Integrand size = 28, antiderivative size = 252 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{3 e^6}+\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{9 e^6}-\frac {10 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^6}+\frac {4 c^3 (d+e x)^{13/2}}{13 e^6} \] Output:
-2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(3/2)/e^6+4/5*(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)^(5/2)/e^6-2/7*(-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)^(7/2)/e^6+8/9*c*(5 *c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(9/2)/e^6-10/11*c^2*(-b*e+2*c*d )*(e*x+d)^(11/2)/e^6+4/13*c^3*(e*x+d)^(13/2)/e^6
Time = 0.30 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.15 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{3/2} \left (-10 c^3 \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )+429 b e^3 \left (35 a^2 e^2+14 a b e (-2 d+3 e x)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-286 c e^2 \left (21 a^2 e^2 (2 d-3 e x)-9 a b e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^2 \left (32 d^3-48 d^2 e x+60 d e^2 x^2-70 e^3 x^3\right )\right )+13 c^2 e \left (44 a e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+5 b \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )\right )}{45045 e^6} \] Input:
Integrate[(b + 2*c*x)*Sqrt[d + e*x]*(a + b*x + c*x^2)^2,x]
Output:
(2*(d + e*x)^(3/2)*(-10*c^3*(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) + 429*b*e^3*(35*a^2*e^2 + 14*a *b*e*(-2*d + 3*e*x) + b^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 286*c*e^2*(21 *a^2*e^2*(2*d - 3*e*x) - 9*a*b*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + b^2*(32 *d^3 - 48*d^2*e*x + 60*d*e^2*x^2 - 70*e^3*x^3)) + 13*c^2*e*(44*a*e*(-16*d^ 3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 5*b*(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))))/(45045*e^6)
Time = 0.65 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1195, 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.
| \(\displaystyle \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {4 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^5}+\frac {(d+e x)^{5/2} (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^5}+\frac {2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^5}+\frac {\sqrt {d+e x} (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac {5 c^2 (d+e x)^{9/2} (2 c d-b e)}{e^5}+\frac {2 c^3 (d+e x)^{11/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^6}+\frac {4 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac {10 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^6}+\frac {4 c^3 (d+e x)^{13/2}}{13 e^6}\) |
Input:
Int[(b + 2*c*x)*Sqrt[d + e*x]*(a + b*x + c*x^2)^2,x]
Output:
(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2))/(3*e^6) + (4* (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 )^(5/2))/(5*e^6) - (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)^(7/2))/(7*e^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(9/2))/(9*e^6) - (10*c^2*(2*c*d - b*e)*(d + e*x)^(11/2) )/(11*e^6) + (4*c^3*(d + e*x)^(13/2))/(13*e^6)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 1.94 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {6 c^{3} x^{5}}{13}+\frac {4 \left (\frac {45 b x}{44}+a \right ) x^{3} c^{2}}{3}+\left (\frac {4}{3} b^{2} x^{3}+\frac {6}{5} a^{2} x +\frac {18}{7} a b \,x^{2}\right ) c +b \left (\frac {3}{7} b^{2} x^{2}+a^{2}+\frac {6}{5} a b x \right )\right ) e^{5}-\frac {4 d \left (\frac {75 c^{3} x^{4}}{143}+\frac {10 x^{2} \left (\frac {35 b x}{33}+a \right ) c^{2}}{7}+\left (\frac {10}{7} b^{2} x^{2}+\frac {18}{7} a b x +a^{2}\right ) c +b^{2} \left (\frac {3 b x}{7}+a \right )\right ) e^{4}}{5}+\frac {48 \left (\frac {350 c^{3} x^{3}}{1287}+\frac {2 \left (\frac {25 b x}{22}+a \right ) x \,c^{2}}{3}+b \left (\frac {2 b x}{3}+a \right ) c +\frac {b^{3}}{6}\right ) d^{2} e^{3}}{35}-\frac {64 d^{3} \left (\frac {75 c^{2} x^{2}}{143}+\left (\frac {15 b x}{11}+a \right ) c +b^{2}\right ) c \,e^{2}}{105}+\frac {128 d^{4} c^{2} \left (\frac {6 c x}{13}+b \right ) e}{231}-\frac {512 d^{5} c^{3}}{3003}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e^{6}}\) | \(237\) |
| derivativedivides | \(\frac {\frac {4 c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {10 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} c +2 c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+4 c \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+2 c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (b e -2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(265\) |
| default | \(\frac {\frac {4 c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {10 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} c +2 c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+4 c \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+2 c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (b e -2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(265\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (6930 e^{5} x^{5} c^{3}+20475 e^{5} x^{4} c^{2} b -6300 x^{4} d \,e^{4} c^{3}+20020 e^{5} a \,c^{2} x^{3}+20020 e^{5} x^{3} b^{2} c -18200 x^{3} d \,e^{4} c^{2} b +5600 d^{2} e^{3} c^{3} x^{3}+38610 e^{5} a b c \,x^{2}-17160 d \,e^{4} a \,c^{2} x^{2}+6435 x^{2} e^{5} b^{3}-17160 x^{2} d \,e^{4} b^{2} c +15600 d^{2} e^{3} b \,c^{2} x^{2}-4800 d^{3} e^{2} c^{3} x^{2}+18018 e^{5} a^{2} c x +18018 e^{5} x a \,b^{2}-30888 d \,e^{4} a b c x +13728 d^{2} e^{3} a \,c^{2} x -5148 x d \,e^{4} b^{3}+13728 d^{2} e^{3} b^{2} c x -12480 d^{3} e^{2} c^{2} b x +3840 c^{3} d^{4} e x +15015 a^{2} b \,e^{5}-12012 d \,e^{4} a^{2} c -12012 a \,b^{2} d \,e^{4}+20592 a b c \,d^{2} e^{3}-9152 d^{3} e^{2} a \,c^{2}+3432 b^{3} d^{2} e^{3}-9152 b^{2} c \,d^{3} e^{2}+8320 b \,c^{2} d^{4} e -2560 d^{5} c^{3}\right )}{45045 e^{6}}\) | \(359\) |
| orering | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (6930 e^{5} x^{5} c^{3}+20475 e^{5} x^{4} c^{2} b -6300 x^{4} d \,e^{4} c^{3}+20020 e^{5} a \,c^{2} x^{3}+20020 e^{5} x^{3} b^{2} c -18200 x^{3} d \,e^{4} c^{2} b +5600 d^{2} e^{3} c^{3} x^{3}+38610 e^{5} a b c \,x^{2}-17160 d \,e^{4} a \,c^{2} x^{2}+6435 x^{2} e^{5} b^{3}-17160 x^{2} d \,e^{4} b^{2} c +15600 d^{2} e^{3} b \,c^{2} x^{2}-4800 d^{3} e^{2} c^{3} x^{2}+18018 e^{5} a^{2} c x +18018 e^{5} x a \,b^{2}-30888 d \,e^{4} a b c x +13728 d^{2} e^{3} a \,c^{2} x -5148 x d \,e^{4} b^{3}+13728 d^{2} e^{3} b^{2} c x -12480 d^{3} e^{2} c^{2} b x +3840 c^{3} d^{4} e x +15015 a^{2} b \,e^{5}-12012 d \,e^{4} a^{2} c -12012 a \,b^{2} d \,e^{4}+20592 a b c \,d^{2} e^{3}-9152 d^{3} e^{2} a \,c^{2}+3432 b^{3} d^{2} e^{3}-9152 b^{2} c \,d^{3} e^{2}+8320 b \,c^{2} d^{4} e -2560 d^{5} c^{3}\right )}{45045 e^{6}}\) | \(359\) |
