Integrand size = 21, antiderivative size = 248 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^3 \, dx=\frac {2 d^3 (c d-b e)^3 (d+e x)^{5/2}}{5 e^7}-\frac {6 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{7/2}}{7 e^7}+\frac {2 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac {2 c^2 (2 c d-b e) (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \]
2/5*d^3*(-b*e+c*d)^3*(e*x+d)^(5/2)/e^7-6/7*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*( e*x+d)^(7/2)/e^7+2/3*d*(-b*e+c*d)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(9 /2)/e^7-2/11*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(11/2)/e ^7+6/13*c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(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 = 0.19 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.93 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{5/2} \left (221 b^3 e^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+51 b^2 c e^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+17 b c^2 e \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )+c^3 \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )}{255255 e^7} \]
(2*(d + e*x)^(5/2)*(221*b^3*e^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105 *e^3*x^3) + 51*b^2*c*e^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*b*c^2*e*(-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) + 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)))/(255255*e^7)
Time = 0.38 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \left (b x+c x^2\right )^3 (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c (d+e x)^{11/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^{9/2} (2 c d-b e) \left (-b^2 e^2+10 b c d e-10 c^2 d^2\right )}{e^6}+\frac {3 d (d+e x)^{7/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}-\frac {3 c^2 (d+e x)^{13/2} (2 c d-b e)}{e^6}+\frac {d^3 (d+e x)^{3/2} (c d-b e)^3}{e^6}-\frac {3 d^2 (d+e x)^{5/2} (c d-b e)^2 (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^{15/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 c (d+e x)^{13/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac {2 (d+e x)^{11/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{11 e^7}+\frac {2 d (d+e x)^{9/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac {2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac {2 d^3 (d+e x)^{5/2} (c d-b e)^3}{5 e^7}-\frac {6 d^2 (d+e x)^{7/2} (c d-b e)^2 (2 c d-b e)}{7 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7}\) |
(2*d^3*(c*d - b*e)^3*(d + e*x)^(5/2))/(5*e^7) - (6*d^2*(c*d - b*e)^2*(2*c* d - b*e)*(d + e*x)^(7/2))/(7*e^7) + (2*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d* e + b^2*e^2)*(d + e*x)^(9/2))/(3*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10* b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d* e + b^2*e^2)*(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)
3.4.55.3.1 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 = 2.15 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.84
| method | result | size |
| pseudoelliptic | \(-\frac {32 \left (-\frac {105 x^{3} \left (\frac {11}{17} c^{3} x^{3}+\frac {11}{5} b \,c^{2} x^{2}+\frac {33}{13} b^{2} c x +b^{3}\right ) e^{6}}{16}+\frac {35 x^{2} \left (\frac {66}{85} c^{3} x^{3}+\frac {33}{13} b \,c^{2} x^{2}+\frac {36}{13} b^{2} c x +b^{3}\right ) d \,e^{5}}{8}-\frac {5 x \,d^{2} \left (\frac {231}{221} c^{3} x^{3}+\frac {42}{13} b \,c^{2} x^{2}+\frac {42}{13} b^{2} c x +b^{3}\right ) e^{4}}{2}+d^{3} \left (\frac {420}{221} c^{3} x^{3}+\frac {70}{13} b \,c^{2} x^{2}+\frac {60}{13} b^{2} c x +b^{3}\right ) e^{3}-\frac {24 \left (\frac {35}{51} c^{2} x^{2}+\frac {5}{3} b c x +b^{2}\right ) c \,d^{4} e^{2}}{13}+\frac {16 c^{2} \left (\frac {10 c x}{17}+b \right ) d^{5} e}{13}-\frac {64 c^{3} d^{6}}{221}\right ) \left (e x +d \right )^{\frac {5}{2}}}{1155 e^{7}}\) | \(208\) |
| derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (-3 c^{3} d +3 \left (b e -c d \right ) c^{2}\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (3 c^{3} d^{2}-9 d \left (b e -c d \right ) c^{2}+3 \left (b e -c d \right )^{2} c \right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-c^{3} d^{3}+9 d^{2} \left (b e -c d \right ) c^{2}-9 d \left (b e -c d \right )^{2} c +\left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-3 d^{3} \left (b e -c d \right ) c^{2}+9 d^{2} \left (b e -c d \right )^{2} c -3 d \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-3 d^{3} \left (b e -c d \right )^{2} c +3 d^{2} \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {2 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\) | \(269\) |
| default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}-\frac {2 \left (3 c^{3} d -3 \left (b e -c d \right ) c^{2}\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {2 \left (-3 c^{3} d^{2}+9 d \left (b e -c d \right ) c^{2}-3 \left (b e -c d \right )^{2} c \right ) \left (e x +d \right )^{\frac {13}{2}}}{13}-\frac {2 \left (c^{3} d^{3}-9 d^{2} \left (b e -c d \right ) c^{2}+9 d \left (b e -c d \right )^{2} c -\left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {2 \left (3 d^{3} \left (b e -c d \right ) c^{2}-9 d^{2} \left (b e -c d \right )^{2} c +3 d \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {2 \left (3 d^{3} \left (b e -c d \right )^{2} c -3 d^{2} \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {2 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\) | \(270\) |
| gosper | \(-\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (-15015 x^{6} c^{3} e^{6}-51051 x^{5} b \,c^{2} e^{6}+12012 x^{5} c^{3} d \,e^{5}-58905 x^{4} b^{2} c \,e^{6}+39270 x^{4} b \,c^{2} d \,e^{5}-9240 x^{4} c^{3} d^{2} e^{4}-23205 x^{3} b^{3} e^{6}+42840 x^{3} b^{2} c d \,e^{5}-28560 x^{3} b \,c^{2} d^{2} e^{4}+6720 x^{3} c^{3} d^{3} e^{3}+15470 x^{2} b^{3} d \,e^{5}-28560 x^{2} b^{2} c \,d^{2} e^{4}+19040 x^{2} b \,c^{2} d^{3} e^{3}-4480 x^{2} c^{3} d^{4} e^{2}-8840 x \,b^{3} d^{2} e^{4}+16320 x \,b^{2} c \,d^{3} e^{3}-10880 x b \,c^{2} d^{4} e^{2}+2560 x \,c^{3} d^{5} e +3536 b^{3} d^{3} e^{3}-6528 b^{2} c \,d^{4} e^{2}+4352 b \,c^{2} d^{5} e -1024 c^{3} d^{6}\right )}{255255 e^{7}}\) | \(286\) |
| trager | \(-\frac {2 \left (-15015 e^{8} c^{3} x^{8}-51051 b \,c^{2} e^{8} x^{7}-18018 c^{3} d \,e^{7} x^{7}-58905 b^{2} c \,e^{8} x^{6}-62832 b \,c^{2} d \,e^{7} x^{6}-231 c^{3} d^{2} e^{6} x^{6}-23205 b^{3} e^{8} x^{5}-74970 b^{2} c d \,e^{7} x^{5}-1071 b \,c^{2} d^{2} e^{6} x^{5}+252 c^{3} d^{3} e^{5} x^{5}-30940 b^{3} d \,e^{7} x^{4}-1785 b^{2} c \,d^{2} e^{6} x^{4}+1190 b \,c^{2} d^{3} e^{5} x^{4}-280 c^{3} d^{4} e^{4} x^{4}-1105 b^{3} d^{2} e^{6} x^{3}+2040 b^{2} c \,d^{3} e^{5} x^{3}-1360 b \,c^{2} d^{4} e^{4} x^{3}+320 c^{3} d^{5} e^{3} x^{3}+1326 b^{3} d^{3} e^{5} x^{2}-2448 b^{2} c \,d^{4} e^{4} x^{2}+1632 b \,c^{2} d^{5} e^{3} x^{2}-384 c^{3} d^{6} e^{2} x^{2}-1768 b^{3} d^{4} e^{4} x +3264 b^{2} c \,d^{5} e^{3} x -2176 b \,c^{2} d^{6} e^{2} x +512 c^{3} d^{7} e x +3536 b^{3} d^{5} e^{3}-6528 b^{2} c \,d^{6} e^{2}+4352 b \,c^{2} d^{7} e -1024 c^{3} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\) | \(402\) |
