Integrand size = 21, antiderivative size = 248 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^3 \, dx=\frac {2 d^3 (c d-b e)^3 (d+e x)^{3/2}}{3 e^7}-\frac {6 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{5/2}}{5 e^7}+\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{9 e^7}+\frac {6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{13/2}}{13 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7} \]
2/3*d^3*(-b*e+c*d)^3*(e*x+d)^(3/2)/e^7-6/5*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*( e*x+d)^(5/2)/e^7+6/7*d*(-b*e+c*d)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(7 /2)/e^7-2/9*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(9/2)/e^7 +6/11*c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(11/2)/e^7-6/13*c^2*(-b*e+2* c*d)*(e*x+d)^(13/2)/e^7+2/15*c^3*(e*x+d)^(15/2)/e^7
Time = 0.17 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.93 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{3/2} \left (143 b^3 e^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+39 b^2 c e^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+15 b c^2 e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+c^3 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7} \]
(2*(d + e*x)^(3/2)*(143*b^3*e^3*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35* e^3*x^3) + 39*b^2*c*e^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e ^3*x^3 + 315*e^4*x^4) + 15*b*c^2*e*(-256*d^5 + 384*d^4*e*x - 480*d^3*e^2*x ^2 + 560*d^2*e^3*x^3 - 630*d*e^4*x^4 + 693*e^5*x^5) + c^3*(1024*d^6 - 1536 *d^5*e*x + 1920*d^4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d *e^5*x^5 + 3003*e^6*x^6)))/(45045*e^7)
Time = 0.36 (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.
| \(\displaystyle \int \left (b x+c x^2\right )^3 \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c (d+e x)^{9/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^{7/2} (2 c d-b e) \left (-b^2 e^2+10 b c d e-10 c^2 d^2\right )}{e^6}+\frac {3 d (d+e x)^{5/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}-\frac {3 c^2 (d+e x)^{11/2} (2 c d-b e)}{e^6}+\frac {d^3 \sqrt {d+e x} (c d-b e)^3}{e^6}-\frac {3 d^2 (d+e x)^{3/2} (c d-b e)^2 (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^{13/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 c (d+e x)^{11/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{9 e^7}+\frac {6 d (d+e x)^{7/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac {6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}+\frac {2 d^3 (d+e x)^{3/2} (c d-b e)^3}{3 e^7}-\frac {6 d^2 (d+e x)^{5/2} (c d-b e)^2 (2 c d-b e)}{5 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7}\) |
(2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2))/(3*e^7) - (6*d^2*(c*d - b*e)^2*(2*c* d - b*e)*(d + e*x)^(5/2))/(5*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)^(7/2))/(7*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)^(9/2))/(9*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(13 /2))/(13*e^7) + (2*c^3*(d + e*x)^(15/2))/(15*e^7)
3.4.56.3.1 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 = 2.32 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.84
| method | result | size |
