Integrand size = 24, antiderivative size = 218 \[ \int (A+B x) \sqrt {d+e x} \left (a+c x^2\right )^2 \, dx=-\frac {2 (B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{3 e^6}+\frac {2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^{5/2}}{5 e^6}-\frac {4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{7/2}}{7 e^6}+\frac {4 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{9/2}}{9 e^6}-\frac {2 c^2 (5 B d-A e) (d+e x)^{11/2}}{11 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6} \] Output:
-2/3*(-A*e+B*d)*(a*e^2+c*d^2)^2*(e*x+d)^(3/2)/e^6+2/5*(a*e^2+c*d^2)*(-4*A* c*d*e+B*a*e^2+5*B*c*d^2)*(e*x+d)^(5/2)/e^6-4/7*c*(-A*a*e^3-3*A*c*d^2*e+3*B *a*d*e^2+5*B*c*d^3)*(e*x+d)^(7/2)/e^6+4/9*c*(-2*A*c*d*e+B*a*e^2+5*B*c*d^2) *(e*x+d)^(9/2)/e^6-2/11*c^2*(-A*e+5*B*d)*(e*x+d)^(11/2)/e^6+2/13*B*c^2*(e* x+d)^(13/2)/e^6
Time = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.98 \[ \int (A+B x) \sqrt {d+e x} \left (a+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{3/2} \left (13 A e \left (1155 a^2 e^4+66 a c e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+B \left (3003 a^2 e^4 (-2 d+3 e x)+286 a c e^2 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )-5 c^2 \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )\right )}{45045 e^6} \] Input:
Integrate[(A + B*x)*Sqrt[d + e*x]*(a + c*x^2)^2,x]
Output:
(2*(d + e*x)^(3/2)*(13*A*e*(1155*a^2*e^4 + 66*a*c*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + c^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)) + B*(3003*a^2*e^4*(-2*d + 3*e*x) + 286*a*c*e^2*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) - 5*c^2*(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))))/(45045 *e^6)
Time = 0.37 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {652, 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.
| \(\displaystyle \int \left (a+c x^2\right )^2 (A+B x) \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {2 c (d+e x)^{7/2} \left (-a B e^2+2 A c d e-5 B c d^2\right )}{e^5}+\frac {(d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^5}+\frac {\sqrt {d+e x} \left (a e^2+c d^2\right )^2 (A e-B d)}{e^5}+\frac {2 c (d+e x)^{5/2} \left (a A e^3-3 a B d e^2+3 A c d^2 e-5 B c d^3\right )}{e^5}+\frac {c^2 (d+e x)^{9/2} (A e-5 B d)}{e^5}+\frac {B c^2 (d+e x)^{11/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 c (d+e x)^{9/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{9 e^6}+\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6}-\frac {4 c (d+e x)^{7/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}-\frac {2 c^2 (d+e x)^{11/2} (5 B d-A e)}{11 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6}\) |
Input:
Int[(A + B*x)*Sqrt[d + e*x]*(a + c*x^2)^2,x]
Output:
