Integrand size = 32, antiderivative size = 326 \[ \int (a+b x)^2 (c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 (b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{5/2}}{5 d^6}+\frac {2 (b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) (c+d x)^{7/2}}{7 d^6}+\frac {2 \left (a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^{9/2}}{9 d^6}+\frac {2 \left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^{11/2}}{11 d^6}+\frac {2 b (b C d-5 b c D+2 a d D) (c+d x)^{13/2}}{13 d^6}+\frac {2 b^2 D (c+d x)^{15/2}}{15 d^6} \] Output:
2/5*(-a*d+b*c)^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(5/2)/d^6+2/7*(-a*d +b*c)*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b*(2*A*d^3-3*B*c*d^2+4*C*c^2*d-5*D*c^3 ))*(d*x+c)^(7/2)/d^6+2/9*(a^2*d^2*(C*d-3*D*c)-2*a*b*d*(-B*d^2+3*C*c*d-6*D* c^2)+b^2*(A*d^3-3*B*c*d^2+6*C*c^2*d-10*D*c^3))*(d*x+c)^(9/2)/d^6+2/11*(a^2 *d^2*D+2*a*b*d*(C*d-4*D*c)-b^2*(-B*d^2+4*C*c*d-10*D*c^2))*(d*x+c)^(11/2)/d ^6+2/13*b*(C*b*d+2*D*a*d-5*D*b*c)*(d*x+c)^(13/2)/d^6+2/15*b^2*D*(d*x+c)^(1 5/2)/d^6
Time = 0.33 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00 \[ \int (a+b x)^2 (c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 (c+d x)^{5/2} \left (13 a^2 d^2 \left (-48 c^3 D+8 c^2 d (11 C+15 D x)-2 c d^2 (99 B+5 x (22 C+21 D x))+d^3 \left (693 A+5 x \left (99 B+77 C x+63 D x^2\right )\right )\right )+2 a b d \left (384 c^4 D-48 c^3 d (13 C+20 D x)+8 c^2 d^2 (143 B+15 x (13 C+14 D x))+5 d^4 x \left (1287 A+7 x \left (143 B+117 C x+99 D x^2\right )\right )-2 c d^3 \left (1287 A+5 x \left (286 B+273 C x+252 D x^2\right )\right )\right )+b^2 \left (-256 c^5 D+128 c^4 d (3 C+5 D x)+7 d^5 x^2 \left (715 A+585 B x+495 C x^2+429 D x^3\right )-16 c^3 d^2 (39 B+10 x (6 C+7 D x))-10 c d^4 x \left (286 A+21 x \left (13 B+12 C x+11 D x^2\right )\right )+8 c^2 d^3 (143 A+15 x (13 B+14 x (C+D x)))\right )\right )}{45045 d^6} \] Input:
Integrate[(a + b*x)^2*(c + d*x)^(3/2)*(A + B*x + C*x^2 + D*x^3),x]
Output:
(2*(c + d*x)^(5/2)*(13*a^2*d^2*(-48*c^3*D + 8*c^2*d*(11*C + 15*D*x) - 2*c* d^2*(99*B + 5*x*(22*C + 21*D*x)) + d^3*(693*A + 5*x*(99*B + 77*C*x + 63*D* x^2))) + 2*a*b*d*(384*c^4*D - 48*c^3*d*(13*C + 20*D*x) + 8*c^2*d^2*(143*B + 15*x*(13*C + 14*D*x)) + 5*d^4*x*(1287*A + 7*x*(143*B + 117*C*x + 99*D*x^ 2)) - 2*c*d^3*(1287*A + 5*x*(286*B + 273*C*x + 252*D*x^2))) + b^2*(-256*c^ 5*D + 128*c^4*d*(3*C + 5*D*x) + 7*d^5*x^2*(715*A + 585*B*x + 495*C*x^2 + 4 29*D*x^3) - 16*c^3*d^2*(39*B + 10*x*(6*C + 7*D*x)) - 10*c*d^4*x*(286*A + 2 1*x*(13*B + 12*C*x + 11*D*x^2)) + 8*c^2*d^3*(143*A + 15*x*(13*B + 14*x*(C + D*x))))))/(45045*d^6)
Time = 0.64 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2123, 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.
