\(\int \frac {(a+b x)^3 (A+B x+C x^2+D x^3)}{(c+d x)^{5/2}} \, dx\) [94]

Optimal result

Integrand size = 32, antiderivative size = 434 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {2 (b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{3 d^7 (c+d x)^{3/2}}+\frac {2 (b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right )}{d^7 \sqrt {c+d x}}-\frac {2 (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) \sqrt {c+d x}}{d^7}+\frac {2 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{3/2}}{3 d^7}+\frac {2 b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{5/2}}{5 d^7}+\frac {2 b^2 (b C d-6 b c D+3 a d D) (c+d x)^{7/2}}{7 d^7}+\frac {2 b^3 D (c+d x)^{9/2}}{9 d^7} \] Output:

2/3*(-a*d+b*c)^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^7/(d*x+c)^(3/2)+2*(-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^7/(d*x+c)^(1/2)-2*(-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)^(1/2)/ 
d^7+2/3*(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)^(3/2)/d^7+2/5*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)^(5/2 
)/d^7+2/7*b^2*(C*b*d+3*D*a*d-6*D*b*c)*(d*x+c)^(7/2)/d^7+2/9*b^3*D*(d*x+c)^ 
(9/2)/d^7
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {2 \left (-105 a^3 d^3 \left (16 c^3 D-8 c^2 d (C-3 D x)+2 c d^2 (B+3 x (-2 C+D x))+d^3 \left (A+3 B x-x^2 (3 C+D x)\right )\right )+63 a^2 b d^2 \left (128 c^4 D+c^3 (-80 C d+192 d D x)+8 c^2 d^2 (5 B+3 x (-5 C+2 D x))+d^4 x \left (-15 A+x \left (15 B+5 C x+3 D x^2\right )\right )-2 c d^3 \left (5 A+x \left (-30 B+15 C x+4 D x^2\right )\right )\right )+b^3 \left (5120 c^6 D-3840 c^5 d (C-2 D x)+384 c^4 d^2 (7 B+5 x (-3 C+D x))+24 c^2 d^4 x (-105 A+x (42 B+5 x (2 C+D x)))-6 c d^5 x^2 (105 A+x (28 B+5 x (3 C+2 D x)))-16 c^3 d^3 (105 A+2 x (-126 B+5 x (9 C+2 D x)))+d^6 x^3 (105 A+x (63 B+5 x (9 C+7 D x)))\right )+9 a b^2 d \left (-1280 c^5 D+128 c^4 d (7 C-15 D x)-16 c^3 d^2 (35 B+6 x (-14 C+5 D x))+d^5 x^2 (105 A+x (35 B+3 x (7 C+5 D x)))+8 c^2 d^3 (35 A+x (-105 B+2 x (21 C+5 D x)))-2 c d^4 x (-210 A+x (105 B+x (28 C+15 D x)))\right )\right )}{315 d^7 (c+d x)^{3/2}} \] Input:

Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]
 

Output:

(2*(-105*a^3*d^3*(16*c^3*D - 8*c^2*d*(C - 3*D*x) + 2*c*d^2*(B + 3*x*(-2*C 
+ D*x)) + d^3*(A + 3*B*x - x^2*(3*C + D*x))) + 63*a^2*b*d^2*(128*c^4*D + c 
^3*(-80*C*d + 192*d*D*x) + 8*c^2*d^2*(5*B + 3*x*(-5*C + 2*D*x)) + d^4*x*(- 
15*A + x*(15*B + 5*C*x + 3*D*x^2)) - 2*c*d^3*(5*A + x*(-30*B + 15*C*x + 4* 
D*x^2))) + b^3*(5120*c^6*D - 3840*c^5*d*(C - 2*D*x) + 384*c^4*d^2*(7*B + 5 
*x*(-3*C + D*x)) + 24*c^2*d^4*x*(-105*A + x*(42*B + 5*x*(2*C + D*x))) - 6* 
c*d^5*x^2*(105*A + x*(28*B + 5*x*(3*C + 2*D*x))) - 16*c^3*d^3*(105*A + 2*x 
*(-126*B + 5*x*(9*C + 2*D*x))) + d^6*x^3*(105*A + x*(63*B + 5*x*(9*C + 7*D 
*x)))) + 9*a*b^2*d*(-1280*c^5*D + 128*c^4*d*(7*C - 15*D*x) - 16*c^3*d^2*(3 
5*B + 6*x*(-14*C + 5*D*x)) + d^5*x^2*(105*A + x*(35*B + 3*x*(7*C + 5*D*x)) 
) + 8*c^2*d^3*(35*A + x*(-105*B + 2*x*(21*C + 5*D*x))) - 2*c*d^4*x*(-210*A 
 + x*(105*B + x*(28*C + 15*D*x))))))/(315*d^7*(c + d*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 434, 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.

