Integrand size = 21, antiderivative size = 248 \[ \int (d+e x)^{5/2} \left (b x+c x^2\right )^3 \, dx=\frac {2 d^3 (c d-b e)^3 (d+e x)^{7/2}}{7 e^7}-\frac {2 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{9/2}}{3 e^7}+\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^7}+\frac {2 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{15/2}}{5 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{17/2}}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7} \]
2/7*d^3*(-b*e+c*d)^3*(e*x+d)^(7/2)/e^7-2/3*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*( e*x+d)^(9/2)/e^7+6/11*d*(-b*e+c*d)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^( 11/2)/e^7-2/13*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(13/2) /e^7+2/5*c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(15/2)/e^7-6/17*c^2*(-b*e +2*c*d)*(e*x+d)^(17/2)/e^7+2/19*c^3*(e*x+d)^(19/2)/e^7
Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.94 \[ \int (d+e x)^{5/2} \left (b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{7/2} \left (1615 b^3 e^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+323 b^2 c e^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+95 b c^2 e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+5 c^3 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )}{4849845 e^7} \]
(2*(d + e*x)^(7/2)*(1615*b^3*e^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 2 31*e^3*x^3) + 323*b^2*c*e^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 18 48*d*e^3*x^3 + 3003*e^4*x^4) + 95*b*c^2*e*(-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) + 5*c^3*(10 24*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)))/(4849845*e^7)
Time = 0.37 (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.
\(\displaystyle \int \left (b x+c x^2\right )^3 (d+e x)^{5/2} \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (\frac {3 c (d+e x)^{13/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^{11/2} (2 c d-b e) \left (-b^2 e^2+10 b c d e-10 c^2 d^2\right )}{e^6}+\frac {3 d (d+e x)^{9/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}-\frac {3 c^2 (d+e x)^{15/2} (2 c d-b e)}{e^6}+\frac {d^3 (d+e x)^{5/2} (c d-b e)^3}{e^6}-\frac {3 d^2 (d+e x)^{7/2} (c d-b e)^2 (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^{17/2}}{e^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 c (d+e x)^{15/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{13/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{13 e^7}+\frac {6 d (d+e x)^{11/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac {6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac {2 d^3 (d+e x)^{7/2} (c d-b e)^3}{7 e^7}-\frac {2 d^2 (d+e x)^{9/2} (c d-b e)^2 (2 c d-b e)}{3 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7}\) |
(2*d^3*(c*d - b*e)^3*(d + e*x)^(7/2))/(7*e^7) - (2*d^2*(c*d - b*e)^2*(2*c* d - b*e)*(d + e*x)^(9/2))/(3*e^7) + (6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d* e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 1 0*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13*e^7) + (2*c*(5*c^2*d^2 - 5*b*c* d*e + b^2*e^2)*(d + e*x)^(15/2))/(5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^ (17/2))/(17*e^7) + (2*c^3*(d + e*x)^(19/2))/(19*e^7)