| \(\displaystyle \int (a+b x)^2 (c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {(c+d x)^{7/2} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^5}+\frac {(c+d x)^{9/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^5}+\frac {(c+d x)^{5/2} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^5}+\frac {(c+d x)^{3/2} (a d-b c)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5}+\frac {b (c+d x)^{11/2} (2 a d D-5 b c D+b C d)}{d^5}+\frac {b^2 D (c+d x)^{13/2}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (c+d x)^{9/2} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{9 d^6}+\frac {2 (c+d x)^{11/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{11 d^6}+\frac {2 (c+d x)^{7/2} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{7 d^6}+\frac {2 (c+d x)^{5/2} (b c-a d)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^6}+\frac {2 b (c+d x)^{13/2} (2 a d D-5 b c D+b C d)}{13 d^6}+\frac {2 b^2 D (c+d x)^{15/2}}{15 d^6}\) |
Input:
Int[(a + b*x)^2*(c + d*x)^(3/2)*(A + B*x + C*x^2 + D*x^3),x]
Output:
(2*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(5/2))/(5*d ^6) + (2*(b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B *c*d^2 + 2*A*d^3 - 5*c^3*D))*(c + d*x)^(7/2))/(7*d^6) + (2*(a^2*d^2*(C*d - 3*c*D) - 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c + d*x)^(9/2))/(9*d^6) + (2*(a^2*d^2*D + 2*a*b*d*( C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c + d*x)^(11/2))/(11*d^6 ) + (2*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(13/2))/(13*d^6) + (2*b^2*D *(c + d*x)^(15/2))/(15*d^6)
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Time = 1.02 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {5 \left (\frac {3}{5} D x^{3}+\frac {9}{13} C \,x^{2}+\frac {9}{11} B x +A \right ) x^{2} b^{2}}{9}+\frac {10 \left (\frac {7}{13} D x^{3}+\frac {7}{11} C \,x^{2}+\frac {7}{9} B x +A \right ) x a b}{7}+a^{2} \left (\frac {5}{11} D x^{3}+\frac {5}{9} C \,x^{2}+\frac {5}{7} B x +A \right )\right ) d^{5}-\frac {4 \left (\frac {5 x \left (\frac {21}{26} D x^{3}+\frac {126}{143} C \,x^{2}+\frac {21}{22} B x +A \right ) b^{2}}{9}+a \left (\frac {140}{143} D x^{3}+\frac {35}{33} C \,x^{2}+\frac {10}{9} B x +A \right ) b +\frac {\left (\frac {35}{33} D x^{2}+\frac {10}{9} C x +B \right ) a^{2}}{2}\right ) c \,d^{4}}{7}+\frac {8 \left (\left (\frac {210}{143} D x^{3}+\frac {210}{143} C \,x^{2}+\frac {15}{11} B x +A \right ) b^{2}+2 \left (\frac {210}{143} D x^{2}+\frac {15}{11} C x +B \right ) a b +a^{2} \left (\frac {15 D x}{11}+C \right )\right ) c^{2} d^{3}}{63}-\frac {16 \left (\left (\frac {70}{39} D x^{2}+\frac {20}{13} C x +B \right ) b^{2}+2 a \left (\frac {20 D x}{13}+C \right ) b +D a^{2}\right ) c^{3} d^{2}}{231}+\frac {128 \left (\left (\frac {5 D x}{3}+C \right ) b +2 D a \right ) c^{4} b d}{3003}-\frac {256 D b^{2} c^{5}}{9009}\right ) \left (x d +c \right )^{\frac {5}{2}}}{5 d^{6}}\) | \(285\) |