\(\displaystyle \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 2123

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

\(\Big \downarrow \) 2009

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

Input:

Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]
 

Output:

(2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^7*(c + d*x)^(3/ 
2)) + (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)))/(d^7*Sqrt[c + d*x]) - (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))*Sqrt[c + d*x])/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*(1 
0*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(3/2))/(3*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)^(5/2))/(5*d^7) + (2*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^(7 
/2))/(7*d^7) + (2*b^3*D*(c + d*x)^(9/2))/(9*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 = 0.62 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.01

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (-x^{3} \left (\frac {1}{3} D x^{3}+\frac {3}{7} C \,x^{2}+\frac {3}{5} B x +A \right ) b^{3}-9 x^{2} \left (\frac {1}{5} C \,x^{2}+\frac {1}{3} B x +\frac {1}{7} D x^{3}+A \right ) a \,b^{2}+9 x \left (-\frac {1}{5} D x^{3}-\frac {1}{3} C \,x^{2}-B x +A \right ) a^{2} b +a^{3} \left (-D x^{3}-3 C \,x^{2}+3 B x +A \right )\right ) d^{6}+6 \left (x^{2} \left (\frac {2}{21} D x^{3}+\frac {1}{7} C \,x^{2}+\frac {4}{15} B x +A \right ) b^{3}-6 x \left (-\frac {1}{14} D x^{3}-\frac {2}{15} C \,x^{2}-\frac {1}{2} B x +A \right ) a \,b^{2}+a^{2} \left (\frac {4}{5} D x^{3}+3 C \,x^{2}-6 B x +A \right ) b +\frac {a^{3} \left (3 D x^{2}-6 C x +B \right )}{3}\right ) c \,d^{5}-24 \left (-x \left (-\frac {1}{21} D x^{3}-\frac {2}{21} C \,x^{2}-\frac {2}{5} B x +A \right ) b^{3}+a \left (\frac {2}{7} D x^{3}+\frac {6}{5} C \,x^{2}-3 B x +A \right ) b^{2}+a^{2} \left (\frac {6}{5} D x^{2}-3 C x +B \right ) b +\frac {a^{3} \left (-3 D x +C \right )}{3}\right ) c^{2} d^{4}+16 c^{3} \left (\left (\frac {4}{21} D x^{3}+\frac {6}{7} C \,x^{2}-\frac {12}{5} B x +A \right ) b^{3}+3 a \left (\frac {6}{7} D x^{2}-\frac {12}{5} C x +B \right ) b^{2}+3 \left (-\frac {12 D x}{5}+C \right ) a^{2} b +a^{3} D\right ) d^{3}-\frac {128 \left (\left (\frac {5}{7} D x^{2}-\frac {15}{7} C x +B \right ) b^{2}+3 \left (-\frac {15 D x}{7}+C \right ) a b +3 D a^{2}\right ) c^{4} b \,d^{2}}{5}+\frac {256 \left (\left (-2 D x +C \right ) b +3 D a \right ) c^{5} b^{2} d}{7}-\frac {1024 D b^{3} c^{6}}{21}\right )}{3 \left (x d +c \right )^{\frac {3}{2}} d^{7}}\)	\(439\)
