\(\int (a+b x)^3 \sqrt {c+d x} (A+B x+C x^2+D x^3) \, dx\) [55]

Optimal result

Integrand size = 32, antiderivative size = 438 \[ \int (a+b x)^3 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {2 (b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{3/2}}{3 d^7}-\frac {2 (b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) (c+d x)^{5/2}}{5 d^7}-\frac {2 (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{7/2}}{7 d^7}+\frac {2 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{9/2}}{9 d^7}+\frac {2 b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{11/2}}{11 d^7}+\frac {2 b^2 (b C d-6 b c D+3 a d D) (c+d x)^{13/2}}{13 d^7}+\frac {2 b^3 D (c+d x)^{15/2}}{15 d^7} \] Output:

-2/3*(-a*d+b*c)^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(3/2)/d^7-2/5*(-a* 
d+b*c)^2*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b*(3*A*d^3-4*B*c*d^2+5*C*c^2*d-6*D* 
c^3))*(d*x+c)^(5/2)/d^7-2/7*(-a*d+b*c)*(a^2*d^2*(C*d-3*D*c)-a*b*d*(-3*B*d^ 
2+8*C*c*d-15*D*c^2)+b^2*(3*A*d^3-6*B*c*d^2+10*C*c^2*d-15*D*c^3))*(d*x+c)^( 
7/2)/d^7+2/9*(a^3*d^3*D+3*a^2*b*d^2*(C*d-4*D*c)-3*a*b^2*d*(-B*d^2+4*C*c*d- 
10*D*c^2)+b^3*(A*d^3-4*B*c*d^2+10*C*c^2*d-20*D*c^3))*(d*x+c)^(9/2)/d^7+2/1 
1*b*(3*a^2*d^2*D+3*a*b*d*(C*d-5*D*c)-b^2*(-B*d^2+5*C*c*d-15*D*c^2))*(d*x+c 
)^(11/2)/d^7+2/13*b^2*(C*b*d+3*D*a*d-6*D*b*c)*(d*x+c)^(13/2)/d^7+2/15*b^3* 
D*(d*x+c)^(15/2)/d^7
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.13 \[ \int (a+b x)^3 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 (c+d x)^{3/2} \left (143 a^3 d^3 \left (-16 c^3 D+24 c^2 d (C+D x)-6 c d^2 (7 B+x (6 C+5 D x))+d^3 (105 A+x (63 B+5 x (9 C+7 D x)))\right )+b^3 \left (1024 c^6 D-256 c^5 d (5 C+6 D x)+7 d^6 x^3 \left (715 A+585 B x+495 C x^2+429 D x^3\right )+128 c^4 d^2 (13 B+15 x (C+D x))+8 c^2 d^4 x \left (429 A+5 x \left (78 B+70 C x+63 D x^2\right )\right )-16 c^3 d^3 \left (143 A+2 x \left (78 B+75 C x+70 D x^2\right )\right )-2 c d^5 x^2 (2145 A+7 x (260 B+9 x (25 C+22 D x)))\right )+39 a^2 b d^2 \left (128 c^4 D-16 c^3 d (11 C+12 D x)+24 c^2 d^2 (11 B+x (11 C+10 D x))+d^4 x (693 A+5 x (99 B+7 x (11 C+9 D x)))-2 c d^3 (231 A+x (198 B+5 x (33 C+28 D x)))\right )+3 a b^2 d \left (-1280 c^5 D+128 c^4 d (13 C+15 D x)-16 c^3 d^2 (143 B+6 x (26 C+25 D x))+5 d^5 x^2 \left (1287 A+7 x \left (143 B+117 C x+99 D x^2\right )\right )+8 c^2 d^3 \left (429 A+x \left (429 B+390 C x+350 D x^2\right )\right )-2 c d^4 x (2574 A+5 x (429 B+7 x (52 C+45 D x)))\right )\right )}{45045 d^7} \] Input:

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

Output:

(2*(c + d*x)^(3/2)*(143*a^3*d^3*(-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)))) + b^ 
3*(1024*c^6*D - 256*c^5*d*(5*C + 6*D*x) + 7*d^6*x^3*(715*A + 585*B*x + 495 
*C*x^2 + 429*D*x^3) + 128*c^4*d^2*(13*B + 15*x*(C + D*x)) + 8*c^2*d^4*x*(4 
29*A + 5*x*(78*B + 70*C*x + 63*D*x^2)) - 16*c^3*d^3*(143*A + 2*x*(78*B + 7 
5*C*x + 70*D*x^2)) - 2*c*d^5*x^2*(2145*A + 7*x*(260*B + 9*x*(25*C + 22*D*x 
)))) + 39*a^2*b*d^2*(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)))) + 3*a*b^2*d*(-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^7)
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 438, 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.

