\(\int (a+b x)^2 \sqrt {c+d x} (A+B x+C x^2+D x^3) \, dx\) [56]

Optimal result

Integrand size = 32, antiderivative size = 326 \[ \int (a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 (b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{3/2}}{3 d^6}+\frac {2 (b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) (c+d x)^{5/2}}{5 d^6}+\frac {2 \left (a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^{7/2}}{7 d^6}+\frac {2 \left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^{9/2}}{9 d^6}+\frac {2 b (b C d-5 b c D+2 a d D) (c+d x)^{11/2}}{11 d^6}+\frac {2 b^2 D (c+d x)^{13/2}}{13 d^6} \] Output:

2/3*(-a*d+b*c)^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(3/2)/d^6+2/5*(-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)^(5/2)/d^6+2/7*(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)^(7/2)/d^6+2/9*(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)^(9/2)/d^6 
+2/11*b*(C*b*d+2*D*a*d-5*D*b*c)*(d*x+c)^(11/2)/d^6+2/13*b^2*D*(d*x+c)^(13/ 
2)/d^6
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.99 \[ \int (a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 (c+d x)^{3/2} \left (143 a^2 d^2 \left (-16 c^3 D+24 c^2 d (C+D x)-6 c d^2 (7 B+x (6 C+5 D x))+d^3 (105 A+x (63 B+5 x (9 C+7 D x)))\right )+26 a b d \left (128 c^4 D-16 c^3 d (11 C+12 D x)+24 c^2 d^2 (11 B+x (11 C+10 D x))+d^4 x (693 A+5 x (99 B+7 x (11 C+9 D x)))-2 c d^3 (231 A+x (198 B+5 x (33 C+28 D x)))\right )+b^2 \left (-1280 c^5 D+128 c^4 d (13 C+15 D x)-16 c^3 d^2 (143 B+6 x (26 C+25 D x))+5 d^5 x^2 \left (1287 A+7 x \left (143 B+117 C x+99 D x^2\right )\right )+8 c^2 d^3 \left (429 A+x \left (429 B+390 C x+350 D x^2\right )\right )-2 c d^4 x (2574 A+5 x (429 B+7 x (52 C+45 D x)))\right )\right )}{45045 d^6} \] Input:

Integrate[(a + b*x)^2*Sqrt[c + d*x]*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

(2*(c + d*x)^(3/2)*(143*a^2*d^2*(-16*c^3*D + 24*c^2*d*(C + D*x) - 6*c*d^2* 
(7*B + x*(6*C + 5*D*x)) + d^3*(105*A + x*(63*B + 5*x*(9*C + 7*D*x)))) + 26 
*a*b*d*(128*c^4*D - 16*c^3*d*(11*C + 12*D*x) + 24*c^2*d^2*(11*B + x*(11*C 
+ 10*D*x)) + d^4*x*(693*A + 5*x*(99*B + 7*x*(11*C + 9*D*x))) - 2*c*d^3*(23 
1*A + x*(198*B + 5*x*(33*C + 28*D*x)))) + b^2*(-1280*c^5*D + 128*c^4*d*(13 
*C + 15*D*x) - 16*c^3*d^2*(143*B + 6*x*(26*C + 25*D*x)) + 5*d^5*x^2*(1287* 
A + 7*x*(143*B + 117*C*x + 99*D*x^2)) + 8*c^2*d^3*(429*A + x*(429*B + 390* 
C*x + 350*D*x^2)) - 2*c*d^4*x*(2574*A + 5*x*(429*B + 7*x*(52*C + 45*D*x))) 
)))/(45045*d^6)
 

Rubi [A] (verified)

Time = 0.60 (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.