| derivativedivides | \(\frac {\frac {2 b^{2} D \left (x d +c \right )^{\frac {15}{2}}}{15}+\frac {2 \left (2 b \left (a d -b c \right ) D+b^{2} \left (C d -3 D c \right )\right ) \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a d -b c \right )^{2} D+2 b \left (a d -b c \right ) \left (C d -3 D c \right )+b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a d -b c \right )^{2} \left (C d -3 D c \right )+2 b \left (a d -b c \right ) \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a d -b c \right )^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+2 b \left (a d -b c \right ) \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (x d +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\) | \(320\) |
| default | \(\frac {\frac {2 b^{2} D \left (x d +c \right )^{\frac {15}{2}}}{15}+\frac {2 \left (2 b \left (a d -b c \right ) D+b^{2} \left (C d -3 D c \right )\right ) \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a d -b c \right )^{2} D+2 b \left (a d -b c \right ) \left (C d -3 D c \right )+b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a d -b c \right )^{2} \left (C d -3 D c \right )+2 b \left (a d -b c \right ) \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a d -b c \right )^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+2 b \left (a d -b c \right ) \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (x d +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\) | \(320\) |
| gosper | \(\frac {2 \left (x d +c \right )^{\frac {5}{2}} \left (3003 D x^{5} b^{2} d^{5}+3465 C \,x^{4} b^{2} d^{5}+6930 D x^{4} a b \,d^{5}-2310 D x^{4} b^{2} c \,d^{4}+4095 B \,x^{3} b^{2} d^{5}+8190 C \,x^{3} a b \,d^{5}-2520 C \,x^{3} b^{2} c \,d^{4}+4095 D x^{3} a^{2} d^{5}-5040 D x^{3} a b c \,d^{4}+1680 D x^{3} b^{2} c^{2} d^{3}+5005 A \,x^{2} b^{2} d^{5}+10010 B \,x^{2} a b \,d^{5}-2730 B \,x^{2} b^{2} c \,d^{4}+5005 C \,x^{2} a^{2} d^{5}-5460 C \,x^{2} a b c \,d^{4}+1680 C \,x^{2} b^{2} c^{2} d^{3}-2730 D x^{2} a^{2} c \,d^{4}+3360 D x^{2} a b \,c^{2} d^{3}-1120 D x^{2} b^{2} c^{3} d^{2}+12870 A a b \,d^{5} x -2860 A x \,b^{2} c \,d^{4}+6435 B \,a^{2} d^{5} x -5720 B x a b c \,d^{4}+1560 B \,b^{2} c^{2} d^{3} x -2860 C x \,a^{2} c \,d^{4}+3120 C a b \,c^{2} d^{3} x -960 C \,b^{2} c^{3} d^{2} x +1560 D a^{2} c^{2} d^{3} x -1920 D a b \,c^{3} d^{2} x +640 D b^{2} c^{4} d x +9009 A \,a^{2} d^{5}-5148 A a b c \,d^{4}+1144 A \,b^{2} c^{2} d^{3}-2574 B \,a^{2} c \,d^{4}+2288 B a b \,c^{2} d^{3}-624 B \,b^{2} c^{3} d^{2}+1144 C \,a^{2} c^{2} d^{3}-1248 C a b \,c^{3} d^{2}+384 C \,b^{2} c^{4} d -624 D a^{2} c^{3} d^{2}+768 D a b \,c^{4} d -256 D b^{2} c^{5}\right )}{45045 d^{6}}\) | \(505\) |
| orering | \(\frac {2 \left (x d +c \right )^{\frac {5}{2}} \left (3003 D x^{5} b^{2} d^{5}+3465 C \,x^{4} b^{2} d^{5}+6930 D x^{4} a b \,d^{5}-2310 D x^{4} b^{2} c \,d^{4}+4095 B \,x^{3} b^{2} d^{5}+8190 C \,x^{3} a b \,d^{5}-2520 C \,x^{3} b^{2} c \,d^{4}+4095 D x^{3} a^{2} d^{5}-5040 D x^{3} a b c \,d^{4}+1680 D x^{3} b^{2} c^{2} d^{3}+5005 A \,x^{2} b^{2} d^{5}+10010 B \,x^{2} a b \,d^{5}-2730 B \,x^{2} b^{2} c \,d^{4}+5005 C \,x^{2} a^{2} d^{5}-5460 C \,x^{2} a b c \,d^{4}+1680 C \,x^{2} b^{2} c^{2} d^{3}-2730 D x^{2} a^{2} c \,d^{4}+3360 D x^{2} a b \,c^{2} d^{3}-1120 D x^{2} b^{2} c^{3} d^{2}+12870 A a b \,d^{5} x -2860 A x \,b^{2} c \,d^{4}+6435 B \,a^{2} d^{5} x -5720 B x a b c \,d^{4}+1560 B \,b^{2} c^{2} d^{3} x -2860 C x \,a^{2} c \,d^{4}+3120 C a b \,c^{2} d^{3} x -960 C \,b^{2} c^{3} d^{2} x +1560 D a^{2} c^{2} d^{3} x -1920 D a b \,c^{3} d^{2} x +640 D b^{2} c^{4} d x +9009 A \,a^{2} d^{5}-5148 A a b c \,d^{4}+1144 A \,b^{2} c^{2} d^{3}-2574 B \,a^{2} c \,d^{4}+2288 B a b \,c^{2} d^{3}-624 B \,b^{2} c^{3} d^{2}+1144 C \,a^{2} c^{2} d^{3}-1248 C a b \,c^{3} d^{2}+384 C \,b^{2} c^{4} d -624 D a^{2} c^{3} d^{2}+768 D a b \,c^{4} d -256 D b^{2} c^{5}\right )}{45045 d^{6}}\) | \(505\) |
| trager | \(\frac {2 \left (3003 d^{7} b^{2} D x^{7}+3465 C \,b^{2} d^{7} x^{6}+6930 D a b \,d^{7} x^{6}+3696 D b^{2} c \,d^{6} x^{6}+4095 B \,b^{2} d^{7} x^{5}+8190 C a b \,d^{7} x^{5}+4410 C \,b^{2} c \,d^{6} x^{5}+4095 D a^{2} d^{7} x^{5}+8820 D a b c \,d^{6} x^{5}+63 D b^{2} c^{2} d^{5} x^{5}+5005 A \,b^{2} d^{7} x^{4}+10010 B a b \,d^{7} x^{4}+5460 B \,b^{2} c \,d^{6} x^{4}+5005 C \,a^{2} d^{7} x^{4}+10920 C a b c \,d^{6} x^{4}+105 C \,b^{2} c^{2} d^{5} x^{4}+5460 D a^{2} c \,d^{6} x^{4}+210 D a b \,c^{2} d^{5} x^{4}-70 D b^{2} c^{3} d^{4} x^{4}+12870 A a b \,d^{7} x^{3}+7150 A \,b^{2} c \,d^{6} x^{3}+6435 B \,a^{2} d^{7} x^{3}+14300 B a b c \,d^{6} x^{3}+195 B \,b^{2} c^{2} d^{5} x^{3}+7150 C \,a^{2} c \,d^{6} x^{3}+390 C a b \,c^{2} d^{5} x^{3}-120 C \,b^{2} c^{3} d^{4} x^{3}+195 D a^{2} c^{2} d^{5} x^{3}-240 D a b \,c^{3} d^{4} x^{3}+80 D b^{2} c^{4} d^{3} x^{3}+9009 A \,a^{2} d^{7} x^{2}+20592 A a b c \,d^{6} x^{2}+429 A \,b^{2} c^{2} d^{5} x^{2}+10296 B \,a^{2} c \,d^{6} x^{2}+858 B a b \,c^{2} d^{5} x^{2}-234 B \,b^{2} c^{3} d^{4} x^{2}+429 C \,a^{2} c^{2} d^{5} x^{2}-468 C a b \,c^{3} d^{4} x^{2}+144 C \,b^{2} c^{4} d^{3} x^{2}-234 D a^{2} c^{3} d^{4} x^{2}+288 D a b \,c^{4} d^{3} x^{2}-96 D b^{2} c^{5} d^{2} x^{2}+18018 A \,a^{2} c \,d^{6} x +2574 A a b \,c^{2} d^{5} x -572 A \,b^{2} c^{3} d^{4} x +1287 B \,a^{2} c^{2} d^{5} x -1144 B a b \,c^{3} d^{4} x +312 B \,b^{2} c^{4} d^{3} x -572 C \,a^{2} c^{3} d^{4} x +624 C a b \,c^{4} d^{3} x -192 C \,b^{2} c^{5} d^{2} x +312 D a^{2} c^{4} d^{3} x -384 D a b \,c^{5} d^{2} x +128 D b^{2} c^{6} d x +9009 A \,a^{2} c^{2} d^{5}-5148 A a b \,c^{3} d^{4}+1144 A \,b^{2} c^{4} d^{3}-2574 B \,a^{2} c^{3} d^{4}+2288 B a b \,c^{4} d^{3}-624 B \,b^{2} c^{5} d^{2}+1144 C \,a^{2} c^{4} d^{3}-1248 C a b \,c^{5} d^{2}+384 C \,b^{2} c^{6} d -624 D a^{2} c^{5} d^{2}+768 D a b \,c^{6} d -256 D b^{2} c^{7}\right ) \sqrt {x d +c}}{45045 d^{6}}\) | \(853\) |
Input:
int((b*x+a)^2*(d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOSE)
Output:
2/5*((5/9*(3/5*D*x^3+9/13*C*x^2+9/11*B*x+A)*x^2*b^2+10/7*(7/13*D*x^3+7/11* C*x^2+7/9*B*x+A)*x*a*b+a^2*(5/11*D*x^3+5/9*C*x^2+5/7*B*x+A))*d^5-4/7*(5/9* x*(21/26*D*x^3+126/143*C*x^2+21/22*B*x+A)*b^2+a*(140/143*D*x^3+35/33*C*x^2 +10/9*B*x+A)*b+1/2*(35/33*D*x^2+10/9*C*x+B)*a^2)*c*d^4+8/63*((210/143*D*x^ 3+210/143*C*x^2+15/11*B*x+A)*b^2+2*(210/143*D*x^2+15/11*C*x+B)*a*b+a^2*(15 /11*D*x+C))*c^2*d^3-16/231*((70/39*D*x^2+20/13*C*x+B)*b^2+2*a*(20/13*D*x+C )*b+D*a^2)*c^3*d^2+128/3003*((5/3*D*x+C)*b+2*D*a)*c^4*b*d-256/9009*D*b^2*c ^5)*(d*x+c)^(5/2)/d^6
Leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (302) = 604\).