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

\(\Big \downarrow \) 2123

\(\displaystyle \int \left (\frac {(c+d x)^{5/2} (b c-a d) \left (-a^2 d^2 (C d-3 c D)+a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )-\left (b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{d^6}+\frac {b (c+d x)^{9/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{d^6}+\frac {(c+d x)^{7/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{d^6}+\frac {(c+d x)^{3/2} (b c-a d)^2 \left (b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )-a d \left (-B d^2-3 c^2 D+2 c C d\right )\right )}{d^6}+\frac {\sqrt {c+d x} (a d-b c)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6}+\frac {b^2 (c+d x)^{11/2} (3 a d D-6 b c D+b C d)}{d^6}+\frac {b^3 D (c+d x)^{13/2}}{d^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 (c+d x)^{7/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{7 d^7}+\frac {2 b (c+d x)^{11/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{11 d^7}+\frac {2 (c+d x)^{9/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{9 d^7}-\frac {2 (c+d x)^{5/2} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{5 d^7}-\frac {2 (c+d x)^{3/2} (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^7}+\frac {2 b^2 (c+d x)^{13/2} (3 a d D-6 b c D+b C d)}{13 d^7}+\frac {2 b^3 D (c+d x)^{15/2}}{15 d^7}\)

Input:

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

Output:

(-2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(3/2))/(3* 
d^7) - (2*(b*c - a*d)^2*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(5*c^2*C*d - 
4*B*c*d^2 + 3*A*d^3 - 6*c^3*D))*(c + d*x)^(5/2))/(5*d^7) - (2*(b*c - a*d)* 
(a^2*d^2*(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^ 
2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))*(c + d*x)^(7/2))/(7*d^7) + (2*(a^ 
3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c*C*d - B*d^2 - 10*c^2* 
D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(9/2))/(9* 
d^7) + (2*b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D) - b^2*(5*c*C*d - B*d^2 - 
15*c^2*D))*(c + d*x)^(11/2))/(11*d^7) + (2*b^2*(b*C*d - 6*b*c*D + 3*a*d*D) 
*(c + d*x)^(13/2))/(13*d^7) + (2*b^3*D*(c + d*x)^(15/2))/(15*d^7)
 

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 = 1.02 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.00