3.4.54.3.1 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 = 2.00 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(-\frac {32 \left (e x +d \right )^{\frac {7}{2}} \left (-\frac {231 x^{3} \left (\frac {13}{19} c^{3} x^{3}+\frac {39}{17} b \,c^{2} x^{2}+\frac {13}{5} b^{2} c x +b^{3}\right ) e^{6}}{16}+\frac {63 x^{2} \left (\frac {286}{323} c^{3} x^{3}+\frac {143}{51} b \,c^{2} x^{2}+\frac {44}{15} b^{2} c x +b^{3}\right ) d \,e^{5}}{8}-\frac {7 x \,d^{2} \left (\frac {429}{323} c^{3} x^{3}+\frac {66}{17} b \,c^{2} x^{2}+\frac {18}{5} b^{2} c x +b^{3}\right ) e^{4}}{2}+d^{3} \left (\frac {924}{323} c^{3} x^{3}+\frac {126}{17} b \,c^{2} x^{2}+\frac {28}{5} b^{2} c x +b^{3}\right ) e^{3}-\frac {8 c \left (\frac {315}{323} c^{2} x^{2}+\frac {35}{17} b c x +b^{2}\right ) d^{4} e^{2}}{5}+\frac {16 \left (\frac {14 c x}{19}+b \right ) c^{2} d^{5} e}{17}-\frac {64 c^{3} d^{6}}{323}\right )}{3003 e^{7}}\) | \(208\) |
derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (-3 c^{3} d +3 \left (b e -c d \right ) c^{2}\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (3 c^{3} d^{2}-9 d \left (b e -c d \right ) c^{2}+3 \left (b e -c d \right )^{2} c \right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (-c^{3} d^{3}+9 d^{2} \left (b e -c d \right ) c^{2}-9 d \left (b e -c d \right )^{2} c +\left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-3 d^{3} \left (b e -c d \right ) c^{2}+9 d^{2} \left (b e -c d \right )^{2} c -3 d \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-3 d^{3} \left (b e -c d \right )^{2} c +3 d^{2} \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {2 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{7}}\) | \(269\) |
default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {19}{2}}}{19}-\frac {2 \left (3 c^{3} d -3 \left (b e -c d \right ) c^{2}\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}-\frac {2 \left (-3 c^{3} d^{2}+9 d \left (b e -c d \right ) c^{2}-3 \left (b e -c d \right )^{2} c \right ) \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {2 \left (c^{3} d^{3}-9 d^{2} \left (b e -c d \right ) c^{2}+9 d \left (b e -c d \right )^{2} c -\left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}-\frac {2 \left (3 d^{3} \left (b e -c d \right ) c^{2}-9 d^{2} \left (b e -c d \right )^{2} c +3 d \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {2 \left (3 d^{3} \left (b e -c d \right )^{2} c -3 d^{2} \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {2 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{7}}\) | \(270\) |
gosper | \(-\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (-255255 x^{6} c^{3} e^{6}-855855 x^{5} b \,c^{2} e^{6}+180180 x^{5} c^{3} d \,e^{5}-969969 x^{4} b^{2} c \,e^{6}+570570 x^{4} b \,c^{2} d \,e^{5}-120120 x^{4} c^{3} d^{2} e^{4}-373065 x^{3} b^{3} e^{6}+596904 x^{3} b^{2} c d \,e^{5}-351120 x^{3} b \,c^{2} d^{2} e^{4}+73920 x^{3} c^{3} d^{3} e^{3}+203490 x^{2} b^{3} d \,e^{5}-325584 x^{2} b^{2} c \,d^{2} e^{4}+191520 x^{2} b \,c^{2} d^{3} e^{3}-40320 x^{2} c^{3} d^{4} e^{2}-90440 x \,b^{3} d^{2} e^{4}+144704 x \,b^{2} c \,d^{3} e^{3}-85120 x b \,c^{2} d^{4} e^{2}+17920 x \,c^{3} d^{5} e +25840 b^{3} d^{3} e^{3}-41344 b^{2} c \,d^{4} e^{2}+24320 b \,c^{2} d^{5} e -5120 c^{3} d^{6}\right )}{4849845 e^{7}}\) | \(286\) |