Time = 0.08 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.89 \[ \int (a+b x)^2 (c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \, {\left (3003 \, D b^{2} d^{7} x^{7} - 256 \, D b^{2} c^{7} + 9009 \, A a^{2} c^{2} d^{5} + 384 \, {\left (2 \, D a b + C b^{2}\right )} c^{6} d - 624 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{5} d^{2} + 1144 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{4} d^{3} - 2574 \, {\left (B a^{2} + 2 \, A a b\right )} c^{3} d^{4} + 231 \, {\left (16 \, D b^{2} c d^{6} + 15 \, {\left (2 \, D a b + C b^{2}\right )} d^{7}\right )} x^{6} + 63 \, {\left (D b^{2} c^{2} d^{5} + 70 \, {\left (2 \, D a b + C b^{2}\right )} c d^{6} + 65 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{7}\right )} x^{5} - 35 \, {\left (2 \, D b^{2} c^{3} d^{4} - 3 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d^{5} - 156 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{6} - 143 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{7}\right )} x^{4} + 5 \, {\left (16 \, D b^{2} c^{4} d^{3} - 24 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d^{4} + 39 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{5} + 1430 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{6} + 1287 \, {\left (B a^{2} + 2 \, A a b\right )} d^{7}\right )} x^{3} - 3 \, {\left (32 \, D b^{2} c^{5} d^{2} - 3003 \, A a^{2} d^{7} - 48 \, {\left (2 \, D a b + C b^{2}\right )} c^{4} d^{3} + 78 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{4} - 143 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{5} - 3432 \, {\left (B a^{2} + 2 \, A a b\right )} c d^{6}\right )} x^{2} + {\left (128 \, D b^{2} c^{6} d + 18018 \, A a^{2} c d^{6} - 192 \, {\left (2 \, D a b + C b^{2}\right )} c^{5} d^{2} + 312 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{4} d^{3} - 572 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{3} d^{4} + 1287 \, {\left (B a^{2} + 2 \, A a b\right )} c^{2} d^{5}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{6}} \] Input:
integrate((b*x+a)^2*(d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas ")
Output:
2/45045*(3003*D*b^2*d^7*x^7 - 256*D*b^2*c^7 + 9009*A*a^2*c^2*d^5 + 384*(2* D*a*b + C*b^2)*c^6*d - 624*(D*a^2 + 2*C*a*b + B*b^2)*c^5*d^2 + 1144*(C*a^2 + 2*B*a*b + A*b^2)*c^4*d^3 - 2574*(B*a^2 + 2*A*a*b)*c^3*d^4 + 231*(16*D*b ^2*c*d^6 + 15*(2*D*a*b + C*b^2)*d^7)*x^6 + 63*(D*b^2*c^2*d^5 + 70*(2*D*a*b + C*b^2)*c*d^6 + 65*(D*a^2 + 2*C*a*b + B*b^2)*d^7)*x^5 - 35*(2*D*b^2*c^3* d^4 - 3*(2*D*a*b + C*b^2)*c^2*d^5 - 156*(D*a^2 + 2*C*a*b + B*b^2)*c*d^6 - 143*(C*a^2 + 2*B*a*b + A*b^2)*d^7)*x^4 + 5*(16*D*b^2*c^4*d^3 - 24*(2*D*a*b + C*b^2)*c^3*d^4 + 39*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^5 + 1430*(C*a^2 + 2 *B*a*b + A*b^2)*c*d^6 + 1287*(B*a^2 + 2*A*a*b)*d^7)*x^3 - 3*(32*D*b^2*c^5* d^2 - 3003*A*a^2*d^7 - 48*(2*D*a*b + C*b^2)*c^4*d^3 + 78*(D*a^2 + 2*C*a*b + B*b^2)*c^3*d^4 - 143*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^5 - 3432*(B*a^2 + 2 *A*a*b)*c*d^6)*x^2 + (128*D*b^2*c^6*d + 18018*A*a^2*c*d^6 - 192*(2*D*a*b + C*b^2)*c^5*d^2 + 312*(D*a^2 + 2*C*a*b + B*b^2)*c^4*d^3 - 572*(C*a^2 + 2*B *a*b + A*b^2)*c^3*d^4 + 1287*(B*a^2 + 2*A*a*b)*c^2*d^5)*x)*sqrt(d*x + c)/d ^6