(-2*(B*d - A*e)*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2))/(3*e^6) + (2*(c*d^2 + a *e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2)*(d + e*x)^(5/2))/(5*e^6) - (4*c*(5 *B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(7/2))/(7*e^6) + (4*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(9/2))/(9*e^6) - (2*c^2* (5*B*d - A*e)*(d + e*x)^(11/2))/(11*e^6) + (2*B*c^2*(d + e*x)^(13/2))/(13* e^6)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Time = 2.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.79
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (\left (\left (\frac {3}{13} B \,x^{5}+\frac {3}{11} A \,x^{4}\right ) c^{2}+\frac {6 \left (\frac {7 B x}{9}+A \right ) x^{2} a c}{7}+a^{2} \left (\frac {3 B x}{5}+A \right )\right ) e^{5}-\frac {24 \left (\frac {35 \left (\frac {45 B x}{52}+A \right ) x^{3} c^{2}}{99}+a x \left (\frac {5 B x}{6}+A \right ) c +\frac {7 a^{2} B}{12}\right ) d \,e^{4}}{35}+\frac {16 c \,d^{2} \left (\frac {5 x^{2} \left (\frac {35 B x}{39}+A \right ) c}{11}+a \left (B x +A \right )\right ) e^{3}}{35}-\frac {64 c \,d^{3} \left (x \left (\frac {25 B x}{26}+A \right ) c +\frac {11 B a}{6}\right ) e^{2}}{385}+\frac {128 c^{2} d^{4} \left (\frac {15 B x}{13}+A \right ) e}{1155}-\frac {256 B \,c^{2} d^{5}}{3003}\right )}{3 e^{6}}\) | \(172\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -B d \right ) c^{2}-4 B \,c^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-4 \left (A e -B d \right ) c^{2} d +B \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (A e -B d \right ) \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )-4 B \left (a \,e^{2}+c \,d^{2}\right ) c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-4 \left (A e -B d \right ) \left (a \,e^{2}+c \,d^{2}\right ) c d +B \left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (A e -B d \right ) \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(234\) |
| default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -B d \right ) c^{2}-4 B \,c^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-4 \left (A e -B d \right ) c^{2} d +B \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (A e -B d \right ) \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )-4 B \left (a \,e^{2}+c \,d^{2}\right ) c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-4 \left (A e -B d \right ) \left (a \,e^{2}+c \,d^{2}\right ) c d +B \left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (A e -B d \right ) \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(234\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3465 B \,x^{5} c^{2} e^{5}+4095 A \,x^{4} c^{2} e^{5}-3150 B \,x^{4} c^{2} d \,e^{4}-3640 A \,x^{3} c^{2} d \,e^{4}+10010 B \,x^{3} a c \,e^{5}+2800 B \,x^{3} c^{2} d^{2} e^{3}+12870 A \,x^{2} a c \,e^{5}+3120 A \,x^{2} c^{2} d^{2} e^{3}-8580 B \,x^{2} a c d \,e^{4}-2400 B \,x^{2} c^{2} d^{3} e^{2}-10296 A x a c d \,e^{4}-2496 A x \,c^{2} d^{3} e^{2}+9009 B x \,a^{2} e^{5}+6864 B x a c \,d^{2} e^{3}+1920 B x \,c^{2} d^{4} e +15015 A \,a^{2} e^{5}+6864 A a c \,d^{2} e^{3}+1664 A \,c^{2} d^{4} e -6006 B \,a^{2} d \,e^{4}-4576 B a c \,d^{3} e^{2}-1280 B \,c^{2} d^{5}\right )}{45045 e^{6}}\) | \(259\) |