trager | \(-\frac {2 \left (-255255 e^{9} c^{3} x^{9}-855855 b \,c^{2} e^{9} x^{8}-585585 c^{3} d \,e^{8} x^{8}-969969 b^{2} c \,e^{9} x^{7}-1996995 b \,c^{2} d \,e^{8} x^{7}-345345 c^{3} d^{2} e^{7} x^{7}-373065 b^{3} e^{9} x^{6}-2313003 b^{2} c d \,e^{8} x^{6}-1206975 b \,c^{2} d^{2} e^{7} x^{6}-1155 c^{3} d^{3} e^{6} x^{6}-915705 b^{3} d \,e^{8} x^{5}-1444779 b^{2} c \,d^{2} e^{7} x^{5}-5985 b \,c^{2} d^{3} e^{6} x^{5}+1260 c^{3} d^{4} e^{5} x^{5}-599165 b^{3} d^{2} e^{7} x^{4}-11305 b^{2} c \,d^{3} e^{6} x^{4}+6650 b \,c^{2} d^{4} e^{5} x^{4}-1400 c^{3} d^{5} e^{4} x^{4}-8075 b^{3} d^{3} e^{6} x^{3}+12920 b^{2} c \,d^{4} e^{5} x^{3}-7600 b \,c^{2} d^{5} e^{4} x^{3}+1600 c^{3} d^{6} e^{3} x^{3}+9690 b^{3} d^{4} e^{5} x^{2}-15504 b^{2} c \,d^{5} e^{4} x^{2}+9120 b \,c^{2} d^{6} e^{3} x^{2}-1920 c^{3} d^{7} e^{2} x^{2}-12920 b^{3} d^{5} e^{4} x +20672 b^{2} c \,d^{6} e^{3} x -12160 b \,c^{2} d^{7} e^{2} x +2560 c^{3} d^{8} e x +25840 b^{3} d^{6} e^{3}-41344 b^{2} c \,d^{7} e^{2}+24320 b \,c^{2} d^{8} e -5120 c^{3} d^{9}\right ) \sqrt {e x +d}}{4849845 e^{7}}\) | \(460\) |
risch | \(-\frac {2 \left (-255255 e^{9} c^{3} x^{9}-855855 b \,c^{2} e^{9} x^{8}-585585 c^{3} d \,e^{8} x^{8}-969969 b^{2} c \,e^{9} x^{7}-1996995 b \,c^{2} d \,e^{8} x^{7}-345345 c^{3} d^{2} e^{7} x^{7}-373065 b^{3} e^{9} x^{6}-2313003 b^{2} c d \,e^{8} x^{6}-1206975 b \,c^{2} d^{2} e^{7} x^{6}-1155 c^{3} d^{3} e^{6} x^{6}-915705 b^{3} d \,e^{8} x^{5}-1444779 b^{2} c \,d^{2} e^{7} x^{5}-5985 b \,c^{2} d^{3} e^{6} x^{5}+1260 c^{3} d^{4} e^{5} x^{5}-599165 b^{3} d^{2} e^{7} x^{4}-11305 b^{2} c \,d^{3} e^{6} x^{4}+6650 b \,c^{2} d^{4} e^{5} x^{4}-1400 c^{3} d^{5} e^{4} x^{4}-8075 b^{3} d^{3} e^{6} x^{3}+12920 b^{2} c \,d^{4} e^{5} x^{3}-7600 b \,c^{2} d^{5} e^{4} x^{3}+1600 c^{3} d^{6} e^{3} x^{3}+9690 b^{3} d^{4} e^{5} x^{2}-15504 b^{2} c \,d^{5} e^{4} x^{2}+9120 b \,c^{2} d^{6} e^{3} x^{2}-1920 c^{3} d^{7} e^{2} x^{2}-12920 b^{3} d^{5} e^{4} x +20672 b^{2} c \,d^{6} e^{3} x -12160 b \,c^{2} d^{7} e^{2} x +2560 c^{3} d^{8} e x +25840 b^{3} d^{6} e^{3}-41344 b^{2} c \,d^{7} e^{2}+24320 b \,c^{2} d^{8} e -5120 c^{3} d^{9}\right ) \sqrt {e x +d}}{4849845 e^{7}}\) | \(460\) |
-32/3003*(e*x+d)^(7/2)*(-231/16*x^3*(13/19*c^3*x^3+39/17*b*c^2*x^2+13/5*b^ 2*c*x+b^3)*e^6+63/8*x^2*(286/323*c^3*x^3+143/51*b*c^2*x^2+44/15*b^2*c*x+b^ 3)*d*e^5-7/2*x*d^2*(429/323*c^3*x^3+66/17*b*c^2*x^2+18/5*b^2*c*x+b^3)*e^4+ d^3*(924/323*c^3*x^3+126/17*b*c^2*x^2+28/5*b^2*c*x+b^3)*e^3-8/5*c*(315/323 *c^2*x^2+35/17*b*c*x+b^2)*d^4*e^2+16/17*(14/19*c*x+b)*c^2*d^5*e-64/323*c^3 *d^6)/e^7
