^{4} + 18 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{2} + 3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{5 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(c**3*(d + e*x)**(17/2)/(17*e**6) + (d + e*x)**(15/2)*(3*b*c* *2*e - 6*c**3*d)/(15*e**6) + (d + e*x)**(13/2)*(3*a*c**2*e**2 + 3*b**2*c*e **2 - 15*b*c**2*d*e + 15*c**3*d**2)/(13*e**6) + (d + e*x)**(11/2)*(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)/(11*e**6) + (d + e*x)**(9/2)*(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)/(9*e**6) + (d + e*x)**(7/2)*(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)/(7*e**6) + (d + e*x)**(5/2)*(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)/(5*e**6))/e, Ne(e, 0)), (d**(3/2)*(a**3*x + 3*a**2*b*x**2/2 + b*c**2*x **6/2 + c**3*x**7/7 + x**5*(3*a*c**2 + 3*b**2*c)/5 + x**4*(6*a*b*c + b**3) /4 + x**3*(3*a**2*c + 3*a*b**2)/3), True))
Time = 0.19 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.42 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{3} - 51051 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 58905 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 23205 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 109395 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 51051 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{255255 \, e^{7}} \]
2/255255*(15015*(e*x + d)^(17/2)*c^3 - 51051*(2*c^3*d - b*c^2*e)*(e*x + d) ^(15/2) + 58905*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^ (13/2) - 23205*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - ( b^3 + 6*a*b*c)*e^3)*(e*x + d)^(11/2) + 85085*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)* (e*x + d)^(9/2) - 109395*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e* x + d)^(7/2) + 51051*(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^ 2*e^4)*(e*x + d)^(5/2))/e^7
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 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
2/765765*(765765*sqrt(e*x + d)*a^3*d^2 + 510510*((e*x + d)^(3/2) - 3*sqrt( e*x + d)*d)*a^3*d + 765765*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*b*d^2 /e + 51051*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^ 2)*a^3 + 153153*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*b^2*d^2/e^2 + 153153*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*c*d^2/e^2 + 306306*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b*d/e + 21879*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b ^3*d^2/e^3 + 131274*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a*b*c*d^2/e^3 + 131274*(5*(e*x + d)^( 7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^ 3)*a*b^2*d/e^2 + 131274*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e* x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^2*c*d/e^2 + 65637*(5*(e*x + d)^ (7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d ^3)*a^2*b/e + 7293*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^2*c*d^ 2/e^4 + 7293*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^( 5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*c^2*d^2/e^4 + 4862*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d ^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^3*d/e^3 + 29172...
Time = 0.07 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.04 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{9\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{13\,e^7}+\frac {2\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{5\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{11\,e^7}+\frac {6\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{7\,e^7} \]
((d + e*x)^(9/2)*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^4 - 6*b^3*d*e^3 + 3 6*a*c^2*d^2*e^2 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e - 36*a*b*c*d*e^3))/(9* e^7) + (2*c^3*(d + e*x)^(17/2))/(17*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e* x)^(15/2))/(15*e^7) + ((d + e*x)^(13/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2* c*e^2 - 30*b*c^2*d*e))/(13*e^7) + (2*(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d* e)^3)/(5*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(11/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(11*e^7) + (6*(b*e - 2*c*d)*(d + e*x)^(7/2)*(a*e^ 2 + c*d^2 - b*d*e)^2)/(7*e^7)