| orering | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3465 B \,x^{5} c^{2} e^{5}+4095 A \,x^{4} c^{2} e^{5}-3150 B \,x^{4} c^{2} d \,e^{4}-3640 A \,x^{3} c^{2} d \,e^{4}+10010 B \,x^{3} a c \,e^{5}+2800 B \,x^{3} c^{2} d^{2} e^{3}+12870 A \,x^{2} a c \,e^{5}+3120 A \,x^{2} c^{2} d^{2} e^{3}-8580 B \,x^{2} a c d \,e^{4}-2400 B \,x^{2} c^{2} d^{3} e^{2}-10296 A x a c d \,e^{4}-2496 A x \,c^{2} d^{3} e^{2}+9009 B x \,a^{2} e^{5}+6864 B x a c \,d^{2} e^{3}+1920 B x \,c^{2} d^{4} e +15015 A \,a^{2} e^{5}+6864 A a c \,d^{2} e^{3}+1664 A \,c^{2} d^{4} e -6006 B \,a^{2} d \,e^{4}-4576 B a c \,d^{3} e^{2}-1280 B \,c^{2} d^{5}\right )}{45045 e^{6}}\) | \(259\) |
| trager | \(\frac {2 \left (3465 B \,c^{2} e^{6} x^{6}+4095 A \,c^{2} e^{6} x^{5}+315 B \,c^{2} d \,e^{5} x^{5}+455 A \,c^{2} d \,e^{5} x^{4}+10010 B a c \,e^{6} x^{4}-350 B \,c^{2} d^{2} e^{4} x^{4}+12870 A a c \,e^{6} x^{3}-520 A \,c^{2} d^{2} e^{4} x^{3}+1430 B a c d \,e^{5} x^{3}+400 B \,c^{2} d^{3} e^{3} x^{3}+2574 A a c d \,e^{5} x^{2}+624 A \,c^{2} d^{3} e^{3} x^{2}+9009 B \,a^{2} e^{6} x^{2}-1716 B a c \,d^{2} e^{4} x^{2}-480 B \,c^{2} d^{4} e^{2} x^{2}+15015 A \,a^{2} e^{6} x -3432 A a c \,d^{2} e^{4} x -832 A \,c^{2} d^{4} e^{2} x +3003 B \,a^{2} d \,e^{5} x +2288 B a c \,d^{3} e^{3} x +640 B \,c^{2} d^{5} e x +15015 A \,a^{2} d \,e^{5}+6864 A a c \,d^{3} e^{3}+1664 A \,c^{2} d^{5} e -6006 B \,a^{2} d^{2} e^{4}-4576 B a c \,d^{4} e^{2}-1280 B \,c^{2} d^{6}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(343\) |
| risch | \(\frac {2 \left (3465 B \,c^{2} e^{6} x^{6}+4095 A \,c^{2} e^{6} x^{5}+315 B \,c^{2} d \,e^{5} x^{5}+455 A \,c^{2} d \,e^{5} x^{4}+10010 B a c \,e^{6} x^{4}-350 B \,c^{2} d^{2} e^{4} x^{4}+12870 A a c \,e^{6} x^{3}-520 A \,c^{2} d^{2} e^{4} x^{3}+1430 B a c d \,e^{5} x^{3}+400 B \,c^{2} d^{3} e^{3} x^{3}+2574 A a c d \,e^{5} x^{2}+624 A \,c^{2} d^{3} e^{3} x^{2}+9009 B \,a^{2} e^{6} x^{2}-1716 B a c \,d^{2} e^{4} x^{2}-480 B \,c^{2} d^{4} e^{2} x^{2}+15015 A \,a^{2} e^{6} x -3432 A a c \,d^{2} e^{4} x -832 A \,c^{2} d^{4} e^{2} x +3003 B \,a^{2} d \,e^{5} x +2288 B a c \,d^{3} e^{3} x +640 B \,c^{2} d^{5} e x +15015 A \,a^{2} d \,e^{5}+6864 A a c \,d^{3} e^{3}+1664 A \,c^{2} d^{5} e -6006 B \,a^{2} d^{2} e^{4}-4576 B a c \,d^{4} e^{2}-1280 B \,c^{2} d^{6}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(343\) |
Input:
int((B*x+A)*(e*x+d)^(1/2)*(c*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
2/3*(e*x+d)^(3/2)*(((3/13*B*x^5+3/11*A*x^4)*c^2+6/7*(7/9*B*x+A)*x^2*a*c+a^ 2*(3/5*B*x+A))*e^5-24/35*(35/99*(45/52*B*x+A)*x^3*c^2+a*x*(5/6*B*x+A)*c+7/ 12*a^2*B)*d*e^4+16/35*c*d^2*(5/11*x^2*(35/39*B*x+A)*c+a*(B*x+A))*e^3-64/38 5*c*d^3*(x*(25/26*B*x+A)*c+11/6*B*a)*e^2+128/1155*c^2*d^4*(15/13*B*x+A)*e- 256/3003*B*c^2*d^5)/e^6