Time = 0.27 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.72 \[ \int (d+e x)^{5/2} \left (b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (255255 \, c^{3} e^{9} x^{9} + 5120 \, c^{3} d^{9} - 24320 \, b c^{2} d^{8} e + 41344 \, b^{2} c d^{7} e^{2} - 25840 \, b^{3} d^{6} e^{3} + 45045 \, {\left (13 \, c^{3} d e^{8} + 19 \, b c^{2} e^{9}\right )} x^{8} + 3003 \, {\left (115 \, c^{3} d^{2} e^{7} + 665 \, b c^{2} d e^{8} + 323 \, b^{2} c e^{9}\right )} x^{7} + 231 \, {\left (5 \, c^{3} d^{3} e^{6} + 5225 \, b c^{2} d^{2} e^{7} + 10013 \, b^{2} c d e^{8} + 1615 \, b^{3} e^{9}\right )} x^{6} - 63 \, {\left (20 \, c^{3} d^{4} e^{5} - 95 \, b c^{2} d^{3} e^{6} - 22933 \, b^{2} c d^{2} e^{7} - 14535 \, b^{3} d e^{8}\right )} x^{5} + 35 \, {\left (40 \, c^{3} d^{5} e^{4} - 190 \, b c^{2} d^{4} e^{5} + 323 \, b^{2} c d^{3} e^{6} + 17119 \, b^{3} d^{2} e^{7}\right )} x^{4} - 5 \, {\left (320 \, c^{3} d^{6} e^{3} - 1520 \, b c^{2} d^{5} e^{4} + 2584 \, b^{2} c d^{4} e^{5} - 1615 \, b^{3} d^{3} e^{6}\right )} x^{3} + 6 \, {\left (320 \, c^{3} d^{7} e^{2} - 1520 \, b c^{2} d^{6} e^{3} + 2584 \, b^{2} c d^{5} e^{4} - 1615 \, b^{3} d^{4} e^{5}\right )} x^{2} - 8 \, {\left (320 \, c^{3} d^{8} e - 1520 \, b c^{2} d^{7} e^{2} + 2584 \, b^{2} c d^{6} e^{3} - 1615 \, b^{3} d^{5} e^{4}\right )} x\right )} \sqrt {e x + d}}{4849845 \, e^{7}} \]
2/4849845*(255255*c^3*e^9*x^9 + 5120*c^3*d^9 - 24320*b*c^2*d^8*e + 41344*b ^2*c*d^7*e^2 - 25840*b^3*d^6*e^3 + 45045*(13*c^3*d*e^8 + 19*b*c^2*e^9)*x^8 + 3003*(115*c^3*d^2*e^7 + 665*b*c^2*d*e^8 + 323*b^2*c*e^9)*x^7 + 231*(5*c ^3*d^3*e^6 + 5225*b*c^2*d^2*e^7 + 10013*b^2*c*d*e^8 + 1615*b^3*e^9)*x^6 - 63*(20*c^3*d^4*e^5 - 95*b*c^2*d^3*e^6 - 22933*b^2*c*d^2*e^7 - 14535*b^3*d* e^8)*x^5 + 35*(40*c^3*d^5*e^4 - 190*b*c^2*d^4*e^5 + 323*b^2*c*d^3*e^6 + 17 119*b^3*d^2*e^7)*x^4 - 5*(320*c^3*d^6*e^3 - 1520*b*c^2*d^5*e^4 + 2584*b^2* c*d^4*e^5 - 1615*b^3*d^3*e^6)*x^3 + 6*(320*c^3*d^7*e^2 - 1520*b*c^2*d^6*e^ 3 + 2584*b^2*c*d^5*e^4 - 1615*b^3*d^4*e^5)*x^2 - 8*(320*c^3*d^8*e - 1520*b *c^2*d^7*e^2 + 2584*b^2*c*d^6*e^3 - 1615*b^3*d^5*e^4)*x)*sqrt(e*x + d)/e^7
Time = 1.11 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.50 \[ \int (d+e x)^{5/2} \left (b x+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {19}{2}}}{19 e^{6}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \left (b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (- 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (- b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{7 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (\frac {b^{3} x^{4}}{4} + \frac {3 b^{2} c x^{5}}{5} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(c**3*(d + e*x)**(19/2)/(19*e**6) + (d + e*x)**(17/2)*(3*b*c* *2*e - 6*c**3*d)/(17*e**6) + (d + e*x)**(15/2)*(3*b**2*c*e**2 - 15*b*c**2* d*e + 15*c**3*d**2)/(15*e**6) + (d + e*x)**(13/2)*(b**3*e**3 - 12*b**2*c*d *e**2 + 30*b*c**2*d**2*e - 20*c**3*d**3)/(13*e**6) + (d + e*x)**(11/2)*(-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)/(11* e**6) + (d + e*x)**(9/2)*(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)/(9*e**6) + (d + e*x)**(7/2)*(-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)/(7*e**6))/e, Ne(e, 0)), (d **(5/2)*(b**3*x**4/4 + 3*b**2*c*x**5/5 + b*c**2*x**6/2 + c**3*x**7/7), Tru e))