| risch | \(-\frac {2 \left (-15015 e^{8} c^{3} x^{8}-51051 b \,c^{2} e^{8} x^{7}-18018 c^{3} d \,e^{7} x^{7}-58905 b^{2} c \,e^{8} x^{6}-62832 b \,c^{2} d \,e^{7} x^{6}-231 c^{3} d^{2} e^{6} x^{6}-23205 b^{3} e^{8} x^{5}-74970 b^{2} c d \,e^{7} x^{5}-1071 b \,c^{2} d^{2} e^{6} x^{5}+252 c^{3} d^{3} e^{5} x^{5}-30940 b^{3} d \,e^{7} x^{4}-1785 b^{2} c \,d^{2} e^{6} x^{4}+1190 b \,c^{2} d^{3} e^{5} x^{4}-280 c^{3} d^{4} e^{4} x^{4}-1105 b^{3} d^{2} e^{6} x^{3}+2040 b^{2} c \,d^{3} e^{5} x^{3}-1360 b \,c^{2} d^{4} e^{4} x^{3}+320 c^{3} d^{5} e^{3} x^{3}+1326 b^{3} d^{3} e^{5} x^{2}-2448 b^{2} c \,d^{4} e^{4} x^{2}+1632 b \,c^{2} d^{5} e^{3} x^{2}-384 c^{3} d^{6} e^{2} x^{2}-1768 b^{3} d^{4} e^{4} x +3264 b^{2} c \,d^{5} e^{3} x -2176 b \,c^{2} d^{6} e^{2} x +512 c^{3} d^{7} e x +3536 b^{3} d^{5} e^{3}-6528 b^{2} c \,d^{6} e^{2}+4352 b \,c^{2} d^{7} e -1024 c^{3} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\) | \(402\) |
-32/1155*(-105/16*x^3*(11/17*c^3*x^3+11/5*b*c^2*x^2+33/13*b^2*c*x+b^3)*e^6 +35/8*x^2*(66/85*c^3*x^3+33/13*b*c^2*x^2+36/13*b^2*c*x+b^3)*d*e^5-5/2*x*d^ 2*(231/221*c^3*x^3+42/13*b*c^2*x^2+42/13*b^2*c*x+b^3)*e^4+d^3*(420/221*c^3 *x^3+70/13*b*c^2*x^2+60/13*b^2*c*x+b^3)*e^3-24/13*(35/51*c^2*x^2+5/3*b*c*x +b^2)*c*d^4*e^2+16/13*c^2*(10/17*c*x+b)*d^5*e-64/221*c^3*d^6)*(e*x+d)^(5/2 )/e^7
Time = 0.26 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.50 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, c^{3} e^{8} x^{8} + 1024 \, c^{3} d^{8} - 4352 \, b c^{2} d^{7} e + 6528 \, b^{2} c d^{6} e^{2} - 3536 \, b^{3} d^{5} e^{3} + 3003 \, {\left (6 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \, {\left (c^{3} d^{2} e^{6} + 272 \, b c^{2} d e^{7} + 255 \, b^{2} c e^{8}\right )} x^{6} - 21 \, {\left (12 \, c^{3} d^{3} e^{5} - 51 \, b c^{2} d^{2} e^{6} - 3570 \, b^{2} c d e^{7} - 1105 \, b^{3} e^{8}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{4} e^{4} - 34 \, b c^{2} d^{3} e^{5} + 51 \, b^{2} c d^{2} e^{6} + 884 \, b^{3} d e^{7}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{5} e^{3} - 272 \, b c^{2} d^{4} e^{4} + 408 \, b^{2} c d^{3} e^{5} - 221 \, b^{3} d^{2} e^{6}\right )} x^{3} + 6 \, {\left (64 \, c^{3} d^{6} e^{2} - 272 \, b c^{2} d^{5} e^{3} + 408 \, b^{2} c d^{4} e^{4} - 221 \, b^{3} d^{3} e^{5}\right )} x^{2} - 8 \, {\left (64 \, c^{3} d^{7} e - 272 \, b c^{2} d^{6} e^{2} + 408 \, b^{2} c d^{5} e^{3} - 221 \, b^{3} d^{4} e^{4}\right )} x\right )} \sqrt {e x + d}}{255255 \, e^{7}} \]
2/255255*(15015*c^3*e^8*x^8 + 1024*c^3*d^8 - 4352*b*c^2*d^7*e + 6528*b^2*c *d^6*e^2 - 3536*b^3*d^5*e^3 + 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*e^8)*x^6 - 21*(12*c^3*d^3*e^5 - 51*b*c^2*d^2*e^6 - 3570*b^2*c*d*e^7 - 1105*b^3*e^8)*x^5 + 35*(8*c^3*d^4*e ^4 - 34*b*c^2*d^3*e^5 + 51*b^2*c*d^2*e^6 + 884*b^3*d*e^7)*x^4 - 5*(64*c^3* d^5*e^3 - 272*b*c^2*d^4*e^4 + 408*b^2*c*d^3*e^5 - 221*b^3*d^2*e^6)*x^3 + 6 *(64*c^3*d^6*e^2 - 272*b*c^2*d^5*e^3 + 408*b^2*c*d^4*e^4 - 221*b^3*d^3*e^5 )*x^2 - 8*(64*c^3*d^7*e - 272*b*c^2*d^6*e^2 + 408*b^2*c*d^5*e^3 - 221*b^3* d^4*e^4)*x)*sqrt(e*x + d)/e^7
Time = 1.01 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.50 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (- 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{5 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (\frac {b^{3} x^{4}}{4} + \frac {3 b^{2} c x^{5}}{5} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