gosper	\(-\frac {2 \left (-35 D x^{6} b^{3} d^{6}-45 C \,x^{5} b^{3} d^{6}-135 D x^{5} a \,b^{2} d^{6}+60 D x^{5} b^{3} c \,d^{5}-63 B \,x^{4} b^{3} d^{6}-189 C \,x^{4} a \,b^{2} d^{6}+90 C \,x^{4} b^{3} c \,d^{5}-189 D x^{4} a^{2} b \,d^{6}+270 D x^{4} a \,b^{2} c \,d^{5}-120 D x^{4} b^{3} c^{2} d^{4}-105 A \,x^{3} b^{3} d^{6}-315 B \,x^{3} a \,b^{2} d^{6}+168 B \,x^{3} b^{3} c \,d^{5}-315 C \,x^{3} a^{2} b \,d^{6}+504 C \,x^{3} a \,b^{2} c \,d^{5}-240 C \,x^{3} b^{3} c^{2} d^{4}-105 D x^{3} a^{3} d^{6}+504 D x^{3} a^{2} b c \,d^{5}-720 D x^{3} a \,b^{2} c^{2} d^{4}+320 D x^{3} b^{3} c^{3} d^{3}-945 A \,x^{2} a \,b^{2} d^{6}+630 A \,x^{2} b^{3} c \,d^{5}-945 B \,x^{2} a^{2} b \,d^{6}+1890 B \,x^{2} a \,b^{2} c \,d^{5}-1008 B \,x^{2} b^{3} c^{2} d^{4}-315 C \,x^{2} a^{3} d^{6}+1890 C \,x^{2} a^{2} b c \,d^{5}-3024 C \,x^{2} a \,b^{2} c^{2} d^{4}+1440 C \,x^{2} b^{3} c^{3} d^{3}+630 D x^{2} a^{3} c \,d^{5}-3024 D x^{2} a^{2} b \,c^{2} d^{4}+4320 D x^{2} a \,b^{2} c^{3} d^{3}-1920 D x^{2} b^{3} c^{4} d^{2}+945 A x \,a^{2} b \,d^{6}-3780 A x a \,b^{2} c \,d^{5}+2520 A x \,b^{3} c^{2} d^{4}+315 B x \,a^{3} d^{6}-3780 B x \,a^{2} b c \,d^{5}+7560 B x a \,b^{2} c^{2} d^{4}-4032 B x \,b^{3} c^{3} d^{3}-1260 C x \,a^{3} c \,d^{5}+7560 C x \,a^{2} b \,c^{2} d^{4}-12096 C x a \,b^{2} c^{3} d^{3}+5760 C x \,b^{3} c^{4} d^{2}+2520 D x \,a^{3} c^{2} d^{4}-12096 D x \,a^{2} b \,c^{3} d^{3}+17280 D x a \,b^{2} c^{4} d^{2}-7680 D x \,b^{3} c^{5} d +105 A \,a^{3} d^{6}+630 A \,a^{2} b c \,d^{5}-2520 A a \,b^{2} c^{2} d^{4}+1680 A \,b^{3} c^{3} d^{3}+210 B \,a^{3} c \,d^{5}-2520 B \,a^{2} b \,c^{2} d^{4}+5040 B a \,b^{2} c^{3} d^{3}-2688 B \,b^{3} c^{4} d^{2}-840 C \,a^{3} c^{2} d^{4}+5040 C \,a^{2} b \,c^{3} d^{3}-8064 C a \,b^{2} c^{4} d^{2}+3840 C \,b^{3} c^{5} d +1680 D a^{3} c^{3} d^{3}-8064 D a^{2} b \,c^{4} d^{2}+11520 D a \,b^{2} c^{5} d -5120 D b^{3} c^{6}\right )}{315 \left (x d +c \right )^{\frac {3}{2}} d^{7}}\)	\(841\)