Time = 0.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.09 \[ \int (d+e x)^{5/2} \left (b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (255255 \, {\left (e x + d\right )}^{\frac {19}{2}} c^{3} - 855855 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 969969 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 373065 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 1322685 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 1616615 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 692835 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{4849845 \, e^{7}} \]
2/4849845*(255255*(e*x + d)^(19/2)*c^3 - 855855*(2*c^3*d - b*c^2*e)*(e*x + d)^(17/2) + 969969*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^2)*(e*x + d)^(15/2) - 373065*(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)^(13/2) + 1322685*(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)^(11/2) - 1616615*(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)^(9/2) + 692835*(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)^(7/2))/e^7
Leaf count of result is larger than twice the leaf count of optimal. 1450 vs. \(2 (220) = 440\).
Time = 0.30 (sec) , antiderivative size = 1450, normalized size of antiderivative = 5.85 \[ \int (d+e x)^{5/2} \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
2/4849845*(138567*(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^3/e^3 + 46189*(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^3/e^4 + 46189*(35*(e*x + d)^(9/2) - 18 0*(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^2/e^3 + 20995*(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^3/e^5 + 62985*(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^2/e^4 + 20995*(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*d/e^3 + 1615*(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)*c^3* d^3/e^6 + 14535*(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 - 6 006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*b*c^2*d^2/e^5 + 14535*(2 31*(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)^(...
Time = 9.30 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.96 \[ \int (d+e x)^{5/2} \left (b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,b^3\,e^3-24\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e-40\,c^3\,d^3\right )}{13\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {{\left (d+e\,x\right )}^{15/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{15\,e^7}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{11\,e^7}-\frac {2\,d^3\,{\left (b\,e-c\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7} \]
((d + e*x)^(13/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^ 2))/(13*e^7) + (2*c^3*(d + e*x)^(19/2))/(19*e^7) - ((12*c^3*d - 6*b*c^2*e) *(d + e*x)^(17/2))/(17*e^7) + ((d + e*x)^(15/2)*(30*c^3*d^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(15*e^7) + ((d + e*x)^(11/2)*(30*c^3*d^4 - 6*b^3*d*e^3 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e))/(11*e^7) - (2*d^3*(b*e - c*d)^3*(d + e *x)^(7/2))/(7*e^7) + (2*d^2*(b*e - c*d)^2*(b*e - 2*c*d)*(d + e*x)^(9/2))/( 3*e^7)