method	result	size
pseudoelliptic	\(\frac {2 \left (x d +c \right )^{\frac {3}{2}} \left (\left (\frac {\left (\frac {3}{5} D x^{3}+\frac {9}{13} C \,x^{2}+\frac {9}{11} B x +A \right ) x^{3} b^{3}}{3}+\frac {9 \left (\frac {7}{13} D x^{3}+\frac {7}{11} C \,x^{2}+\frac {7}{9} B x +A \right ) x^{2} a \,b^{2}}{7}+\frac {9 x \left (\frac {5}{11} D x^{3}+\frac {5}{9} C \,x^{2}+\frac {5}{7} B x +A \right ) a^{2} b}{5}+a^{3} \left (\frac {1}{3} D x^{3}+\frac {3}{7} C \,x^{2}+\frac {3}{5} B x +A \right )\right ) d^{6}-\frac {6 \left (\frac {5 \left (\frac {42}{65} D x^{3}+\frac {105}{143} C \,x^{2}+\frac {28}{33} B x +A \right ) x^{2} b^{3}}{21}+\frac {6 \left (\frac {175}{286} D x^{3}+\frac {70}{99} C \,x^{2}+\frac {5}{6} B x +A \right ) x a \,b^{2}}{7}+a^{2} \left (\frac {20}{33} D x^{3}+\frac {5}{7} C \,x^{2}+\frac {6}{7} B x +A \right ) b +\frac {\left (\frac {5}{7} D x^{2}+\frac {6}{7} C x +B \right ) a^{3}}{3}\right ) c \,d^{5}}{5}+\frac {24 \left (\frac {x \left (\frac {105}{143} D x^{3}+\frac {350}{429} C \,x^{2}+\frac {10}{11} B x +A \right ) b^{3}}{3}+a \left (\frac {350}{429} D x^{3}+\frac {10}{11} C \,x^{2}+B x +A \right ) b^{2}+a^{2} \left (\frac {10}{11} D x^{2}+C x +B \right ) b +\frac {a^{3} \left (D x +C \right )}{3}\right ) c^{2} d^{4}}{35}-\frac {16 \left (\left (\frac {140}{143} D x^{3}+\frac {150}{143} C \,x^{2}+\frac {12}{11} B x +A \right ) b^{3}+3 \left (\frac {150}{143} D x^{2}+\frac {12}{11} C x +B \right ) a \,b^{2}+3 \left (\frac {12 D x}{11}+C \right ) a^{2} b +a^{3} D\right ) c^{3} d^{3}}{105}+\frac {128 c^{4} b \left (\left (\frac {15}{13} D x^{2}+\frac {15}{13} C x +B \right ) b^{2}+3 \left (\frac {15 D x}{13}+C \right ) a b +3 D a^{2}\right ) d^{2}}{1155}-\frac {256 \left (\left (\frac {6 D x}{5}+C \right ) b +3 D a \right ) c^{5} b^{2} d}{3003}+\frac {1024 D b^{3} c^{6}}{15015}\right )}{3 d^{7}}\)	\(437\)
derivativedivides	\(\frac {\frac {2 b^{3} D \left (x d +c \right )^{\frac {15}{2}}}{15}+\frac {2 \left (3 \left (a d -b c \right ) b^{2} D+b^{3} \left (C d -3 D c \right )\right ) \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (a d -b c \right )^{2} b D+3 \left (a d -b c \right ) b^{2} \left (C d -3 D c \right )+b^{3} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a d -b c \right )^{3} D+3 \left (a d -b c \right )^{2} b \left (C d -3 D c \right )+3 \left (a d -b c \right ) b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{3} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a d -b c \right )^{3} \left (C d -3 D c \right )+3 \left (a d -b c \right )^{2} b \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+3 \left (a d -b c \right ) b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a d -b c \right )^{3} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+3 \left (a d -b c \right )^{2} b \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right )^{3} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (x d +c \right )^{\frac {3}{2}}}{3}}{d^{7}}\)	\(441\)
default	\(\frac {\frac {2 b^{3} D \left (x d +c \right )^{\frac {15}{2}}}{15}+\frac {2 \left (3 \left (a d -b c \right ) b^{2} D+b^{3} \left (C d -3 D c \right )\right ) \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (a d -b c \right )^{2} b D+3 \left (a d -b c \right ) b^{2} \left (C d -3 D c \right )+b^{3} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a d -b c \right )^{3} D+3 \left (a d -b c \right )^{2} b \left (C d -3 D c \right )+3 \left (a d -b c \right ) b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{3} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a d -b c \right )^{3} \left (C d -3 D c \right )+3 \left (a d -b c \right )^{2} b \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+3 \left (a d -b c \right ) b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a d -b c \right )^{3} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+3 \left (a d -b c \right )^{2} b \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right )^{3} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (x d +c \right )^{\frac {3}{2}}}{3}}{d^{7}}\)	\(441\)