trager	\(-\frac {2 \left (-35 D x^{6} b^{3} d^{6}-45 C \,x^{5} b^{3} d^{6}-135 D x^{5} a \,b^{2} d^{6}+60 D x^{5} b^{3} c \,d^{5}-63 B \,x^{4} b^{3} d^{6}-189 C \,x^{4} a \,b^{2} d^{6}+90 C \,x^{4} b^{3} c \,d^{5}-189 D x^{4} a^{2} b \,d^{6}+270 D x^{4} a \,b^{2} c \,d^{5}-120 D x^{4} b^{3} c^{2} d^{4}-105 A \,x^{3} b^{3} d^{6}-315 B \,x^{3} a \,b^{2} d^{6}+168 B \,x^{3} b^{3} c \,d^{5}-315 C \,x^{3} a^{2} b \,d^{6}+504 C \,x^{3} a \,b^{2} c \,d^{5}-240 C \,x^{3} b^{3} c^{2} d^{4}-105 D x^{3} a^{3} d^{6}+504 D x^{3} a^{2} b c \,d^{5}-720 D x^{3} a \,b^{2} c^{2} d^{4}+320 D x^{3} b^{3} c^{3} d^{3}-945 A \,x^{2} a \,b^{2} d^{6}+630 A \,x^{2} b^{3} c \,d^{5}-945 B \,x^{2} a^{2} b \,d^{6}+1890 B \,x^{2} a \,b^{2} c \,d^{5}-1008 B \,x^{2} b^{3} c^{2} d^{4}-315 C \,x^{2} a^{3} d^{6}+1890 C \,x^{2} a^{2} b c \,d^{5}-3024 C \,x^{2} a \,b^{2} c^{2} d^{4}+1440 C \,x^{2} b^{3} c^{3} d^{3}+630 D x^{2} a^{3} c \,d^{5}-3024 D x^{2} a^{2} b \,c^{2} d^{4}+4320 D x^{2} a \,b^{2} c^{3} d^{3}-1920 D x^{2} b^{3} c^{4} d^{2}+945 A x \,a^{2} b \,d^{6}-3780 A x a \,b^{2} c \,d^{5}+2520 A x \,b^{3} c^{2} d^{4}+315 B x \,a^{3} d^{6}-3780 B x \,a^{2} b c \,d^{5}+7560 B x a \,b^{2} c^{2} d^{4}-4032 B x \,b^{3} c^{3} d^{3}-1260 C x \,a^{3} c \,d^{5}+7560 C x \,a^{2} b \,c^{2} d^{4}-12096 C x a \,b^{2} c^{3} d^{3}+5760 C x \,b^{3} c^{4} d^{2}+2520 D x \,a^{3} c^{2} d^{4}-12096 D x \,a^{2} b \,c^{3} d^{3}+17280 D x a \,b^{2} c^{4} d^{2}-7680 D x \,b^{3} c^{5} d +105 A \,a^{3} d^{6}+630 A \,a^{2} b c \,d^{5}-2520 A a \,b^{2} c^{2} d^{4}+1680 A \,b^{3} c^{3} d^{3}+210 B \,a^{3} c \,d^{5}-2520 B \,a^{2} b \,c^{2} d^{4}+5040 B a \,b^{2} c^{3} d^{3}-2688 B \,b^{3} c^{4} d^{2}-840 C \,a^{3} c^{2} d^{4}+5040 C \,a^{2} b \,c^{3} d^{3}-8064 C a \,b^{2} c^{4} d^{2}+3840 C \,b^{3} c^{5} d +1680 D a^{3} c^{3} d^{3}-8064 D a^{2} b \,c^{4} d^{2}+11520 D a \,b^{2} c^{5} d -5120 D b^{3} c^{6}\right )}{315 \left (x d +c \right )^{\frac {3}{2}} d^{7}}\)	\(841\)
orering	\(-\frac {2 \left (-35 D x^{6} b^{3} d^{6}-45 C \,x^{5} b^{3} d^{6}-135 D x^{5} a \,b^{2} d^{6}+60 D x^{5} b^{3} c \,d^{5}-63 B \,x^{4} b^{3} d^{6}-189 C \,x^{4} a \,b^{2} d^{6}+90 C \,x^{4} b^{3} c \,d^{5}-189 D x^{4} a^{2} b \,d^{6}+270 D x^{4} a \,b^{2} c \,d^{5}-120 D x^{4} b^{3} c^{2} d^{4}-105 A \,x^{3} b^{3} d^{6}-315 B \,x^{3} a \,b^{2} d^{6}+168 B \,x^{3} b^{3} c \,d^{5}-315 C \,x^{3} a^{2} b \,d^{6}+504 C \,x^{3} a \,b^{2} c \,d^{5}-240 C \,x^{3} b^{3} c^{2} d^{4}-105 D x^{3} a^{3} d^{6}+504 D x^{3} a^{2} b c \,d^{5}-720 D x^{3} a \,b^{2} c^{2} d^{4}+320 D x^{3} b^{3} c^{3} d^{3}-945 A \,x^{2} a \,b^{2} d^{6}+630 A \,x^{2} b^{3} c \,d^{5}-945 B \,x^{2} a^{2} b \,d^{6}+1890 B \,x^{2} a \,b^{2} c \,d^{5}-1008 B \,x^{2} b^{3} c^{2} d^{4}-315 