Time = 0.08 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.47 \[ \int (A+B x) \sqrt {d+e x} \left (a+c x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, B c^{2} e^{6} x^{6} - 1280 \, B c^{2} d^{6} + 1664 \, A c^{2} d^{5} e - 4576 \, B a c d^{4} e^{2} + 6864 \, A a c d^{3} e^{3} - 6006 \, B a^{2} d^{2} e^{4} + 15015 \, A a^{2} d e^{5} + 315 \, {\left (B c^{2} d e^{5} + 13 \, A c^{2} e^{6}\right )} x^{5} - 35 \, {\left (10 \, B c^{2} d^{2} e^{4} - 13 \, A c^{2} d e^{5} - 286 \, B a c e^{6}\right )} x^{4} + 10 \, {\left (40 \, B c^{2} d^{3} e^{3} - 52 \, A c^{2} d^{2} e^{4} + 143 \, B a c d e^{5} + 1287 \, A a c e^{6}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{4} e^{2} - 208 \, A c^{2} d^{3} e^{3} + 572 \, B a c d^{2} e^{4} - 858 \, A a c d e^{5} - 3003 \, B a^{2} e^{6}\right )} x^{2} + {\left (640 \, B c^{2} d^{5} e - 832 \, A c^{2} d^{4} e^{2} + 2288 \, B a c d^{3} e^{3} - 3432 \, A a c d^{2} e^{4} + 3003 \, B a^{2} d e^{5} + 15015 \, A a^{2} e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)*(c*x^2+a)^2,x, algorithm="fricas")
Output:
2/45045*(3465*B*c^2*e^6*x^6 - 1280*B*c^2*d^6 + 1664*A*c^2*d^5*e - 4576*B*a *c*d^4*e^2 + 6864*A*a*c*d^3*e^3 - 6006*B*a^2*d^2*e^4 + 15015*A*a^2*d*e^5 + 315*(B*c^2*d*e^5 + 13*A*c^2*e^6)*x^5 - 35*(10*B*c^2*d^2*e^4 - 13*A*c^2*d* e^5 - 286*B*a*c*e^6)*x^4 + 10*(40*B*c^2*d^3*e^3 - 52*A*c^2*d^2*e^4 + 143*B *a*c*d*e^5 + 1287*A*a*c*e^6)*x^3 - 3*(160*B*c^2*d^4*e^2 - 208*A*c^2*d^3*e^ 3 + 572*B*a*c*d^2*e^4 - 858*A*a*c*d*e^5 - 3003*B*a^2*e^6)*x^2 + (640*B*c^2 *d^5*e - 832*A*c^2*d^4*e^2 + 2288*B*a*c*d^3*e^3 - 3432*A*a*c*d^2*e^4 + 300 3*B*a^2*d*e^5 + 15015*A*a^2*e^6)*x)*sqrt(e*x + d)/e^6
Time = 1.00 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.71 \[ \int (A+B x) \sqrt {d+e x} \left (a+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (A c^{2} e - 5 B c^{2} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (- 4 A c^{2} d e + 2 B a c e^{2} + 10 B c^{2} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 A a c e^{3} + 6 A c^{2} d^{2} e - 6 B a c d e^{2} - 10 B c^{2} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 4 A a c d e^{3} - 4 A c^{2} d^{3} e + B a^{2} e^{4} + 6 B a c d^{2} e^{2} + 5 B c^{2} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{2} e^{5} + 2 A a c d^{2} e^{3} + A c^{2} d^{4} e - B a^{2} d e^{4} - 2 B a c d^{3} e^{2} - B c^{2} d^{5}\right )}{3 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (A a^{2} x + \frac {2 A a c x^{3}}{3} + \frac {A c^{2} x^{5}}{5} + \frac {B a^{2} x^{2}}{2} + \frac {B a c x^{4}}{2} + \frac {B c^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((B*x+A)*(e*x+d)**(1/2)*(c*x**2+a)**2,x)
Output:
Piecewise((2*(B*c**2*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(A*c* *2*e - 5*B*c**2*d)/(11*e**5) + (d + e*x)**(9/2)*(-4*A*c**2*d*e + 2*B*a*c*e **2 + 10*B*c**2*d**2)/(9*e**5) + (d + e*x)**(7/2)*(2*A*a*c*e**3 + 6*A*c**2 *d**2*e - 6*B*a*c*d*e**2 - 10*B*c**2*d**3)/(7*e**5) + (d + e*x)**(5/2)*(-4 *A*a*c*d*e**3 - 4*A*c**2*d**3*e + B*a**2*e**4 + 6*B*a*c*d**2*e**2 + 5*B*c* *2*d**4)/(5*e**5) + (d + e*x)**(3/2)*(A*a**2*e**5 + 2*A*a*c*d**2*e**3 + A* c**2*d**4*e - B*a**2*d*e**4 - 2*B*a*c*d**3*e**2 - B*c**2*d**5)/(3*e**5))/e , Ne(e, 0)), (sqrt(d)*(A*a**2*x + 2*A*a*c*x**3/3 + A*c**2*x**5/5 + B*a**2* x**2/2 + B*a*c*x**4/2 + B*c**2*x**6/6), True))
Time = 0.05 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.14 \[ \int (A+B x) \sqrt {d+e x} \left (a+c x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} B c^{2} - 4095 \, {\left (5 \, B c^{2} d - A c^{2} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 10010 \, {\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 12870 \, {\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 15015 \, {\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{6}} \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)*(c*x^2+a)^2,x, algorithm="maxima")
Output:
2/45045*(3465*(e*x + d)^(13/2)*B*c^2 - 4095*(5*B*c^2*d - A*c^2*e)*(e*x + d )^(11/2) + 10010*(5*B*c^2*d^2 - 2*A*c^2*d*e + B*a*c*e^2)*(e*x + d)^(9/2) - 12870*(5*B*c^2*d^3 - 3*A*c^2*d^2*e + 3*B*a*c*d*e^2 - A*a*c*e^3)*(e*x + d) ^(7/2) + 9009*(5*B*c^2*d^4 - 4*A*c^2*d^3*e + 6*B*a*c*d^2*e^2 - 4*A*a*c*d*e ^3 + B*a^2*e^4)*(e*x + d)^(5/2) - 15015*(B*c^2*d^5 - A*c^2*d^4*e + 2*B*a*c *d^3*e^2 - 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 - A*a^2*e^5)*(e*x + d)^(3/2))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (194) = 388\).
Time = 0.13 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.90 \[ \int (A+B x) \sqrt {d+e x} \left (a+c x^2\right )^2 \, dx =\text {Too large to display} \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)*(c*x^2+a)^2,x, algorithm="giac")
Output:
2/45045*(45045*sqrt(e*x + d)*A*a^2*d + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^2 + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a^2*d/e + 6 006*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a* c*d/e^2 + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d )*d^2)*B*a^2/e + 2574*(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*a*c*d/e^3 + 2574*(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*a*c/e^2 + 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)*A*c^2*d/e^ 4 + 286*(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*a*c/e^3 + 65*(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/e^5 + 65*(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*sq rt(e*x + d)*d^5)*A*c^2/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 + 3003*sqrt(e*x + d)*d^6)*B*c^2/e^ 5)/e