\(\displaystyle \int (a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx\)

\(\Big \downarrow \) 2123

\(\displaystyle \int \left (\frac {(c+d x)^{5/2} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^5}+\frac {(c+d x)^{7/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^5}+\frac {(c+d x)^{3/2} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^5}+\frac {\sqrt {c+d x} (a d-b c)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5}+\frac {b (c+d x)^{9/2} (2 a d D-5 b c D+b C d)}{d^5}+\frac {b^2 D (c+d x)^{11/2}}{d^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 (c+d x)^{7/2} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{7 d^6}+\frac {2 (c+d x)^{9/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{9 d^6}+\frac {2 (c+d x)^{5/2} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{5 d^6}+\frac {2 (c+d x)^{3/2} (b c-a d)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^6}+\frac {2 b (c+d x)^{11/2} (2 a d D-5 b c D+b C d)}{11 d^6}+\frac {2 b^2 D (c+d x)^{13/2}}{13 d^6}\)

Input:

Int[(a + b*x)^2*Sqrt[c + d*x]*(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)^(3/2))/(3*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)^(5/2))/(5*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)^(7/2))/(7*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)^(9/2))/(9*d^6) 
+ (2*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(11/2))/(11*d^6) + (2*b^2*D*( 
c + d*x)^(13/2))/(13*d^6)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2123

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])
 