Time = 1.71 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.97 \[ \int (a+b x)^2 (c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\begin {cases} \frac {2 \left (\frac {D b^{2} \left (c + d x\right )^{\frac {15}{2}}}{15 d^{5}} + \frac {\left (c + d x\right )^{\frac {13}{2}} \left (C b^{2} d + 2 D a b d - 5 D b^{2} c\right )}{13 d^{5}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \left (B b^{2} d^{2} + 2 C a b d^{2} - 4 C b^{2} c d + D a^{2} d^{2} - 8 D a b c d + 10 D b^{2} c^{2}\right )}{11 d^{5}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (A b^{2} d^{3} + 2 B a b d^{3} - 3 B b^{2} c d^{2} + C a^{2} d^{3} - 6 C a b c d^{2} + 6 C b^{2} c^{2} d - 3 D a^{2} c d^{2} + 12 D a b c^{2} d - 10 D b^{2} c^{3}\right )}{9 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (2 A a b d^{4} - 2 A b^{2} c d^{3} + B a^{2} d^{4} - 4 B a b c d^{3} + 3 B b^{2} c^{2} d^{2} - 2 C a^{2} c d^{3} + 6 C a b c^{2} d^{2} - 4 C b^{2} c^{3} d + 3 D a^{2} c^{2} d^{2} - 8 D a b c^{3} d + 5 D b^{2} c^{4}\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (A a^{2} d^{5} - 2 A a b c d^{4} + A b^{2} c^{2} d^{3} - B a^{2} c d^{4} + 2 B a b c^{2} d^{3} - B b^{2} c^{3} d^{2} + C a^{2} c^{2} d^{3} - 2 C a b c^{3} d^{2} + C b^{2} c^{4} d - D a^{2} c^{3} d^{2} + 2 D a b c^{4} d - D b^{2} c^{5}\right )}{5 d^{5}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (A a^{2} x + \frac {D b^{2} x^{6}}{6} + \frac {x^{5} \left (C b^{2} + 2 D a b\right )}{5} + \frac {x^{4} \left (B b^{2} + 2 C a b + D a^{2}\right )}{4} + \frac {x^{3} \left (A b^{2} + 2 B a b + C a^{2}\right )}{3} + \frac {x^{2} \cdot \left (2 A a b + B a^{2}\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**2*(d*x+c)**(3/2)*(D*x**3+C*x**2+B*x+A),x)
Output:
Piecewise((2*(D*b**2*(c + d*x)**(15/2)/(15*d**5) + (c + d*x)**(13/2)*(C*b* *2*d + 2*D*a*b*d - 5*D*b**2*c)/(13*d**5) + (c + d*x)**(11/2)*(B*b**2*d**2 + 2*C*a*b*d**2 - 4*C*b**2*c*d + D*a**2*d**2 - 8*D*a*b*c*d + 10*D*b**2*c**2 )/(11*d**5) + (c + d*x)**(9/2)*(A*b**2*d**3 + 2*B*a*b*d**3 - 3*B*b**2*c*d* *2 + C*a**2*d**3 - 6*C*a*b*c*d**2 + 6*C*b**2*c**2*d - 3*D*a**2*c*d**2 + 12 *D*a*b*c**2*d - 10*D*b**2*c**3)/(9*d**5) + (c + d*x)**(7/2)*(2*A*a*b*d**4 - 2*A*b**2*c*d**3 + B*a**2*d**4 - 4*B*a*b*c*d**3 + 3*B*b**2*c**2*d**2 - 2* C*a**2*c*d**3 + 6*C*a*b*c**2*d**2 - 4*C*b**2*c**3*d + 3*D*a**2*c**2*d**2 - 8*D*a*b*c**3*d + 5*D*b**2*c**4)/(7*d**5) + (c + d*x)**(5/2)*(A*a**2*d**5 - 2*A*a*b*c*d**4 + A*b**2*c**2*d**3 - B*a**2*c*d**4 + 2*B*a*b*c**2*d**3 - B*b**2*c**3*d**2 + C*a**2*c**2*d**3 - 2*C*a*b*c**3*d**2 + C*b**2*c**4*d - D*a**2*c**3*d**2 + 2*D*a*b*c**4*d - D*b**2*c**5)/(5*d**5))/d, Ne(d, 0)), ( c**(3/2)*(A*a**2*x + D*b**2*x**6/6 + x**5*(C*b**2 + 2*D*a*b)/5 + x**4*(B*b **2 + 2*C*a*b + D*a**2)/4 + x**3*(A*b**2 + 2*B*a*b + C*a**2)/3 + x**2*(2*A *a*b + B*a**2)/2), True))