| pseudoelliptic | \(-\frac {32 \left (e x +d \right )^{\frac {3}{2}} \left (-\frac {35 \left (\frac {3}{5} c^{3} x^{3}+\frac {27}{13} b \,c^{2} x^{2}+\frac {27}{11} b^{2} c x +b^{3}\right ) x^{3} e^{6}}{16}+\frac {15 x^{2} \left (\frac {42}{65} c^{3} x^{3}+\frac {315}{143} b \,c^{2} x^{2}+\frac {28}{11} b^{2} c x +b^{3}\right ) d \,e^{5}}{8}-\frac {3 x \,d^{2} \left (\frac {105}{143} c^{3} x^{3}+\frac {350}{143} b \,c^{2} x^{2}+\frac {30}{11} b^{2} c x +b^{3}\right ) e^{4}}{2}+d^{3} \left (\frac {140}{143} c^{3} x^{3}+\frac {450}{143} b \,c^{2} x^{2}+\frac {36}{11} b^{2} c x +b^{3}\right ) e^{3}-\frac {24 c \left (\frac {5}{13} c^{2} x^{2}+\frac {15}{13} b c x +b^{2}\right ) d^{4} e^{2}}{11}+\frac {240 \left (\frac {2 c x}{5}+b \right ) c^{2} d^{5} e}{143}-\frac {64 c^{3} d^{6}}{143}\right )}{315 e^{7}}\) | \(208\) |
| derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (-3 c^{3} d +3 \left (b e -c d \right ) c^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 c^{3} d^{2}-9 d \left (b e -c d \right ) c^{2}+3 \left (b e -c d \right )^{2} c \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-c^{3} d^{3}+9 d^{2} \left (b e -c d \right ) c^{2}-9 d \left (b e -c d \right )^{2} c +\left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-3 d^{3} \left (b e -c d \right ) c^{2}+9 d^{2} \left (b e -c d \right )^{2} c -3 d \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-3 d^{3} \left (b e -c d \right )^{2} c +3 d^{2} \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{7}}\) | \(269\) |
| default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {2 \left (3 c^{3} d -3 \left (b e -c d \right ) c^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}-\frac {2 \left (-3 c^{3} d^{2}+9 d \left (b e -c d \right ) c^{2}-3 \left (b e -c d \right )^{2} c \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {2 \left (c^{3} d^{3}-9 d^{2} \left (b e -c d \right ) c^{2}+9 d \left (b e -c d \right )^{2} c -\left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {2 \left (3 d^{3} \left (b e -c d \right ) c^{2}-9 d^{2} \left (b e -c d \right )^{2} c +3 d \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {2 \left (3 d^{3} \left (b e -c d \right )^{2} c -3 d^{2} \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{7}}\) | \(270\) |
| gosper | \(-\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (-3003 x^{6} c^{3} e^{6}-10395 x^{5} b \,c^{2} e^{6}+2772 x^{5} c^{3} d \,e^{5}-12285 x^{4} b^{2} c \,e^{6}+9450 x^{4} b \,c^{2} d \,e^{5}-2520 x^{4} c^{3} d^{2} e^{4}-5005 x^{3} b^{3} e^{6}+10920 x^{3} b^{2} c d \,e^{5}-8400 x^{3} b \,c^{2} d^{2} e^{4}+2240 x^{3} c^{3} d^{3} e^{3}+4290 x^{2} b^{3} d \,e^{5}-9360 x^{2} b^{2} c \,d^{2} e^{4}+7200 x^{2} b \,c^{2} d^{3} e^{3}-1920 x^{2} c^{3} d^{4} e^{2}-3432 x \,b^{3} d^{2} e^{4}+7488 x \,b^{2} c \,d^{3} e^{3}-5760 x b \,c^{2} d^{4} e^{2}+1536 x \,c^{3} d^{5} e +2288 b^{3} d^{3} e^{3}-4992 b^{2} c \,d^{4} e^{2}+3840 b \,c^{2} d^{5} e -1024 c^{3} d^{6}\right )}{45045 e^{7}}\) | \(286\) |
| trager | \(-\frac {2 \left (-3003 c^{3} e^{7} x^{7}-10395 b \,c^{2} e^{7} x^{6}-231 c^{3} d \,e^{6} x^{6}-12285 b^{2} c \,e^{7} x^{5}-945 b \,c^{2} d \,e^{6} x^{5}+252 c^{3} d^{2} e^{5} x^{5}-5005 b^{3} e^{7} x^{4}-1365 b^{2} c d \,e^{6} x^{4}+1050 b \,c^{2} d^{2} e^{5} x^{4}-280 c^{3} d^{3} e^{4} x^{4}-715 b^{3} d \,e^{6} x^{3}+1560 b^{2} c \,d^{2} e^{5} x^{3}-1200 b \,c^{2} d^{3} e^{4} x^{3}+320 c^{3} d^{4} e^{3} x^{3}+858 b^{3} d^{2} e^{5} x^{2}-1872 b^{2} c \,d^{3} e^{4} x^{2}+1440 b \,c^{2} d^{4} e^{3} x^{2}-384 c^{3} d^{5} e^{2} x^{2}-1144 b^{3} d^{3} e^{4} x +2496 b^{2} c \,d^{4} e^{3} x -1920 b \,c^{2} d^{5} e^{2} x +512 c^{3} d^{6} e x +2288 b^{3} d^{4} e^{3}-4992 b^{2} c \,d^{5} e^{2}+3840 b \,c^{2} d^{6} e -1024 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{7}}\) | \(344\) |