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\frac {2 \left (x d +c \right )^{\frac {3}{2}} \left (\left (\frac {3 \left (\frac {7}{13} D x^{3}+\frac {7}{11} C \,x^{2}+\frac {7}{9} B x +A \right ) x^{2} b^{2}}{7}+\frac {6 x \left (\frac {5}{11} D x^{3}+\frac {5}{9} C \,x^{2}+\frac {5}{7} B x +A \right ) a b}{5}+a^{2} \left (\frac {1}{3} D x^{3}+\frac {3}{7} C \,x^{2}+\frac {3}{5} B x +A \right )\right ) d^{5}-\frac {4 c \left (\frac {3 \left (\frac {175}{286} D x^{3}+\frac {70}{99} C \,x^{2}+\frac {5}{6} B x +A \right ) x \,b^{2}}{7}+a \left (\frac {20}{33} D x^{3}+\frac {5}{7} C \,x^{2}+\frac {6}{7} B x +A \right ) b +\frac {\left (\frac {5}{7} D x^{2}+\frac {6}{7} C x +B \right ) a^{2}}{2}\right ) d^{4}}{5}+\frac {8 \left (\left (\frac {350}{429} D x^{3}+\frac {10}{11} C \,x^{2}+B x +A \right ) b^{2}+2 \left (\frac {10}{11} D x^{2}+C x +B \right ) a b +a^{2} \left (D x +C \right )\right ) c^{2} d^{3}}{35}-\frac {16 c^{3} \left (\left (\frac {150}{143} D x^{2}+\frac {12}{11} C x +B \right ) b^{2}+2 \left (\frac {12 D x}{11}+C \right ) a b +D a^{2}\right ) d^{2}}{105}+\frac {128 c^{4} b \left (\left (\frac {15 D x}{13}+C \right ) b +2 D a \right ) d}{1155}-\frac {256 D b^{2} c^{5}}{3003}\right )}{3 d^{6}}\)	\(282\)
derivativedivides	\(\frac {\frac {2 b^{2} D \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 b \left (a d -b c \right ) D+b^{2} \left (C d -3 D c \right )\right ) \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a d -b c \right )^{2} D+2 b \left (a d -b c \right ) \left (C d -3 D c \right )+b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (x d +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a d -b c \right )^{2} \left (C d -3 D c \right )+2 b \left (a d -b c \right ) \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a d -b c \right )^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+2 b \left (a d -b c \right ) \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right )^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (x d +c \right )^{\frac {3}{2}}}{3}}{d^{6}}\)	\(320\)
default	\(\frac {\frac {2 b^{2} D \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 b \left (a d -b c \right ) D+b^{2} \left (C d -3 D c \right )\right ) \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a d -b c \right )^{2} D+2 b \left (a d -b c \right ) \left (C d -3 D c \right )+b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (x d +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a d -b c \right )^{2} \left (C d -3 D c \right )+2 b \left (a d -b c \right ) \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a d -b c \right )^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+2 b \left (a d -b c \right ) \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right )^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (x d +c \right )^{\frac {3}{2}}}{3}}{d^{6}}\)	\(320\)
gosper	\(\frac {2 \left (x d +c \right )^{\frac {3}{2}} \left (3465 D x^{5} b^{2} d^{5}+4095 C \,x^{4} b^{2} d^{5}+8190 D x^{4} a b \,d^{5}-3150 D x^{4} b^{2} c \,d^{4}+5005 B \,x^{3} b^{2} d^{5}+10010 C \,x^{3} a b \,d^{5}-3640 C \,x^{3} b^{2} c \,d^{4}+5005 D x^{3} a^{2} d^{5}-7280 D x^{3} a b c \,d^{4}+2800 D x^{3} b^{2} c^{2} d^{3}+6435 A \,x^{2} b^{2} d^{5}+12870 B \,x^{2} a b \,d^{5}-4290 B \,x^{2} b^{2} c \,d^{4}+6435 C \,x^{2} a^{2} d^{5}-8580 C \,x^{2} a b c \,d^{4}+3120 C \,x^{2} b^{2} c^{2} d^{3}-4290 D x^{2} a^{2} c \,d^{4}+6240 D x^{2} a b \,c^{2} d^{3}-2400 D x^{2} b^{2} c^{3} d^{2}+18018 A a b \,d^{5} x -5148 A x \,b^{2} c \,d^{4}+9009 B \,a^{2} d^{5} x -10296 B x a b c \,d^{4}+3432 B \,b^{2} c^{2} d^{3} x -5148 C x \,a^{2} c \,d^{4}+6864 C a b \,c^{2} d^{3} x -2496 C \,b^{2} c^{3} d^{2} x +3432 D a^{2} c^{2} d^{3} x -4992 D a b \,c^{3} d^{2} x +1920 D b^{2} c^{4} d x +15015 A \,a^{2} d^{5}-12012 A a b c \,d^{4}+3432 A \,b^{2} c^{2} d^{3}-6006 B \,a^{2} c \,d^{4}+6864 B a b \,c^{2} d^{3}-2288 B \,b^{2} c^{3} d^{2}+3432 C \,a^{2} c^{2} d^{3}-4576 C a b \,c^{3} d^{2}+1664 C \,b^{2} c^{4} d -2288 D a^{2} c^{3} d^{2}+3328 D a b \,c^{4} d -1280 D b^{2} c^{5}\right )}{45045 d^{6}}\)	\(505\)
orering	\(\frac {2 \left (x d +c \right )^{\frac {3}{2}} \left (3465 D x^{5} b^{2} d^{5}+4095 C \,x^{4} b^{2} d^{5}+8190 D x^{4} a b \,d^{5}-3150 D x^{4} b^{2} c \,d^{4}+5005 B \,x^{3} b^{2} d^{5}+10010 C \,x^{3} a b \,d^{5}-3640 C \,x^{3} b^{2} c \,d^{4}+5005 D x^{3} a^{2} d^{5}-7280 D x^{3} a b c \,d^{4}+2800 D x^{3} b^{2} c^{2} d^{3}+6435 A \,x^{2} b^{2} d^{5}+12870 B \,x^{2} a b \,d^{5}-4290 B \,x^{2} b^{2} c \,d^{4}+6435 C \,x^{2} a^{2} d^{5}-8580 C \,x^{2} a b c \,d^{4}+3120 C \,x^{2} b^{2} c^{2} d^{3}-4290 D x^{2} a^{2} c \,d^{4}+6240 D x^{2} a b \,c^{2} d^{3}-2400 D x^{2} b^{2} c^{3} d^{2}+18018 A a b \,d^{5} x -5148 A x \,b^{2} c \,d^{4}+9009 B \,a^{2} d^{5} x -10296 B x a b c \,d^{4}+3432 B \,b^{2} c^{2} d^{3} x -5148 C x \,a^{2} c \,d^{4}+6864 C a b \,c^{2} d^{3} x -2496 C \,b^{2} c^{3} d^{2} x +3432 D a^{2} c^{2} d^{3} x -4992 D a b \,c^{3} d^{2} x +1920 D b^{2} c^{4} d x +15015 A \,a^{2} d^{5}-12012 A a b c \,d^{4}+3432 A \,b^{2} c^{2} d^{3}-6006 B \,a^{2} c \,d^{4}+6864 B a b \,c^{2} d^{3}-2288 B \,b^{2} c^{3} d^{2}+3432 C \,a^{2} c^{2} d^{3}-4576 C a b \,c^{3} d^{2}+1664 C \,b^{2} c^{4} d -2288 D a^{2} c^{3} d^{2}+3328 D a b \,c^{4} d -1280 D b^{2} c^{5}\right )}{45045 d^{6}}\)	\(505\)
trager	\(\frac {2 \left (3465 b^{2} D d^{6} x^{6}+4095 C \,b^{2} d^{6} x^{5}+8190 D a b \,d^{6} x^{5}+315 D b^{2} c \,d^{5} x^{5}+5005 B \,b^{2} d^{6} x^{4}+10010 C a b \,d^{6} x^{4}+455 C \,b^{2} c \,d^{5} x^{4}+5005 D a^{2} d^{6} x^{4}+910 D a b c \,d^{5} x^{4}-350 D b^{2} c^{2} d^{4} x^{4}+6435 A \,b^{2} d^{6} x^{3}+12870 B a b \,d^{6} x^{3}+715 B \,b^{2} c \,d^{5} x^{3}+6435 C \,a^{2} d^{6} x^{3}+1430 C a b c \,d^{5} x^{3}-520 C \,b^{2} c^{2} d^{4} x^{3}+715 D a^{2} c \,d^{5} x^{3}-1040 D a b \,c^{2} d^{4} x^{3}+400 D b^{2} c^{3} d^{3} x^{3}+18018 A a b \,d^{6} x^{2}+1287 A \,b^{2} c \,d^{5} x^{2}+9009 B \,a^{2} d^{6} x^{2}+2574 B a b c \,d^{5} x^{2}-858 B \,b^{2} c^{2} d^{4} x^{2}+1287 C \,a^{2} c \,d^{5} x^{2}-1716 C a b \,c^{2} d^{4} x^{2}+624 C \,b^{2} c^{3} d^{3} x^{2}-858 D a^{2} c^{2} d^{4} x^{2}+1248 D a b \,c^{3} d^{3} x^{2}-480 D b^{2} c^{4} d^{2} x^{2}+15015 A \,a^{2} d^{6} x +6006 A a b c \,d^{5} x -1716 A \,b^{2} c^{2} d^{4} x +3003 B \,a^{2} c \,d^{5} x -3432 B a b \,c^{2} d^{4} x +1144 B \,b^{2} c^{3} d^{3} x -1716 C \,a^{2} c^{2} d^{4} x +2288 C a b \,c^{3} d^{3} x -832 C \,b^{2} c^{4} d^{2} x +1144 D a^{2} c^{3} d^{3} x -1664 D a b \,c^{4} d^{2} x +640 D b^{2} c^{5} d x +15015 A \,a^{2} c \,d^{5}-12012 A a b \,c^{2} d^{4}+3432 A \,b^{2} c^{3} d^{3}-6006 B \,a^{2} c^{2} d^{4}+6864 B a b \,c^{3} d^{3}-2288 B \,b^{2} c^{4} d^{2}+3432 C \,a^{2} c^{3} d^{3}-4576 C a b \,c^{4} d^{2}+1664 C \,b^{2} c^{5} d -2288 D a^{2} c^{4} d^{2}+3328 D a b \,c^{5} d -1280 D b^{2} c^{6}\right ) \sqrt {x d +c}}{45045 d^{6}}\)	\(677\)