gosper	\(\frac {2 \left (x d +c \right )^{\frac {3}{2}} \left (3003 D x^{6} b^{3} d^{6}+3465 C \,x^{5} b^{3} d^{6}+10395 D x^{5} a \,b^{2} d^{6}-2772 D x^{5} b^{3} c \,d^{5}+4095 B \,x^{4} b^{3} d^{6}+12285 C \,x^{4} a \,b^{2} d^{6}-3150 C \,x^{4} b^{3} c \,d^{5}+12285 D x^{4} a^{2} b \,d^{6}-9450 D x^{4} a \,b^{2} c \,d^{5}+2520 D x^{4} b^{3} c^{2} d^{4}+5005 A \,x^{3} b^{3} d^{6}+15015 B \,x^{3} a \,b^{2} d^{6}-3640 B \,x^{3} b^{3} c \,d^{5}+15015 C \,x^{3} a^{2} b \,d^{6}-10920 C \,x^{3} a \,b^{2} c \,d^{5}+2800 C \,x^{3} b^{3} c^{2} d^{4}+5005 D x^{3} a^{3} d^{6}-10920 D x^{3} a^{2} b c \,d^{5}+8400 D x^{3} a \,b^{2} c^{2} d^{4}-2240 D x^{3} b^{3} c^{3} d^{3}+19305 A \,x^{2} a \,b^{2} d^{6}-4290 A \,x^{2} b^{3} c \,d^{5}+19305 B \,x^{2} a^{2} b \,d^{6}-12870 B \,x^{2} a \,b^{2} c \,d^{5}+3120 B \,x^{2} b^{3} c^{2} d^{4}+6435 C \,x^{2} a^{3} d^{6}-12870 C \,x^{2} a^{2} b c \,d^{5}+9360 C \,x^{2} a \,b^{2} c^{2} d^{4}-2400 C \,x^{2} b^{3} c^{3} d^{3}-4290 D x^{2} a^{3} c \,d^{5}+9360 D x^{2} a^{2} b \,c^{2} d^{4}-7200 D x^{2} a \,b^{2} c^{3} d^{3}+1920 D x^{2} b^{3} c^{4} d^{2}+27027 A x \,a^{2} b \,d^{6}-15444 A x a \,b^{2} c \,d^{5}+3432 A x \,b^{3} c^{2} d^{4}+9009 B x \,a^{3} d^{6}-15444 B x \,a^{2} b c \,d^{5}+10296 B x a \,b^{2} c^{2} d^{4}-2496 B x \,b^{3} c^{3} d^{3}-5148 C x \,a^{3} c \,d^{5}+10296 C x \,a^{2} b \,c^{2} d^{4}-7488 C x a \,b^{2} c^{3} d^{3}+1920 C x \,b^{3} c^{4} d^{2}+3432 D x \,a^{3} c^{2} d^{4}-7488 D x \,a^{2} b \,c^{3} d^{3}+5760 D x a \,b^{2} c^{4} d^{2}-1536 D x \,b^{3} c^{5} d +15015 A \,a^{3} d^{6}-18018 A \,a^{2} b c \,d^{5}+10296 A a \,b^{2} c^{2} d^{4}-2288 A \,b^{3} c^{3} d^{3}-6006 B \,a^{3} c \,d^{5}+10296 B \,a^{2} b \,c^{2} d^{4}-6864 B a \,b^{2} c^{3} d^{3}+1664 B \,b^{3} c^{4} d^{2}+3432 C \,a^{3} c^{2} d^{4}-6864 C \,a^{2} b \,c^{3} d^{3}+4992 C a \,b^{2} c^{4} d^{2}-1280 C \,b^{3} c^{5} d -2288 D a^{3} c^{3} d^{3}+4992 D a^{2} b \,c^{4} d^{2}-3840 D a \,b^{2} c^{5} d +1024 D b^{3} c^{6}\right )}{45045 d^{7}}\)	\(841\)
orering	\(\frac {2 \left (x d +c \right )^{\frac {3}{2}} \left (3003 D x^{6} b^{3} d^{6}+3465 C \,x^{5} b^{3} d^{6}+10395 D x^{5} a \,b^{2} d^{6}-2772 D x^{5} b^{3} c \,d^{5}+4095 B \,x^{4} b^{3} d^{6}+12285 C \,x^{4} a \,b^{2} d^{6}-3150 C \,x^{4} b^{3} c \,d^{5}+12285 D x^{4} a^{2} b \,d^{6}-9450 D x^{4} a \,b^{2} c \,d^{5}+2520 D x^{4} b^{3} c^{2} d^{4}+5005 A \,x^{3} b^{3} d^{6}+15015 B \,x^{3} a \,b^{2} d^{6}-3640 B \,x^{3} b^{3} c \,d^{5}+15015 C \,x^{3} a^{2} b \,d^{6}-10920 C \,x^{3} a \,b^{2} c \,d^{5}+2800 C \,x^{3} b^{3} c^{2} d^{4}+5005 D x^{3} a^{3} d^{6}-10920 D x^{3} a^{2} b c \,d^{5}+8400 D x^{3} a \,b^{2} c^{2} d^{4}-2240 D x^{3} b^{3} c^{3} d^{3}+19305 A \,x^{2} a \,b^{2} d^{6}-4290 A \,x^{2} b^{3} c \,d^{5}+19305 B \,x^{2} a^{2} b \,d^{6}-12870 B \,x^{2} a \,b^{2} c \,d^{5}+3120 B \,x^{2} b^{3} c^{2} d^{4}+6435 C \,x^{2} a^{3} d^{6}-12870 C \,x^{2} a^{2} b c \,d^{5}+9360 C \,x^{2} a \,b^{2} c^{2} d^{4}-2400 C \,x^{2} b^{3} c^{3} d^{3}-4290 D x^{2} a^{3} c \,d^{5}+9360 D x^{2} a^{2} b \,c^{2} d^{4}-7200 D x^{2} a \,b^{2} c^{3} d^{3}+1920 D x^{2} b^{3} c^{4} d^{2}+27027 A x \,a^{2} b \,d^{6}-15444 A x a \,b^{2} c \,d^{5}+3432 A x \,b^{3} c^{2} d^{4}+9009 B x \,a^{3} d^{6}-15444 B x \,a^{2} b c \,d^{5}+10296 B x a \,b^{2} c^{2} d^{4}-2496 B x \,b^{3} c^{3} d^{3}-5148 C x \,a^{3} c \,d^{5}+10296 C x \,a^{2} b \,c^{2} d^{4}-7488 C x a \,b^{2} c^{3} d^{3}+1920 C x \,b^{3} c^{4} d^{2}+3432 D x \,a^{3} c^{2} d^{4}-7488 D x \,a^{2} b \,c^{3} d^{3}+5760 D x a \,b^{2} c^{4} d^{2}-1536 D x \,b^{3} c^{5} d +15015 A \,a^{3} d^{6}-18018 A \,a^{2} b c \,d^{5}+10296 A a \,b^{2} c^{2} d^{4}-2288 A \,b^{3} c^{3} d^{3}-6006 B \,a^{3} c \,d^{5}+10296 B \,a^{2} b \,c^{2} d^{4}-6864 B a \,b^{2} c^{3} d^{3}+1664 B \,b^{3} c^{4} d^{2}+3432 C \,a^{3} c^{2} d^{4}-6864 C \,a^{2} b \,c^{3} d^{3}+4992 C a \,b^{2} c^{4} d^{2}-1280 C \,b^{3} c^{5} d -2288 D a^{3} c^{3} d^{3}+4992 D a^{2} b \,c^{4} d^{2}-3840 D a \,b^{2} c^{5} d +1024 D b^{3} c^{6}\right )}{45045 d^{7}}\)	\(841\)
trager	\(\text {Expression too large to display}\)	\(1085\)