| trager | \(\frac {2 \left (6930 c^{3} e^{6} x^{6}+20475 b \,c^{2} e^{6} x^{5}+630 c^{3} d \,e^{5} x^{5}+20020 a \,c^{2} e^{6} x^{4}+20020 b^{2} c \,e^{6} x^{4}+2275 b \,c^{2} d \,e^{5} x^{4}-700 c^{3} d^{2} e^{4} x^{4}+38610 a b c \,e^{6} x^{3}+2860 a \,c^{2} d \,e^{5} x^{3}+6435 b^{3} e^{6} x^{3}+2860 b^{2} c d \,e^{5} x^{3}-2600 b \,c^{2} d^{2} e^{4} x^{3}+800 c^{3} d^{3} e^{3} x^{3}+18018 a^{2} c \,e^{6} x^{2}+18018 a \,b^{2} e^{6} x^{2}+7722 a b c d \,e^{5} x^{2}-3432 x^{2} a \,c^{2} d^{2} e^{4}+1287 b^{3} d \,e^{5} x^{2}-3432 b^{2} c \,d^{2} e^{4} x^{2}+3120 b \,c^{2} d^{3} e^{3} x^{2}-960 c^{3} d^{4} e^{2} x^{2}+15015 a^{2} b \,e^{6} x +6006 x \,a^{2} c d \,e^{5}+6006 a \,b^{2} d \,e^{5} x -10296 a b c \,d^{2} e^{4} x +4576 x a \,c^{2} d^{3} e^{3}-1716 b^{3} d^{2} e^{4} x +4576 b^{2} c \,d^{3} e^{3} x -4160 b \,c^{2} d^{4} e^{2} x +1280 c^{3} d^{5} e x +15015 a^{2} b d \,e^{5}-12012 d^{2} e^{4} a^{2} c -12012 a \,b^{2} d^{2} e^{4}+20592 a b c \,d^{3} e^{3}-9152 d^{4} e^{2} a \,c^{2}+3432 b^{3} d^{3} e^{3}-9152 b^{2} c \,d^{4} e^{2}+8320 b \,c^{2} d^{5} e -2560 d^{6} c^{3}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(487\) |
| risch | \(\frac {2 \left (6930 c^{3} e^{6} x^{6}+20475 b \,c^{2} e^{6} x^{5}+630 c^{3} d \,e^{5} x^{5}+20020 a \,c^{2} e^{6} x^{4}+20020 b^{2} c \,e^{6} x^{4}+2275 b \,c^{2} d \,e^{5} x^{4}-700 c^{3} d^{2} e^{4} x^{4}+38610 a b c \,e^{6} x^{3}+2860 a \,c^{2} d \,e^{5} x^{3}+6435 b^{3} e^{6} x^{3}+2860 b^{2} c d \,e^{5} x^{3}-2600 b \,c^{2} d^{2} e^{4} x^{3}+800 c^{3} d^{3} e^{3} x^{3}+18018 a^{2} c \,e^{6} x^{2}+18018 a \,b^{2} e^{6} x^{2}+7722 a b c d \,e^{5} x^{2}-3432 x^{2} a \,c^{2} d^{2} e^{4}+1287 b^{3} d \,e^{5} x^{2}-3432 b^{2} c \,d^{2} e^{4} x^{2}+3120 b \,c^{2} d^{3} e^{3} x^{2}-960 c^{3} d^{4} e^{2} x^{2}+15015 a^{2} b \,e^{6} x +6006 x \,a^{2} c d \,e^{5}+6006 a \,b^{2} d \,e^{5} x -10296 a b c \,d^{2} e^{4} x +4576 x a \,c^{2} d^{3} e^{3}-1716 b^{3} d^{2} e^{4} x +4576 b^{2} c \,d^{3} e^{3} x -4160 b \,c^{2} d^{4} e^{2} x +1280 c^{3} d^{5} e x +15015 a^{2} b d \,e^{5}-12012 d^{2} e^{4} a^{2} c -12012 a \,b^{2} d^{2} e^{4}+20592 a b c \,d^{3} e^{3}-9152 d^{4} e^{2} a \,c^{2}+3432 b^{3} d^{3} e^{3}-9152 b^{2} c \,d^{4} e^{2}+8320 b \,c^{2} d^{5} e -2560 d^{6} c^{3}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(487\) |
Input:
int((2*c*x+b)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
2/3*((6/13*c^3*x^5+4/3*(45/44*b*x+a)*x^3*c^2+(4/3*b^2*x^3+6/5*a^2*x+18/7*a *b*x^2)*c+b*(3/7*b^2*x^2+a^2+6/5*a*b*x))*e^5-4/5*d*(75/143*c^3*x^4+10/7*x^ 2*(35/33*b*x+a)*c^2+(10/7*b^2*x^2+18/7*a*b*x+a^2)*c+b^2*(3/7*b*x+a))*e^4+4 8/35*(350/1287*c^3*x^3+2/3*(25/22*b*x+a)*x*c^2+b*(2/3*b*x+a)*c+1/6*b^3)*d^ 2*e^3-64/105*d^3*(75/143*c^2*x^2+(15/11*b*x+a)*c+b^2)*c*e^2+128/231*d^4*c^ 2*(6/13*c*x+b)*e-512/3003*d^5*c^3)*(e*x+d)^(3/2)/e^6