| risch | \(-\frac {2 \left (-3003 c^{3} e^{7} x^{7}-10395 b \,c^{2} e^{7} x^{6}-231 c^{3} d \,e^{6} x^{6}-12285 b^{2} c \,e^{7} x^{5}-945 b \,c^{2} d \,e^{6} x^{5}+252 c^{3} d^{2} e^{5} x^{5}-5005 b^{3} e^{7} x^{4}-1365 b^{2} c d \,e^{6} x^{4}+1050 b \,c^{2} d^{2} e^{5} x^{4}-280 c^{3} d^{3} e^{4} x^{4}-715 b^{3} d \,e^{6} x^{3}+1560 b^{2} c \,d^{2} e^{5} x^{3}-1200 b \,c^{2} d^{3} e^{4} x^{3}+320 c^{3} d^{4} e^{3} x^{3}+858 b^{3} d^{2} e^{5} x^{2}-1872 b^{2} c \,d^{3} e^{4} x^{2}+1440 b \,c^{2} d^{4} e^{3} x^{2}-384 c^{3} d^{5} e^{2} x^{2}-1144 b^{3} d^{3} e^{4} x +2496 b^{2} c \,d^{4} e^{3} x -1920 b \,c^{2} d^{5} e^{2} x +512 c^{3} d^{6} e x +2288 b^{3} d^{4} e^{3}-4992 b^{2} c \,d^{5} e^{2}+3840 b \,c^{2} d^{6} e -1024 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{7}}\) | \(344\) |
-32/315*(e*x+d)^(3/2)*(-35/16*(3/5*c^3*x^3+27/13*b*c^2*x^2+27/11*b^2*c*x+b ^3)*x^3*e^6+15/8*x^2*(42/65*c^3*x^3+315/143*b*c^2*x^2+28/11*b^2*c*x+b^3)*d *e^5-3/2*x*d^2*(105/143*c^3*x^3+350/143*b*c^2*x^2+30/11*b^2*c*x+b^3)*e^4+d ^3*(140/143*c^3*x^3+450/143*b*c^2*x^2+36/11*b^2*c*x+b^3)*e^3-24/11*c*(5/13 *c^2*x^2+15/13*b*c*x+b^2)*d^4*e^2+240/143*(2/5*c*x+b)*c^2*d^5*e-64/143*c^3 *d^6)/e^7
Time = 0.47 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.29 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (3003 \, c^{3} e^{7} x^{7} + 1024 \, c^{3} d^{7} - 3840 \, b c^{2} d^{6} e + 4992 \, b^{2} c d^{5} e^{2} - 2288 \, b^{3} d^{4} e^{3} + 231 \, {\left (c^{3} d e^{6} + 45 \, b c^{2} e^{7}\right )} x^{6} - 63 \, {\left (4 \, c^{3} d^{2} e^{5} - 15 \, b c^{2} d e^{6} - 195 \, b^{2} c e^{7}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{3} e^{4} - 30 \, b c^{2} d^{2} e^{5} + 39 \, b^{2} c d e^{6} + 143 \, b^{3} e^{7}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{4} e^{3} - 240 \, b c^{2} d^{3} e^{4} + 312 \, b^{2} c d^{2} e^{5} - 143 \, b^{3} d e^{6}\right )} x^{3} + 6 \, {\left (64 \, c^{3} d^{5} e^{2} - 240 \, b c^{2} d^{4} e^{3} + 312 \, b^{2} c d^{3} e^{4} - 143 \, b^{3} d^{2} e^{5}\right )} x^{2} - 8 \, {\left (64 \, c^{3} d^{6} e - 240 \, b c^{2} d^{5} e^{2} + 312 \, b^{2} c d^{4} e^{3} - 143 \, b^{3} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \]
2/45045*(3003*c^3*e^7*x^7 + 1024*c^3*d^7 - 3840*b*c^2*d^6*e + 4992*b^2*c*d ^5*e^2 - 2288*b^3*d^4*e^3 + 231*(c^3*d*e^6 + 45*b*c^2*e^7)*x^6 - 63*(4*c^3 *d^2*e^5 - 15*b*c^2*d*e^6 - 195*b^2*c*e^7)*x^5 + 35*(8*c^3*d^3*e^4 - 30*b* c^2*d^2*e^5 + 39*b^2*c*d*e^6 + 143*b^3*e^7)*x^4 - 5*(64*c^3*d^4*e^3 - 240* b*c^2*d^3*e^4 + 312*b^2*c*d^2*e^5 - 143*b^3*d*e^6)*x^3 + 6*(64*c^3*d^5*e^2 - 240*b*c^2*d^4*e^3 + 312*b^2*c*d^3*e^4 - 143*b^3*d^2*e^5)*x^2 - 8*(64*c^ 3*d^6*e - 240*b*c^2*d^5*e^2 + 312*b^2*c*d^4*e^3 - 143*b^3*d^3*e^4)*x)*sqrt (e*x + d)/e^7
Time = 0.89 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.50 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (- 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{3 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (\frac {b^{3} x^{4}}{4} + \frac {3 b^{2} c x^{5}}{5} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(c**3*(d + e*x)**(15/2)/(15*e**6) + (d + e*x)**(13/2)*(3*b*c* *2*e - 6*c**3*d)/(13*e**6) + (d + e*x)**(11/2)*(3*b**2*c*e**2 - 15*b*c**2* d*e + 15*c**3*d**2)/(11*e**6) + (d + e*x)**(9/2)*(b**3*e**3 - 12*b**2*c*d* e**2 + 30*b*c**2*d**2*e - 20*c**3*d**3)/(9*e**6) + (d + e*x)**(7/2)*(-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)/(7*e**6 ) + (d + e*x)**(5/2)*(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)/(5*e**6) + (d + e*x)**(3/2)*(-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)/(3*e**6))/e, Ne(e, 0)), (sqrt( d)*(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 = 0.22 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.09 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} - 10395 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 5005 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 19305 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 27027 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{7}} \]