Input:

int((b*x+a)^3*(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)*((1/3*(3/5*D*x^3+9/13*C*x^2+9/11*B*x+A)*x^3*b^3+9/7*(7/1 
3*D*x^3+7/11*C*x^2+7/9*B*x+A)*x^2*a*b^2+9/5*x*(5/11*D*x^3+5/9*C*x^2+5/7*B* 
x+A)*a^2*b+a^3*(1/3*D*x^3+3/7*C*x^2+3/5*B*x+A))*d^6-6/5*(5/21*(42/65*D*x^3 
+105/143*C*x^2+28/33*B*x+A)*x^2*b^3+6/7*(175/286*D*x^3+70/99*C*x^2+5/6*B*x 
+A)*x*a*b^2+a^2*(20/33*D*x^3+5/7*C*x^2+6/7*B*x+A)*b+1/3*(5/7*D*x^2+6/7*C*x 
+B)*a^3)*c*d^5+24/35*(1/3*x*(105/143*D*x^3+350/429*C*x^2+10/11*B*x+A)*b^3+ 
a*(350/429*D*x^3+10/11*C*x^2+B*x+A)*b^2+a^2*(10/11*D*x^2+C*x+B)*b+1/3*a^3* 
(D*x+C))*c^2*d^4-16/105*((140/143*D*x^3+150/143*C*x^2+12/11*B*x+A)*b^3+3*( 
150/143*D*x^2+12/11*C*x+B)*a*b^2+3*(12/11*D*x+C)*a^2*b+a^3*D)*c^3*d^3+128/ 
1155*c^4*b*((15/13*D*x^2+15/13*C*x+B)*b^2+3*(15/13*D*x+C)*a*b+3*D*a^2)*d^2 
-256/3003*((6/5*D*x+C)*b+3*D*a)*c^5*b^2*d+1024/15015*D*b^3*c^6)/d^7
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 782, normalized size of antiderivative = 1.79 \[ \int (a+b x)^3 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

2/45045*(3003*D*b^3*d^7*x^7 + 1024*D*b^3*c^7 + 15015*A*a^3*c*d^6 - 1280*(3 
*D*a*b^2 + C*b^3)*c^6*d + 1664*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^5*d^2 - 2 
288*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^4*d^3 + 3432*(C*a^3 + 3*B*a^ 
2*b + 3*A*a*b^2)*c^3*d^4 - 6006*(B*a^3 + 3*A*a^2*b)*c^2*d^5 + 231*(D*b^3*c 
*d^6 + 15*(3*D*a*b^2 + C*b^3)*d^7)*x^6 - 63*(4*D*b^3*c^2*d^5 - 5*(3*D*a*b^ 
2 + C*b^3)*c*d^6 - 65*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7)*x^5 + 35*(8*D*b 
^3*c^3*d^4 - 10*(3*D*a*b^2 + C*b^3)*c^2*d^5 + 13*(3*D*a^2*b + 3*C*a*b^2 + 
B*b^3)*c*d^6 + 143*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^7)*x^4 - 5*(6 
4*D*b^3*c^4*d^3 - 80*(3*D*a*b^2 + C*b^3)*c^3*d^4 + 104*(3*D*a^2*b + 3*C*a* 
b^2 + B*b^3)*c^2*d^5 - 143*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c*d^6 - 
 1287*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^7)*x^3 + 3*(128*D*b^3*c^5*d^2 - 16 
0*(3*D*a*b^2 + C*b^3)*c^4*d^3 + 208*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^3*d^ 
4 - 286*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^5 + 429*(C*a^3 + 3*B 
*a^2*b + 3*A*a*b^2)*c*d^6 + 3003*(B*a^3 + 3*A*a^2*b)*d^7)*x^2 - (512*D*b^3 
*c^6*d - 15015*A*a^3*d^7 - 640*(3*D*a*b^2 + C*b^3)*c^5*d^2 + 832*(3*D*a^2* 
b + 3*C*a*b^2 + B*b^3)*c^4*d^3 - 1144*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b 
^3)*c^3*d^4 + 1716*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^5 - 3003*(B*a^3 + 
 3*A*a^2*b)*c*d^6)*x)*sqrt(d*x + c)/d^7
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1028 vs. \(2 (456) = 912\).