Time = 0.07 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.58 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (6930 \, c^{3} e^{6} x^{6} - 2560 \, c^{3} d^{6} + 8320 \, b c^{2} d^{5} e + 15015 \, a^{2} b d e^{5} - 9152 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 3432 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 12012 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 315 \, {\left (2 \, c^{3} d e^{5} + 65 \, b c^{2} e^{6}\right )} x^{5} - 35 \, {\left (20 \, c^{3} d^{2} e^{4} - 65 \, b c^{2} d e^{5} - 572 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 5 \, {\left (160 \, c^{3} d^{3} e^{3} - 520 \, b c^{2} d^{2} e^{4} + 572 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + 1287 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 3 \, {\left (320 \, c^{3} d^{4} e^{2} - 1040 \, b c^{2} d^{3} e^{3} + 1144 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 429 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} - 6006 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + {\left (1280 \, c^{3} d^{5} e - 4160 \, b c^{2} d^{4} e^{2} + 15015 \, a^{2} b e^{6} + 4576 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 1716 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 6006 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \] Input:
integrate((2*c*x+b)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
2/45045*(6930*c^3*e^6*x^6 - 2560*c^3*d^6 + 8320*b*c^2*d^5*e + 15015*a^2*b* d*e^5 - 9152*(b^2*c + a*c^2)*d^4*e^2 + 3432*(b^3 + 6*a*b*c)*d^3*e^3 - 1201 2*(a*b^2 + a^2*c)*d^2*e^4 + 315*(2*c^3*d*e^5 + 65*b*c^2*e^6)*x^5 - 35*(20* c^3*d^2*e^4 - 65*b*c^2*d*e^5 - 572*(b^2*c + a*c^2)*e^6)*x^4 + 5*(160*c^3*d ^3*e^3 - 520*b*c^2*d^2*e^4 + 572*(b^2*c + a*c^2)*d*e^5 + 1287*(b^3 + 6*a*b *c)*e^6)*x^3 - 3*(320*c^3*d^4*e^2 - 1040*b*c^2*d^3*e^3 + 1144*(b^2*c + a*c ^2)*d^2*e^4 - 429*(b^3 + 6*a*b*c)*d*e^5 - 6006*(a*b^2 + a^2*c)*e^6)*x^2 + (1280*c^3*d^5*e - 4160*b*c^2*d^4*e^2 + 15015*a^2*b*e^6 + 4576*(b^2*c + a*c ^2)*d^3*e^3 - 1716*(b^3 + 6*a*b*c)*d^2*e^4 + 6006*(a*b^2 + a^2*c)*d*e^5)*x )*sqrt(e*x + d)/e^6
Time = 1.43 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.68 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 c^{3} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (5 b c^{2} e - 10 c^{3} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (4 a c^{2} e^{2} + 4 b^{2} c e^{2} - 20 b c^{2} d e + 20 c^{3} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 a^{2} c e^{4} + 2 a b^{2} e^{4} - 12 a b c d e^{3} + 12 a c^{2} d^{2} e^{2} - 2 b^{3} d e^{3} + 12 b^{2} c d^{2} e^{2} - 20 b c^{2} d^{3} e + 10 c^{3} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{2} b e^{5} - 2 a^{2} c d e^{4} - 2 a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 4 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 4 b^{2} c d^{3} e^{2} + 5 b c^{2} d^{4} e - 2 c^{3} d^{5}\right )}{3 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\sqrt {d} \left (a + b x + c x^{2}\right )^{3}}{3} & \text {otherwise} \end {cases} \] Input:
integrate((2*c*x+b)*(e*x+d)**(1/2)*(c*x**2+b*x+a)**2,x)
Output:
Piecewise((2*(2*c**3*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(5*b* c**2*e - 10*c**3*d)/(11*e**5) + (d + e*x)**(9/2)*(4*a*c**2*e**2 + 4*b**2*c *e**2 - 20*b*c**2*d*e + 20*c**3*d**2)/(9*e**5) + (d + e*x)**(7/2)*(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)/(7*e**5) + (d + e*x)**(5/2)*(2*a**2*c*e**4 + 2*a*b**2*e** 4 - 12*a*b*c*d*e**3 + 12*a*c**2*d**2*e**2 - 2*b**3*d*e**3 + 12*b**2*c*d**2 *e**2 - 20*b*c**2*d**3*e + 10*c**3*d**4)/(5*e**5) + (d + e*x)**(3/2)*(a**2 *b*e**5 - 2*a**2*c*d*e**4 - 2*a*b**2*d*e**4 + 6*a*b*c*d**2*e**3 - 4*a*c**2 *d**3*e**2 + b**3*d**2*e**3 - 4*b**2*c*d**3*e**2 + 5*b*c**2*d**4*e - 2*c** 3*d**5)/(3*e**5))/e, Ne(e, 0)), (sqrt(d)*(a + b*x + c*x**2)**3/3, True))
Time = 0.04 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.22 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (6930 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{3} - 20475 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 20020 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 6435 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 18018 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 15015 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{6}} \] Input:
integrate((2*c*x+b)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
2/45045*(6930*(e*x + d)^(13/2)*c^3 - 20475*(2*c^3*d - b*c^2*e)*(e*x + d)^( 11/2) + 20020*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(9 /2) - 6435*(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)^(7/2) + 18018*(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)^(5/2) - 15015*(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) ^(3/2))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 914 vs. \(2 (228) = 456\).
Time = 0.17 (sec) , antiderivative size = 914, normalized size of antiderivative = 3.63 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((2*c*x+b)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
2/45045*(45045*sqrt(e*x + d)*a^2*b*d + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*b + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b^2*d/e + 3 0030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*c*d/e + 3003*(3*(e*x + d)^( 5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^3*d/e^2 + 18018*(3*( e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*b*c*d/e^2 + 6006*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a *b^2/e + 6006*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d) *d^2)*a^2*c/e + 5148*(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^2*c*d/e^3 + 5148*(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*c^2*d/e^3 + 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/e^2 + 7722*(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/ e^2 + 715*(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*c^2*d/e^4 + 572 *(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 4 20*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^2*c/e^3 + 572*(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/e^3 + 130*(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...