Input:

int((b*x+a)^2*(d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOSE)
 

Output:

2/3*(d*x+c)^(3/2)*((3/7*(7/13*D*x^3+7/11*C*x^2+7/9*B*x+A)*x^2*b^2+6/5*x*(5 
/11*D*x^3+5/9*C*x^2+5/7*B*x+A)*a*b+a^2*(1/3*D*x^3+3/7*C*x^2+3/5*B*x+A))*d^ 
5-4/5*c*(3/7*(175/286*D*x^3+70/99*C*x^2+5/6*B*x+A)*x*b^2+a*(20/33*D*x^3+5/ 
7*C*x^2+6/7*B*x+A)*b+1/2*(5/7*D*x^2+6/7*C*x+B)*a^2)*d^4+8/35*((350/429*D*x 
^3+10/11*C*x^2+B*x+A)*b^2+2*(10/11*D*x^2+C*x+B)*a*b+a^2*(D*x+C))*c^2*d^3-1 
6/105*c^3*((150/143*D*x^2+12/11*C*x+B)*b^2+2*(12/11*D*x+C)*a*b+D*a^2)*d^2+ 
128/1155*c^4*b*((15/13*D*x+C)*b+2*D*a)*d-256/3003*D*b^2*c^5)/d^6
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.53 \[ \int (a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \, {\left (3465 \, D b^{2} d^{6} x^{6} - 1280 \, D b^{2} c^{6} + 15015 \, A a^{2} c d^{5} + 1664 \, {\left (2 \, D a b + C b^{2}\right )} c^{5} d - 2288 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{4} d^{2} + 3432 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{3} d^{3} - 6006 \, {\left (B a^{2} + 2 \, A a b\right )} c^{2} d^{4} + 315 \, {\left (D b^{2} c d^{5} + 13 \, {\left (2 \, D a b + C b^{2}\right )} d^{6}\right )} x^{5} - 35 \, {\left (10 \, D b^{2} c^{2} d^{4} - 13 \, {\left (2 \, D a b + C b^{2}\right )} c d^{5} - 143 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{6}\right )} x^{4} + 5 \, {\left (80 \, D b^{2} c^{3} d^{3} - 104 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d^{4} + 143 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{5} + 1287 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{6}\right )} x^{3} - 3 \, {\left (160 \, D b^{2} c^{4} d^{2} - 208 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d^{3} + 286 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{4} - 429 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{5} - 3003 \, {\left (B a^{2} + 2 \, A a b\right )} d^{6}\right )} x^{2} + {\left (640 \, D b^{2} c^{5} d + 15015 \, A a^{2} d^{6} - 832 \, {\left (2 \, D a b + C b^{2}\right )} c^{4} d^{2} + 1144 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{3} - 1716 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{4} + 3003 \, {\left (B a^{2} + 2 \, A a b\right )} c d^{5}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{6}} \] Input:

integrate((b*x+a)^2*(d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas 
")
 

Output:

2/45045*(3465*D*b^2*d^6*x^6 - 1280*D*b^2*c^6 + 15015*A*a^2*c*d^5 + 1664*(2 
*D*a*b + C*b^2)*c^5*d - 2288*(D*a^2 + 2*C*a*b + B*b^2)*c^4*d^2 + 3432*(C*a 
^2 + 2*B*a*b + A*b^2)*c^3*d^3 - 6006*(B*a^2 + 2*A*a*b)*c^2*d^4 + 315*(D*b^ 
2*c*d^5 + 13*(2*D*a*b + C*b^2)*d^6)*x^5 - 35*(10*D*b^2*c^2*d^4 - 13*(2*D*a 
*b + C*b^2)*c*d^5 - 143*(D*a^2 + 2*C*a*b + B*b^2)*d^6)*x^4 + 5*(80*D*b^2*c 
^3*d^3 - 104*(2*D*a*b + C*b^2)*c^2*d^4 + 143*(D*a^2 + 2*C*a*b + B*b^2)*c*d 
^5 + 1287*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*x^3 - 3*(160*D*b^2*c^4*d^2 - 208* 
(2*D*a*b + C*b^2)*c^3*d^3 + 286*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^4 - 429*(C 
*a^2 + 2*B*a*b + A*b^2)*c*d^5 - 3003*(B*a^2 + 2*A*a*b)*d^6)*x^2 + (640*D*b 
^2*c^5*d + 15015*A*a^2*d^6 - 832*(2*D*a*b + C*b^2)*c^4*d^2 + 1144*(D*a^2 + 
 2*C*a*b + B*b^2)*c^3*d^3 - 1716*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^4 + 3003* 
(B*a^2 + 2*A*a*b)*c*d^5)*x)*sqrt(d*x + c)/d^6
 

Sympy [A] (verification not implemented)