Time = 2.19 (sec) , antiderivative size = 1028, normalized size of antiderivative = 2.35 \[ \int (a+b x)^3 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((2*(D*b**3*(c + d*x)**(15/2)/(15*d**6) + (c + d*x)**(13/2)*(C*b* 
*3*d + 3*D*a*b**2*d - 6*D*b**3*c)/(13*d**6) + (c + d*x)**(11/2)*(B*b**3*d* 
*2 + 3*C*a*b**2*d**2 - 5*C*b**3*c*d + 3*D*a**2*b*d**2 - 15*D*a*b**2*c*d + 
15*D*b**3*c**2)/(11*d**6) + (c + d*x)**(9/2)*(A*b**3*d**3 + 3*B*a*b**2*d** 
3 - 4*B*b**3*c*d**2 + 3*C*a**2*b*d**3 - 12*C*a*b**2*c*d**2 + 10*C*b**3*c** 
2*d + D*a**3*d**3 - 12*D*a**2*b*c*d**2 + 30*D*a*b**2*c**2*d - 20*D*b**3*c* 
*3)/(9*d**6) + (c + d*x)**(7/2)*(3*A*a*b**2*d**4 - 3*A*b**3*c*d**3 + 3*B*a 
**2*b*d**4 - 9*B*a*b**2*c*d**3 + 6*B*b**3*c**2*d**2 + C*a**3*d**4 - 9*C*a* 
*2*b*c*d**3 + 18*C*a*b**2*c**2*d**2 - 10*C*b**3*c**3*d - 3*D*a**3*c*d**3 + 
 18*D*a**2*b*c**2*d**2 - 30*D*a*b**2*c**3*d + 15*D*b**3*c**4)/(7*d**6) + ( 
c + d*x)**(5/2)*(3*A*a**2*b*d**5 - 6*A*a*b**2*c*d**4 + 3*A*b**3*c**2*d**3 
+ B*a**3*d**5 - 6*B*a**2*b*c*d**4 + 9*B*a*b**2*c**2*d**3 - 4*B*b**3*c**3*d 
**2 - 2*C*a**3*c*d**4 + 9*C*a**2*b*c**2*d**3 - 12*C*a*b**2*c**3*d**2 + 5*C 
*b**3*c**4*d + 3*D*a**3*c**2*d**3 - 12*D*a**2*b*c**3*d**2 + 15*D*a*b**2*c* 
*4*d - 6*D*b**3*c**5)/(5*d**6) + (c + d*x)**(3/2)*(A*a**3*d**6 - 3*A*a**2* 
b*c*d**5 + 3*A*a*b**2*c**2*d**4 - A*b**3*c**3*d**3 - B*a**3*c*d**5 + 3*B*a 
**2*b*c**2*d**4 - 3*B*a*b**2*c**3*d**3 + B*b**3*c**4*d**2 + C*a**3*c**2*d* 
*4 - 3*C*a**2*b*c**3*d**3 + 3*C*a*b**2*c**4*d**2 - C*b**3*c**5*d - D*a**3* 
c**3*d**3 + 3*D*a**2*b*c**4*d**2 - 3*D*a*b**2*c**5*d + D*b**3*c**6)/(3*d** 
6))/d, Ne(d, 0)), (sqrt(c)*(A*a**3*x + D*b**3*x**7/7 + x**6*(C*b**3 + 3...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.42 \[ \int (a+b x)^3 \sqrt {c+d x} \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^{3} - 3465 \, {\left (6 \, D b^{3} c - {\left (3 \, D a b^{2} + C b^{3}\right )} d\right )} {\left (d x + c\right )}^{\frac {13}{2}} + 4095 \, {\left (15 \, D b^{3} c^{2} - 5 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c d + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} - 5005 \, {\left (20 \, D b^{3} c^{3} - 10 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d + 4 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{2} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 6435 \, {\left (15 \, D b^{3} c^{4} - 10 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d + 6 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{2} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{4}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 9009 \, {\left (6 \, D b^{3} c^{5} - 5 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d + 4 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{2} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{3} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{4} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 15015 \, {\left (D b^{3} c^{6} + A a^{3} d^{6} - {\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} - {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5}\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{45045 \, d^{7}} \] Input:

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

Output:

2/45045*(3003*(d*x + c)^(15/2)*D*b^3 - 3465*(6*D*b^3*c - (3*D*a*b^2 + C*b^ 
3)*d)*(d*x + c)^(13/2) + 4095*(15*D*b^3*c^2 - 5*(3*D*a*b^2 + C*b^3)*c*d + 
(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^2)*(d*x + c)^(11/2) - 5005*(20*D*b^3*c^3 
 - 10*(3*D*a*b^2 + C*b^3)*c^2*d + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^2 
- (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^3)*(d*x + c)^(9/2) + 6435*(15* 
D*b^3*c^4 - 10*(3*D*a*b^2 + C*b^3)*c^3*d + 6*(3*D*a^2*b + 3*C*a*b^2 + B*b^ 
3)*c^2*d^2 - 3*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c*d^3 + (C*a^3 + 3* 
B*a^2*b + 3*A*a*b^2)*d^4)*(d*x + c)^(7/2) - 9009*(6*D*b^3*c^5 - 5*(3*D*a*b 
^2 + C*b^3)*c^4*d + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^3*d^2 - 3*(D*a^3 + 
 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^3 + 2*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2 
)*c*d^4 - (B*a^3 + 3*A*a^2*b)*d^5)*(d*x + c)^(5/2) + 15015*(D*b^3*c^6 + A* 
a^3*d^6 - (3*D*a*b^2 + C*b^3)*c^5*d + (3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4* 
d^2 - (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 + (C*a^3 + 3*B*a^2*b 
 + 3*A*a*b^2)*c^2*d^4 - (B*a^3 + 3*A*a^2*b)*c*d^5)*(d*x + c)^(3/2))/d^7
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1913 vs. \(2 (412) = 824\).