2/45045*(3003*(e*x + d)^(15/2)*c^3 - 10395*(2*c^3*d - b*c^2*e)*(e*x + d)^( 13/2) + 12285*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^2)*(e*x + d)^(11/2) - 500 5*(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)^(9/2) + 19305*(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)^(7/2) - 27027*(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)^(5/2) + 15015*(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)^(3/2))/e^7
Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (220) = 440\).
Time = 0.29 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.51 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (\frac {1287 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b^{3} d}{e^{3}} + \frac {429 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{2} c d}{e^{4}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{3}}{e^{3}} + \frac {195 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b c^{2} d}{e^{5}} + \frac {195 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b^{2} c}{e^{4}} + \frac {15 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} c^{3} d}{e^{6}} + \frac {45 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} b c^{2}}{e^{5}} + \frac {7 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} - 3465 \, {\left (e x + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (e x + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {e x + d} d^{7}\right )} c^{3}}{e^{6}}\right )}}{45045 \, e} \]
2/45045*(1287*(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/e^3 + 429*(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/e^4 + 143*(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/e^3 + 195*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 99 0*(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/e^5 + 195*(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 + 115 5*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b^2*c/e^4 + 15*(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 + 30 03*sqrt(e*x + d)*d^6)*c^3*d/e^6 + 45*(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/e^5 + 7*(429*(e*x + d)^(15/2) - 3465*(e*x + d)^(13/2)*d + 12285*(e*x + d)^(11/2)*d^2 - 25025*(e*x + d)^(9/2)*d^3 + 32175*(e*x + d)^(7/2)*d^4 - 27 027*(e*x + d)^(5/2)*d^5 + 15015*(e*x + d)^(3/2)*d^6 - 6435*sqrt(e*x + d)*d ^7)*c^3/e^6)/e
Time = 9.36 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.96 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,b^3\,e^3-24\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e-40\,c^3\,d^3\right )}{9\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{7\,e^7}-\frac {2\,d^3\,{\left (b\,e-c\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}+\frac {6\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7} \]
((d + e*x)^(9/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2 ))/(9*e^7) + (2*c^3*(d + e*x)^(15/2))/(15*e^7) - ((12*c^3*d - 6*b*c^2*e)*( d + e*x)^(13/2))/(13*e^7) + ((d + e*x)^(11/2)*(30*c^3*d^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(11*e^7) + ((d + e*x)^(7/2)*(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))/(7*e^7) - (2*d^3*(b*e - c*d)^3*(d + e*x)^ (3/2))/(3*e^7) + (6*d^2*(b*e - c*d)^2*(b*e - 2*c*d)*(d + e*x)^(5/2))/(5*e^ 7)