Time = 1.73 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.97 \[ \int (a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\begin {cases} \frac {2 \left (\frac {D b^{2} \left (c + d x\right )^{\frac {13}{2}}}{13 d^{5}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \left (C b^{2} d + 2 D a b d - 5 D b^{2} c\right )}{11 d^{5}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (B b^{2} d^{2} + 2 C a b d^{2} - 4 C b^{2} c d + D a^{2} d^{2} - 8 D a b c d + 10 D b^{2} c^{2}\right )}{9 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (A b^{2} d^{3} + 2 B a b d^{3} - 3 B b^{2} c d^{2} + C a^{2} d^{3} - 6 C a b c d^{2} + 6 C b^{2} c^{2} d - 3 D a^{2} c d^{2} + 12 D a b c^{2} d - 10 D b^{2} c^{3}\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (2 A a b d^{4} - 2 A b^{2} c d^{3} + B a^{2} d^{4} - 4 B a b c d^{3} + 3 B b^{2} c^{2} d^{2} - 2 C a^{2} c d^{3} + 6 C a b c^{2} d^{2} - 4 C b^{2} c^{3} d + 3 D a^{2} c^{2} d^{2} - 8 D a b c^{3} d + 5 D b^{2} c^{4}\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (A a^{2} d^{5} - 2 A a b c d^{4} + A b^{2} c^{2} d^{3} - B a^{2} c d^{4} + 2 B a b c^{2} d^{3} - B b^{2} c^{3} d^{2} + C a^{2} c^{2} d^{3} - 2 C a b c^{3} d^{2} + C b^{2} c^{4} d - D a^{2} c^{3} d^{2} + 2 D a b c^{4} d - D b^{2} c^{5}\right )}{3 d^{5}}\right )}{d} & \text {for}\: d \neq 0 \\\sqrt {c} \left (A a^{2} x + \frac {D b^{2} x^{6}}{6} + \frac {x^{5} \left (C b^{2} + 2 D a b\right )}{5} + \frac {x^{4} \left (B b^{2} + 2 C a b + D a^{2}\right )}{4} + \frac {x^{3} \left (A b^{2} + 2 B a b + C a^{2}\right )}{3} + \frac {x^{2} \cdot \left (2 A a b + B a^{2}\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((b*x+a)**2*(d*x+c)**(1/2)*(D*x**3+C*x**2+B*x+A),x)
 

Output:

Piecewise((2*(D*b**2*(c + d*x)**(13/2)/(13*d**5) + (c + d*x)**(11/2)*(C*b* 
*2*d + 2*D*a*b*d - 5*D*b**2*c)/(11*d**5) + (c + d*x)**(9/2)*(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) 
/(9*d**5) + (c + d*x)**(7/2)*(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)/(7*d**5) + (c + d*x)**(5/2)*(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)/(5*d**5) + (c + d*x)**(3/2)*(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)/(3*d**5))/d, Ne(d, 0)), (sq 
rt(c)*(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))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.19 \[ \int (a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \, {\left (3465 \, {\left (d x + c\right )}^{\frac {13}{2}} D b^{2} - 4095 \, {\left (5 \, D b^{2} c - {\left (2 \, D a b + C b^{2}\right )} d\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 5005 \, {\left (10 \, D b^{2} c^{2} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 6435 \, {\left (10 \, D b^{2} c^{3} - 6 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 9009 \, {\left (5 \, D b^{2} c^{4} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{2} - 2 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{3} + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}} - 15015 \, {\left (D b^{2} c^{5} - A a^{2} d^{5} - {\left (2 \, D a b + C b^{2}\right )} c^{4} d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} + {\left (B a^{2} + 2 \, A a b\right )} c d^{4}\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{45045 \, d^{6}} \] Input:

integrate((b*x+a)^2*(d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima 
")
 

Output:

2/45045*(3465*(d*x + c)^(13/2)*D*b^2 - 4095*(5*D*b^2*c - (2*D*a*b + C*b^2) 
*d)*(d*x + c)^(11/2) + 5005*(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)^(9/2) - 6435*(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)^(7/2) + 9009*(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)^(5/2) - 15015*(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)^(3/2))/d^6
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1269 vs. \(2 (302) = 604\).

Time = 0.13 (sec) , antiderivative size = 1269, normalized size of antiderivative = 3.89 \[ \int (a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^2*(d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
 

Output:

2/45045*(45045*sqrt(d*x + c)*A*a^2*c + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x 
 + c)*c)*A*a^2 + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*B*a^2*c/d + 3 
0030*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*A*a*b*c/d + 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/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/d^2 
 + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)* 
A*b^2*c/d^2 + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x 
 + c)*c^2)*B*a^2/d + 6006*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*s 
qrt(d*x + c)*c^2)*A*a*b/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/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/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/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)*C*a^2/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)*B*a*b/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 
^3)*A*b^2/d^2 + 286*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x 
 + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*D*a*b*c 
/d^4 + 143*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^...
 