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*b**2*c*e**2 - 15*b*c**2* d*e + 15*c**3*d**2)/(13*e**6) + (d + e*x)**(11/2)*(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* 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*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)*(-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)*(b**3*x**4/4 + 3*b**2*c*x**5/5 + b*c**2*x**6/2 + c**3*x**7/7), True) )
Time = 0.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.09 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{3} - 51051 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 58905 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 23205 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 109395 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 51051 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{255255 \, e^{7}} \]
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*e^2)*(e*x + d)^(13/2) - 2 3205*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e*x + d)^(1 1/2) + 85085*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e *x + d)^(9/2) - 109395*(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3* d^2*e^3)*(e*x + d)^(7/2) + 51051*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^ 2 - b^3*d^3*e^3)*(e*x + d)^(5/2))/e^7
Leaf count of result is larger than twice the leaf count of optimal. 1012 vs. \(2 (220) = 440\).
Time = 0.30 (sec) , antiderivative size = 1012, normalized size of antiderivative = 4.08 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
2/765765*(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 + 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 + 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 + 3315*(63*(e*x + d)^(11/2) - 385*(e*x + d)^( 9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b*c^2*d^2/e^5 + 6630*(63*(e*x + d)^( 11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^( 5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b^2*c*d/e^4 + 1105*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d ^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b^3/e^3 + 255*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 50 05*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d ^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c^3*d^2/e^6 + 1530 *(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^ 2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^( 3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*b*c^2*d/e^5 + 765*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/ 2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqr...
Time = 9.39 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.96 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,b^3\,e^3-24\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e-40\,c^3\,d^3\right )}{11\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{13\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{9\,e^7}-\frac {2\,d^3\,{\left (b\,e-c\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}+\frac {6\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7} \]
((d + e*x)^(11/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^ 2))/(11*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*b^2*c*e^2 - 30*b*c^2*d*e))/(13*e^7) + ((d + e*x)^(9/2)*(30*c^3*d^4 - 6*b^3*d*e^3 + 3 6*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e))/(9*e^7) - (2*d^3*(b*e - c*d)^3*(d + e*x )^(5/2))/(5*e^7) + (6*d^2*(b*e - c*d)^2*(b*e - 2*c*d)*(d + e*x)^(7/2))/(7* e^7)