Time = 0.14 (sec) , antiderivative size = 1913, normalized size of antiderivative = 4.37 \[ \int (a+b x)^3 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^3*(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^3*c + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x 
 + c)*c)*A*a^3 + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*B*a^3*c/d + 4 
5045*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*A*a^2*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^3*c/d^2 + 9009*( 
3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*B*a^2*b*c 
/d^2 + 9009*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c 
^2)*A*a*b^2*c/d^2 + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sq 
rt(d*x + c)*c^2)*B*a^3/d + 9009*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c 
+ 15*sqrt(d*x + c)*c^2)*A*a^2*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^3*c/d^3 + 38 
61*(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*b*c/d^3 + 3861*(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^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)*A*b^3*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^3/d^2 + 3861 
*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*s 
qrt(d*x + c)*c^3)*B*a^2*b/d^2 + 3861*(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*a*b^2/d^2 + 429*(3 
5*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 4...
 

Mupad [B] (verification not implemented)

Time = 3.88 (sec) , antiderivative size = 770, normalized size of antiderivative = 1.76 \[ \int (a+b x)^3 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(70*a^3*(c + d*x)^(9/2)*D - 270*a^3*c*(c + d*x)^(7/2)*D - 210*a^3*c^3*(c + 
 d*x)^(3/2)*D + 378*a^3*c^2*(c + d*x)^(5/2)*D)/(315*d^4) + ((6*a*b^2*(c + 
d*x)^(13/2)*D)/13 - 2*a*b^2*c^5*(c + d*x)^(3/2)*D + 6*a*b^2*c^4*(c + d*x)^ 
(5/2)*D - (60*a*b^2*c^3*(c + d*x)^(7/2)*D)/7 + (20*a*b^2*c^2*(c + d*x)^(9/ 
2)*D)/3 - (30*a*b^2*c*(c + d*x)^(11/2)*D)/11)/d^6 + ((2*b^3*(c + d*x)^(15/ 
2)*D)/15 - (12*b^3*c*(c + d*x)^(13/2)*D)/13 + (2*b^3*c^6*(c + d*x)^(3/2)*D 
)/3 - (12*b^3*c^5*(c + d*x)^(5/2)*D)/5 + (30*b^3*c^4*(c + d*x)^(7/2)*D)/7 
- (40*b^3*c^3*(c + d*x)^(9/2)*D)/9 + (30*b^3*c^2*(c + d*x)^(11/2)*D)/11)/d 
^7 + ((6*a^2*b*(c + d*x)^(11/2)*D)/11 + 2*a^2*b*c^4*(c + d*x)^(3/2)*D - (2 
4*a^2*b*c^3*(c + d*x)^(5/2)*D)/5 + (36*a^2*b*c^2*(c + d*x)^(7/2)*D)/7 - (8 
*a^2*b*c*(c + d*x)^(9/2)*D)/3)/d^5 + (2*C*(c + d*x)^(7/2)*(a^3*d^3 - 10*b^ 
3*c^3 + 18*a*b^2*c^2*d - 9*a^2*b*c*d^2))/(7*d^6) + (2*A*b^3*(c + d*x)^(9/2 
))/(9*d^4) + (2*B*b^3*(c + d*x)^(11/2))/(11*d^5) + (2*C*b^3*(c + d*x)^(13/ 
2))/(13*d^6) + (2*A*(a*d - b*c)^3*(c + d*x)^(3/2))/(3*d^4) + (6*A*b*(a*d - 
 b*c)^2*(c + d*x)^(5/2))/(5*d^4) + (6*A*b^2*(a*d - b*c)*(c + d*x)^(7/2))/( 
7*d^4) + (2*B*b^2*(3*a*d - 4*b*c)*(c + d*x)^(9/2))/(9*d^5) - (2*B*c*(a*d - 
 b*c)^3*(c + d*x)^(3/2))/(3*d^5) + (2*C*b^2*(3*a*d - 5*b*c)*(c + d*x)^(11/ 
2))/(11*d^6) + (6*B*b*(c + d*x)^(7/2)*(a^2*d^2 + 2*b^2*c^2 - 3*a*b*c*d))/( 
7*d^5) + (2*C*b*(c + d*x)^(9/2)*(3*a^2*d^2 + 10*b^2*c^2 - 12*a*b*c*d))/(9* 
d^6) + (2*B*(a*d - b*c)^2*(a*d - 4*b*c)*(c + d*x)^(5/2))/(5*d^5) + (2*C...
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.35 \[ \int (a+b x)^3 \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \sqrt {d x +c}\, \left (3003 b^{3} d^{7} x^{7}+10395 a \,b^{2} d^{7} x^{6}+3696 b^{3} c \,d^{6} x^{6}+12285 a^{2} b \,d^{7} x^{5}+13230 a \,b^{2} c \,d^{6} x^{5}+4095 b^{4} d^{6} x^{5}+63 b^{3} c^{2} d^{5} x^{5}+5005 a^{3} d^{7} x^{4}+16380 a^{2} b c \,d^{6} x^{4}+20020 a \,b^{3} d^{6} x^{4}+315 a \,b^{2} c^{2} d^{5} x^{4}+455 b^{4} c \,d^{5} x^{4}-70 b^{3} c^{3} d^{4} x^{4}+7150 a^{3} c \,d^{6} x^{3}+38610 a^{2} b^{2} d^{6} x^{3}+585 a^{2} b \,c^{2} d^{5} x^{3}+2860 a \,b^{3} c \,d^{5} x^{3}-360 a \,b^{2} c^{3} d^{4} x^{3}-520 b^{4} c^{2} d^{4} x^{3}+80 b^{3} c^{4} d^{3} x^{3}+36036 a^{3} b \,d^{6} x^{2}+429 a^{3} c^{2} d^{5} x^{2}+7722 a^{2} b^{2} c \,d^{5} x^{2}-702 a^{2} b \,c^{3} d^{4} x^{2}-3432 a \,b^{3} c^{2} d^{4} x^{2}+432 a \,b^{2} c^{4} d^{3} x^{2}+624 b^{4} c^{3} d^{3} x^{2}-96 b^{3} c^{5} d^{2} x^{2}+15015 a^{4} d^{6} x +12012 a^{3} b c \,d^{5} x -572 a^{3} c^{3} d^{4} x -10296 a^{2} b^{2} c^{2} d^{4} x +936 a^{2} b \,c^{4} d^{3} x +4576 a \,b^{3} c^{3} d^{3} x -576 a \,b^{2} c^{5} d^{2} x -832 b^{4} c^{4} d^{2} x +128 b^{3} c^{6} d x +15015 a^{4} c \,d^{5}-24024 a^{3} b \,c^{2} d^{4}+1144 a^{3} c^{4} d^{3}+20592 a^{2} b^{2} c^{3} d^{3}-1872 a^{2} b \,c^{5} d^{2}-9152 a \,b^{3} c^{4} d^{2}+1152 a \,b^{2} c^{6} d +1664 b^{4} c^{5} d -256 b^{3} c^{7}\right )}{45045 d^{6}} \] Input:

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

Output:

(2*sqrt(c + d*x)*(15015*a**4*c*d**5 + 15015*a**4*d**6*x - 24024*a**3*b*c** 
2*d**4 + 12012*a**3*b*c*d**5*x + 36036*a**3*b*d**6*x**2 + 1144*a**3*c**4*d 
**3 - 572*a**3*c**3*d**4*x + 429*a**3*c**2*d**5*x**2 + 7150*a**3*c*d**6*x* 
*3 + 5005*a**3*d**7*x**4 + 20592*a**2*b**2*c**3*d**3 - 10296*a**2*b**2*c** 
2*d**4*x + 7722*a**2*b**2*c*d**5*x**2 + 38610*a**2*b**2*d**6*x**3 - 1872*a 
**2*b*c**5*d**2 + 936*a**2*b*c**4*d**3*x - 702*a**2*b*c**3*d**4*x**2 + 585 
*a**2*b*c**2*d**5*x**3 + 16380*a**2*b*c*d**6*x**4 + 12285*a**2*b*d**7*x**5 
 - 9152*a*b**3*c**4*d**2 + 4576*a*b**3*c**3*d**3*x - 3432*a*b**3*c**2*d**4 
*x**2 + 2860*a*b**3*c*d**5*x**3 + 20020*a*b**3*d**6*x**4 + 1152*a*b**2*c** 
6*d - 576*a*b**2*c**5*d**2*x + 432*a*b**2*c**4*d**3*x**2 - 360*a*b**2*c**3 
*d**4*x**3 + 315*a*b**2*c**2*d**5*x**4 + 13230*a*b**2*c*d**6*x**5 + 10395* 
a*b**2*d**7*x**6 + 1664*b**4*c**5*d - 832*b**4*c**4*d**2*x + 624*b**4*c**3 
*d**3*x**2 - 520*b**4*c**2*d**4*x**3 + 455*b**4*c*d**5*x**4 + 4095*b**4*d* 
*6*x**5 - 256*b**3*c**7 + 128*b**3*c**6*d*x - 96*b**3*c**5*d**2*x**2 + 80* 
b**3*c**4*d**3*x**3 - 70*b**3*c**3*d**4*x**4 + 63*b**3*c**2*d**5*x**5 + 36 
96*b**3*c*d**6*x**6 + 3003*b**3*d**7*x**7))/(45045*d**6)