Time = 0.04 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.19 \[ \int (a+b x)^2 (c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \, {\left (3003 \, {\left (d x + c\right )}^{\frac {15}{2}} D b^{2} - 3465 \, {\left (5 \, D b^{2} c - {\left (2 \, D a b + C b^{2}\right )} d\right )} {\left (d x + c\right )}^{\frac {13}{2}} + 4095 \, {\left (10 \, D b^{2} c^{2} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} - 5005 \, {\left (10 \, D b^{2} c^{3} - 6 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{3}\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 6435 \, {\left (5 \, D b^{2} c^{4} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{2} - 2 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{3} + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 9009 \, {\left (D b^{2} c^{5} - A a^{2} d^{5} - {\left (2 \, D a b + C b^{2}\right )} c^{4} d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} + {\left (B a^{2} + 2 \, A a b\right )} c d^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{45045 \, d^{6}} \] Input:
integrate((b*x+a)^2*(d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima ")
Output:
2/45045*(3003*(d*x + c)^(15/2)*D*b^2 - 3465*(5*D*b^2*c - (2*D*a*b + C*b^2) *d)*(d*x + c)^(13/2) + 4095*(10*D*b^2*c^2 - 4*(2*D*a*b + C*b^2)*c*d + (D*a ^2 + 2*C*a*b + B*b^2)*d^2)*(d*x + c)^(11/2) - 5005*(10*D*b^2*c^3 - 6*(2*D* a*b + C*b^2)*c^2*d + 3*(D*a^2 + 2*C*a*b + B*b^2)*c*d^2 - (C*a^2 + 2*B*a*b + A*b^2)*d^3)*(d*x + c)^(9/2) + 6435*(5*D*b^2*c^4 - 4*(2*D*a*b + C*b^2)*c^ 3*d + 3*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^2 - 2*(C*a^2 + 2*B*a*b + A*b^2)*c* d^3 + (B*a^2 + 2*A*a*b)*d^4)*(d*x + c)^(7/2) - 9009*(D*b^2*c^5 - A*a^2*d^5 - (2*D*a*b + C*b^2)*c^4*d + (D*a^2 + 2*C*a*b + B*b^2)*c^3*d^2 - (C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 + (B*a^2 + 2*A*a*b)*c*d^4)*(d*x + c)^(5/2))/d^6
Leaf count of result is larger than twice the leaf count of optimal. 2150 vs. \(2 (302) = 604\).
Time = 0.15 (sec) , antiderivative size = 2150, normalized size of antiderivative = 6.60 \[ \int (a+b x)^2 (c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^2*(d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
Output:
2/45045*(45045*sqrt(d*x + c)*A*a^2*c^2 + 30030*((d*x + c)^(3/2) - 3*sqrt(d *x + c)*c)*A*a^2*c + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*B*a^2*c^2 /d + 30030*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*A*a*b*c^2/d + 3003*(3*(d* x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*A*a^2 + 3003*( 3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*C*a^2*c^2 /d^2 + 6006*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c ^2)*B*a*b*c^2/d^2 + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sq rt(d*x + c)*c^2)*A*b^2*c^2/d^2 + 6006*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3 /2)*c + 15*sqrt(d*x + c)*c^2)*B*a^2*c/d + 12012*(3*(d*x + c)^(5/2) - 10*(d *x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*A*a*b*c/d + 1287*(5*(d*x + c)^(7/2 ) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)* D*a^2*c^2/d^3 + 2574*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*C*a*b*c^2/d^3 + 1287*(5*(d*x + c)^(7 /2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3 )*B*b^2*c^2/d^3 + 2574*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*C*a^2*c/d^2 + 5148*(5*(d*x + c)^(7 /2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3 )*B*a*b*c/d^2 + 2574*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*A*b^2*c/d^2 + 1287*(5*(d*x + c)^(7/2 ) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^...