Time = 12.01 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.06 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^{5/2}\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{5\,e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{9\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{7\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{3\,e^6} \] Input:
int((b + 2*c*x)*(d + e*x)^(1/2)*(a + b*x + c*x^2)^2,x)
Output:
((d + e*x)^(5/2)*(20*c^3*d^4 + 4*a*b^2*e^4 + 4*a^2*c*e^4 - 4*b^3*d*e^3 + 2 4*a*c^2*d^2*e^2 + 24*b^2*c*d^2*e^2 - 40*b*c^2*d^3*e - 24*a*b*c*d*e^3))/(5* e^6) + (4*c^3*(d + e*x)^(13/2))/(13*e^6) - ((20*c^3*d - 10*b*c^2*e)*(d + e *x)^(11/2))/(11*e^6) + ((d + e*x)^(9/2)*(40*c^3*d^2 + 8*a*c^2*e^2 + 8*b^2* c*e^2 - 40*b*c^2*d*e))/(9*e^6) + (2*(b*e - 2*c*d)*(d + e*x)^(7/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(7*e^6) + (2*(b*e - 2*c*d)*(d + e *x)^(3/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(3*e^6)
Time = 0.22 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.92 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \sqrt {e x +d}\, \left (6930 c^{3} e^{6} x^{6}+20475 b \,c^{2} e^{6} x^{5}+630 c^{3} d \,e^{5} x^{5}+20020 a \,c^{2} e^{6} x^{4}+20020 b^{2} c \,e^{6} x^{4}+2275 b \,c^{2} d \,e^{5} x^{4}-700 c^{3} d^{2} e^{4} x^{4}+38610 a b c \,e^{6} x^{3}+2860 a \,c^{2} d \,e^{5} x^{3}+6435 b^{3} e^{6} x^{3}+2860 b^{2} c d \,e^{5} x^{3}-2600 b \,c^{2} d^{2} e^{4} x^{3}+800 c^{3} d^{3} e^{3} x^{3}+18018 a^{2} c \,e^{6} x^{2}+18018 a \,b^{2} e^{6} x^{2}+7722 a b c d \,e^{5} x^{2}-3432 a \,c^{2} d^{2} e^{4} x^{2}+1287 b^{3} d \,e^{5} x^{2}-3432 b^{2} c \,d^{2} e^{4} x^{2}+3120 b \,c^{2} d^{3} e^{3} x^{2}-960 c^{3} d^{4} e^{2} x^{2}+15015 a^{2} b \,e^{6} x +6006 a^{2} c d \,e^{5} x +6006 a \,b^{2} d \,e^{5} x -10296 a b c \,d^{2} e^{4} x +4576 a \,c^{2} d^{3} e^{3} x -1716 b^{3} d^{2} e^{4} x +4576 b^{2} c \,d^{3} e^{3} x -4160 b \,c^{2} d^{4} e^{2} x +1280 c^{3} d^{5} e x +15015 a^{2} b d \,e^{5}-12012 a^{2} c \,d^{2} e^{4}-12012 a \,b^{2} d^{2} e^{4}+20592 a b c \,d^{3} e^{3}-9152 a \,c^{2} d^{4} e^{2}+3432 b^{3} d^{3} e^{3}-9152 b^{2} c \,d^{4} e^{2}+8320 b \,c^{2} d^{5} e -2560 c^{3} d^{6}\right )}{45045 e^{6}} \] Input:
int((2*c*x+b)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^2,x)
Output:
(2*sqrt(d + e*x)*(15015*a**2*b*d*e**5 + 15015*a**2*b*e**6*x - 12012*a**2*c *d**2*e**4 + 6006*a**2*c*d*e**5*x + 18018*a**2*c*e**6*x**2 - 12012*a*b**2* d**2*e**4 + 6006*a*b**2*d*e**5*x + 18018*a*b**2*e**6*x**2 + 20592*a*b*c*d* *3*e**3 - 10296*a*b*c*d**2*e**4*x + 7722*a*b*c*d*e**5*x**2 + 38610*a*b*c*e **6*x**3 - 9152*a*c**2*d**4*e**2 + 4576*a*c**2*d**3*e**3*x - 3432*a*c**2*d **2*e**4*x**2 + 2860*a*c**2*d*e**5*x**3 + 20020*a*c**2*e**6*x**4 + 3432*b* *3*d**3*e**3 - 1716*b**3*d**2*e**4*x + 1287*b**3*d*e**5*x**2 + 6435*b**3*e **6*x**3 - 9152*b**2*c*d**4*e**2 + 4576*b**2*c*d**3*e**3*x - 3432*b**2*c*d **2*e**4*x**2 + 2860*b**2*c*d*e**5*x**3 + 20020*b**2*c*e**6*x**4 + 8320*b* c**2*d**5*e - 4160*b*c**2*d**4*e**2*x + 3120*b*c**2*d**3*e**3*x**2 - 2600* b*c**2*d**2*e**4*x**3 + 2275*b*c**2*d*e**5*x**4 + 20475*b*c**2*e**6*x**5 - 2560*c**3*d**6 + 1280*c**3*d**5*e*x - 960*c**3*d**4*e**2*x**2 + 800*c**3* d**3*e**3*x**3 - 700*c**3*d**2*e**4*x**4 + 630*c**3*d*e**5*x**5 + 6930*c** 3*e**6*x**6))/(45045*e**6)