Time = 0.06 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.90 \[ \int (A+B x) \sqrt {d+e x} \left (a+c x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^{9/2}\,\left (20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e+4\,B\,a\,c\,e^2\right )}{9\,e^6}+\frac {4\,c\,{\left (d+e\,x\right )}^{7/2}\,\left (-5\,B\,c\,d^3+3\,A\,c\,d^2\,e-3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{7\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {2\,\left (c\,d^2+a\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (5\,B\,c\,d^2-4\,A\,c\,d\,e+B\,a\,e^2\right )}{5\,e^6}+\frac {2\,c^2\,\left (A\,e-5\,B\,d\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,\left (A\,e-B\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6} \] Input:
int((a + c*x^2)^2*(A + B*x)*(d + e*x)^(1/2),x)
Output:
((d + e*x)^(9/2)*(20*B*c^2*d^2 + 4*B*a*c*e^2 - 8*A*c^2*d*e))/(9*e^6) + (4* c*(d + e*x)^(7/2)*(A*a*e^3 - 5*B*c*d^3 - 3*B*a*d*e^2 + 3*A*c*d^2*e))/(7*e^ 6) + (2*B*c^2*(d + e*x)^(13/2))/(13*e^6) + (2*(a*e^2 + c*d^2)*(d + e*x)^(5 /2)*(B*a*e^2 + 5*B*c*d^2 - 4*A*c*d*e))/(5*e^6) + (2*c^2*(A*e - 5*B*d)*(d + e*x)^(11/2))/(11*e^6) + (2*(a*e^2 + c*d^2)^2*(A*e - B*d)*(d + e*x)^(3/2)) /(3*e^6)
Time = 0.26 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.57 \[ \int (A+B x) \sqrt {d+e x} \left (a+c x^2\right )^2 \, dx=\frac {2 \sqrt {e x +d}\, \left (3465 b \,c^{2} e^{6} x^{6}+4095 a \,c^{2} e^{6} x^{5}+315 b \,c^{2} d \,e^{5} x^{5}+10010 a b c \,e^{6} x^{4}+455 a \,c^{2} d \,e^{5} x^{4}-350 b \,c^{2} d^{2} e^{4} x^{4}+12870 a^{2} c \,e^{6} x^{3}+1430 a b c d \,e^{5} x^{3}-520 a \,c^{2} d^{2} e^{4} x^{3}+400 b \,c^{2} d^{3} e^{3} x^{3}+9009 a^{2} b \,e^{6} x^{2}+2574 a^{2} c d \,e^{5} x^{2}-1716 a b c \,d^{2} e^{4} x^{2}+624 a \,c^{2} d^{3} e^{3} x^{2}-480 b \,c^{2} d^{4} e^{2} x^{2}+15015 a^{3} e^{6} x +3003 a^{2} b d \,e^{5} x -3432 a^{2} c \,d^{2} e^{4} x +2288 a b c \,d^{3} e^{3} x -832 a \,c^{2} d^{4} e^{2} x +640 b \,c^{2} d^{5} e x +15015 a^{3} d \,e^{5}-6006 a^{2} b \,d^{2} e^{4}+6864 a^{2} c \,d^{3} e^{3}-4576 a b c \,d^{4} e^{2}+1664 a \,c^{2} d^{5} e -1280 b \,c^{2} d^{6}\right )}{45045 e^{6}} \] Input:
int((B*x+A)*(e*x+d)^(1/2)*(c*x^2+a)^2,x)
Output:
(2*sqrt(d + e*x)*(15015*a**3*d*e**5 + 15015*a**3*e**6*x - 6006*a**2*b*d**2 *e**4 + 3003*a**2*b*d*e**5*x + 9009*a**2*b*e**6*x**2 + 6864*a**2*c*d**3*e* *3 - 3432*a**2*c*d**2*e**4*x + 2574*a**2*c*d*e**5*x**2 + 12870*a**2*c*e**6 *x**3 - 4576*a*b*c*d**4*e**2 + 2288*a*b*c*d**3*e**3*x - 1716*a*b*c*d**2*e* *4*x**2 + 1430*a*b*c*d*e**5*x**3 + 10010*a*b*c*e**6*x**4 + 1664*a*c**2*d** 5*e - 832*a*c**2*d**4*e**2*x + 624*a*c**2*d**3*e**3*x**2 - 520*a*c**2*d**2 *e**4*x**3 + 455*a*c**2*d*e**5*x**4 + 4095*a*c**2*e**6*x**5 - 1280*b*c**2* d**6 + 640*b*c**2*d**5*e*x - 480*b*c**2*d**4*e**2*x**2 + 400*b*c**2*d**3*e **3*x**3 - 350*b*c**2*d**2*e**4*x**4 + 315*b*c**2*d*e**5*x**5 + 3465*b*c** 2*e**6*x**6))/(45045*e**6)