Timed out. \[ \int (a+b x)^2 (c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int {\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \] Input:
int((a + b*x)^2*(c + d*x)^(3/2)*(A + B*x + C*x^2 + x^3*D),x)
Output:
int((a + b*x)^2*(c + d*x)^(3/2)*(A + B*x + C*x^2 + x^3*D), x)
Time = 0.14 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.48 \[ \int (a+b x)^2 (c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \sqrt {d x +c}\, \left (3003 b^{2} d^{7} x^{7}+6930 a b \,d^{7} x^{6}+7161 b^{2} c \,d^{6} x^{6}+4095 a^{2} d^{7} x^{5}+17010 a b c \,d^{6} x^{5}+4095 b^{3} d^{6} x^{5}+4473 b^{2} c^{2} d^{5} x^{5}+10465 a^{2} c \,d^{6} x^{4}+15015 a \,b^{2} d^{6} x^{4}+11130 a b \,c^{2} d^{5} x^{4}+5460 b^{3} c \,d^{5} x^{4}+35 b^{2} c^{3} d^{4} x^{4}+19305 a^{2} b \,d^{6} x^{3}+7345 a^{2} c^{2} d^{5} x^{3}+21450 a \,b^{2} c \,d^{5} x^{3}+150 a b \,c^{3} d^{4} x^{3}+195 b^{3} c^{2} d^{4} x^{3}-40 b^{2} c^{4} d^{3} x^{3}+9009 a^{3} d^{6} x^{2}+30888 a^{2} b c \,d^{5} x^{2}+195 a^{2} c^{3} d^{4} x^{2}+1287 a \,b^{2} c^{2} d^{4} x^{2}-180 a b \,c^{4} d^{3} x^{2}-234 b^{3} c^{3} d^{3} x^{2}+48 b^{2} c^{5} d^{2} x^{2}+18018 a^{3} c \,d^{5} x +3861 a^{2} b \,c^{2} d^{4} x -260 a^{2} c^{4} d^{3} x -1716 a \,b^{2} c^{3} d^{3} x +240 a b \,c^{5} d^{2} x +312 b^{3} c^{4} d^{2} x -64 b^{2} c^{6} d x +9009 a^{3} c^{2} d^{4}-7722 a^{2} b \,c^{3} d^{3}+520 a^{2} c^{5} d^{2}+3432 a \,b^{2} c^{4} d^{2}-480 a b \,c^{6} d -624 b^{3} c^{5} d +128 b^{2} c^{7}\right )}{45045 d^{5}} \] Input:
int((b*x+a)^2*(d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A),x)
Output:
(2*sqrt(c + d*x)*(9009*a**3*c**2*d**4 + 18018*a**3*c*d**5*x + 9009*a**3*d* *6*x**2 - 7722*a**2*b*c**3*d**3 + 3861*a**2*b*c**2*d**4*x + 30888*a**2*b*c *d**5*x**2 + 19305*a**2*b*d**6*x**3 + 520*a**2*c**5*d**2 - 260*a**2*c**4*d **3*x + 195*a**2*c**3*d**4*x**2 + 7345*a**2*c**2*d**5*x**3 + 10465*a**2*c* d**6*x**4 + 4095*a**2*d**7*x**5 + 3432*a*b**2*c**4*d**2 - 1716*a*b**2*c**3 *d**3*x + 1287*a*b**2*c**2*d**4*x**2 + 21450*a*b**2*c*d**5*x**3 + 15015*a* b**2*d**6*x**4 - 480*a*b*c**6*d + 240*a*b*c**5*d**2*x - 180*a*b*c**4*d**3* x**2 + 150*a*b*c**3*d**4*x**3 + 11130*a*b*c**2*d**5*x**4 + 17010*a*b*c*d** 6*x**5 + 6930*a*b*d**7*x**6 - 624*b**3*c**5*d + 312*b**3*c**4*d**2*x - 234 *b**3*c**3*d**3*x**2 + 195*b**3*c**2*d**4*x**3 + 5460*b**3*c*d**5*x**4 + 4 095*b**3*d**6*x**5 + 128*b**2*c**7 - 64*b**2*c**6*d*x + 48*b**2*c**5*d**2* x**2 - 40*b**2*c**4*d**3*x**3 + 35*b**2*c**3*d**4*x**4 + 4473*b**2*c**2*d* *5*x**5 + 7161*b**2*c*d**6*x**6 + 3003*b**2*d**7*x**7))/(45045*d**5)