Mupad [B] (verification not implemented)

Time = 3.63 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.57 \[ \int (a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2\,A\,{\left (c+d\,x\right )}^{3/2}\,\left (15\,b^2\,{\left (c+d\,x\right )}^2+35\,a^2\,d^2+35\,b^2\,c^2-42\,b^2\,c\,\left (c+d\,x\right )+42\,a\,b\,d\,\left (c+d\,x\right )-70\,a\,b\,c\,d\right )}{105\,d^3}+\frac {2\,B\,b^2\,{\left (c+d\,x\right )}^{9/2}}{9\,d^4}+\frac {2\,C\,b^2\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}+\frac {2\,{\left (c+d\,x\right )}^{3/2}\,D\,\left (3465\,b^2\,{\left (c+d\,x\right )}^5-15015\,b^2\,c^5+5005\,a^2\,d^2\,{\left (c+d\,x\right )}^3+50050\,b^2\,c^2\,{\left (c+d\,x\right )}^3-64350\,b^2\,c^3\,{\left (c+d\,x\right )}^2-15015\,a^2\,c^3\,d^2-20475\,b^2\,c\,{\left (c+d\,x\right )}^4+45045\,b^2\,c^4\,\left (c+d\,x\right )+30030\,a\,b\,c^4\,d-19305\,a^2\,c\,d^2\,{\left (c+d\,x\right )}^2+27027\,a^2\,c^2\,d^2\,\left (c+d\,x\right )+8190\,a\,b\,d\,{\left (c+d\,x\right )}^4-40040\,a\,b\,c\,d\,{\left (c+d\,x\right )}^3-72072\,a\,b\,c^3\,d\,\left (c+d\,x\right )+77220\,a\,b\,c^2\,d\,{\left (c+d\,x\right )}^2\right )}{45045\,d^6}+\frac {2\,B\,{\left (c+d\,x\right )}^{5/2}\,\left (a^2\,d^2-4\,a\,b\,c\,d+3\,b^2\,c^2\right )}{5\,d^4}+\frac {2\,C\,{\left (c+d\,x\right )}^{7/2}\,\left (a^2\,d^2-6\,a\,b\,c\,d+6\,b^2\,c^2\right )}{7\,d^5}-\frac {2\,B\,c\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{3\,d^4}-\frac {4\,C\,c\,{\left (c+d\,x\right )}^{5/2}\,\left (a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{5\,d^5}+\frac {2\,C\,c^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{3\,d^5}+\frac {2\,B\,b\,\left (2\,a\,d-3\,b\,c\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}+\frac {4\,C\,b\,\left (a\,d-2\,b\,c\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5} \] Input:

int((a + b*x)^2*(c + d*x)^(1/2)*(A + B*x + C*x^2 + x^3*D),x)
 

Output:

(2*A*(c + d*x)^(3/2)*(15*b^2*(c + d*x)^2 + 35*a^2*d^2 + 35*b^2*c^2 - 42*b^ 
2*c*(c + d*x) + 42*a*b*d*(c + d*x) - 70*a*b*c*d))/(105*d^3) + (2*B*b^2*(c 
+ d*x)^(9/2))/(9*d^4) + (2*C*b^2*(c + d*x)^(11/2))/(11*d^5) + (2*(c + d*x) 
^(3/2)*D*(3465*b^2*(c + d*x)^5 - 15015*b^2*c^5 + 5005*a^2*d^2*(c + d*x)^3 
+ 50050*b^2*c^2*(c + d*x)^3 - 64350*b^2*c^3*(c + d*x)^2 - 15015*a^2*c^3*d^ 
2 - 20475*b^2*c*(c + d*x)^4 + 45045*b^2*c^4*(c + d*x) + 30030*a*b*c^4*d - 
19305*a^2*c*d^2*(c + d*x)^2 + 27027*a^2*c^2*d^2*(c + d*x) + 8190*a*b*d*(c 
+ d*x)^4 - 40040*a*b*c*d*(c + d*x)^3 - 72072*a*b*c^3*d*(c + d*x) + 77220*a 
*b*c^2*d*(c + d*x)^2))/(45045*d^6) + (2*B*(c + d*x)^(5/2)*(a^2*d^2 + 3*b^2 
*c^2 - 4*a*b*c*d))/(5*d^4) + (2*C*(c + d*x)^(7/2)*(a^2*d^2 + 6*b^2*c^2 - 6 
*a*b*c*d))/(7*d^5) - (2*B*c*(a*d - b*c)^2*(c + d*x)^(3/2))/(3*d^4) - (4*C* 
c*(c + d*x)^(5/2)*(a^2*d^2 + 2*b^2*c^2 - 3*a*b*c*d))/(5*d^5) + (2*C*c^2*(a 
*d - b*c)^2*(c + d*x)^(3/2))/(3*d^5) + (2*B*b*(2*a*d - 3*b*c)*(c + d*x)^(7 
/2))/(7*d^4) + (4*C*b*(a*d - 2*b*c)*(c + d*x)^(9/2))/(9*d^5)
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.17 \[ \int (a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \sqrt {d x +c}\, \left (3465 b^{2} d^{6} x^{6}+8190 a b \,d^{6} x^{5}+4410 b^{2} c \,d^{5} x^{5}+5005 a^{2} d^{6} x^{4}+10920 a b c \,d^{5} x^{4}+5005 b^{3} d^{5} x^{4}+105 b^{2} c^{2} d^{4} x^{4}+7150 a^{2} c \,d^{5} x^{3}+19305 a \,b^{2} d^{5} x^{3}+390 a b \,c^{2} d^{4} x^{3}+715 b^{3} c \,d^{4} x^{3}-120 b^{2} c^{3} d^{3} x^{3}+27027 a^{2} b \,d^{5} x^{2}+429 a^{2} c^{2} d^{4} x^{2}+3861 a \,b^{2} c \,d^{4} x^{2}-468 a b \,c^{3} d^{3} x^{2}-858 b^{3} c^{2} d^{3} x^{2}+144 b^{2} c^{4} d^{2} x^{2}+15015 a^{3} d^{5} x +9009 a^{2} b c \,d^{4} x -572 a^{2} c^{3} d^{3} x -5148 a \,b^{2} c^{2} d^{3} x +624 a b \,c^{4} d^{2} x +1144 b^{3} c^{3} d^{2} x -192 b^{2} c^{5} d x +15015 a^{3} c \,d^{4}-18018 a^{2} b \,c^{2} d^{3}+1144 a^{2} c^{4} d^{2}+10296 a \,b^{2} c^{3} d^{2}-1248 a b \,c^{5} d -2288 b^{3} c^{4} d +384 b^{2} c^{6}\right )}{45045 d^{5}} \] Input:

int((b*x+a)^2*(d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A),x)
 

Output:

(2*sqrt(c + d*x)*(15015*a**3*c*d**4 + 15015*a**3*d**5*x - 18018*a**2*b*c** 
2*d**3 + 9009*a**2*b*c*d**4*x + 27027*a**2*b*d**5*x**2 + 1144*a**2*c**4*d* 
*2 - 572*a**2*c**3*d**3*x + 429*a**2*c**2*d**4*x**2 + 7150*a**2*c*d**5*x** 
3 + 5005*a**2*d**6*x**4 + 10296*a*b**2*c**3*d**2 - 5148*a*b**2*c**2*d**3*x 
 + 3861*a*b**2*c*d**4*x**2 + 19305*a*b**2*d**5*x**3 - 1248*a*b*c**5*d + 62 
4*a*b*c**4*d**2*x - 468*a*b*c**3*d**3*x**2 + 390*a*b*c**2*d**4*x**3 + 1092 
0*a*b*c*d**5*x**4 + 8190*a*b*d**6*x**5 - 2288*b**3*c**4*d + 1144*b**3*c**3 
*d**2*x - 858*b**3*c**2*d**3*x**2 + 715*b**3*c*d**4*x**3 + 5005*b**3*d**5* 
x**4 + 384*b**2*c**6 - 192*b**2*c**5*d*x + 144*b**2*c**4*d**2*x**2 - 120*b 
**2*c**3*d**3*x**3 + 105*b**2*c**2*d**4*x**4 + 4410*b**2*c*d**5*x**5 + 346 
5*b**2*d**6*x**6))/(45045*d**5)