C \,x^{2} a^{3} d^{6}+1890 C \,x^{2} a^{2} b c \,d^{5}-3024 C \,x^{2} a \,b^{2} c^{2} d^{4}+1440 C \,x^{2} b^{3} c^{3} d^{3}+630 D x^{2} a^{3} c \,d^{5}-3024 D x^{2} a^{2} b \,c^{2} d^{4}+4320 D x^{2} a \,b^{2} c^{3} d^{3}-1920 D x^{2} b^{3} c^{4} d^{2}+945 A x \,a^{2} b \,d^{6}-3780 A x a \,b^{2} c \,d^{5}+2520 A x \,b^{3} c^{2} d^{4}+315 B x \,a^{3} d^{6}-3780 B x \,a^{2} b c \,d^{5}+7560 B x a \,b^{2} c^{2} d^{4}-4032 B x \,b^{3} c^{3} d^{3}-1260 C x \,a^{3} c \,d^{5}+7560 C x \,a^{2} b \,c^{2} d^{4}-12096 C x a \,b^{2} c^{3} d^{3}+5760 C x \,b^{3} c^{4} d^{2}+2520 D x \,a^{3} c^{2} d^{4}-12096 D x \,a^{2} b \,c^{3} d^{3}+17280 D x a \,b^{2} c^{4} d^{2}-7680 D x \,b^{3} c^{5} d +105 A \,a^{3} d^{6}+630 A \,a^{2} b c \,d^{5}-2520 A a \,b^{2} c^{2} d^{4}+1680 A \,b^{3} c^{3} d^{3}+210 B \,a^{3} c \,d^{5}-2520 B \,a^{2} b \,c^{2} d^{4}+5040 B a \,b^{2} c^{3} d^{3}-2688 B \,b^{3} c^{4} d^{2}-840 C \,a^{3} c^{2} d^{4}+5040 C \,a^{2} b \,c^{3} d^{3}-8064 C a \,b^{2} c^{4} d^{2}+3840 C \,b^{3} c^{5} d +1680 D a^{3} c^{3} d^{3}-8064 D a^{2} b \,c^{4} d^{2}+11520 D a \,b^{2} c^{5} d -5120 D b^{3} c^{6}\right )}{315 \left (x d +c \right )^{\frac {3}{2}} d^{7}}\)	\(841\)
derivativedivides	\(\frac {-18 C \,a^{2} b c \,d^{3} \sqrt {x d +c}+36 C a \,b^{2} c^{2} d^{2} \sqrt {x d +c}+30 D b^{3} c^{4} \sqrt {x d +c}+2 C \,a^{3} d^{4} \sqrt {x d +c}+\frac {2 C \,b^{3} d \left (x d +c \right )^{\frac {7}{2}}}{7}-\frac {12 D b^{3} c \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 B \,b^{3} d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}+6 D b^{3} c^{2} \left (x d +c \right )^{\frac {5}{2}}+\frac {2 A \,b^{3} d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {8 B \,b^{3} c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {20 C \,b^{3} c^{2} d \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {6 D a \,b^{2} d \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {6 C a \,b^{2} d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}-8 C a \,b^{2} c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}-8 D a^{2} b c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}+20 D a \,b^{2} c^{2} d \left (x d +c \right )^{\frac {3}{2}}+36 D a^{2} b \,c^{2} d^{2} \sqrt {x d +c}-60 D a \,b^{2} c^{3} d \sqrt {x d +c}+\frac {2 D a^{3} d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {40 D b^{3} c^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-18 B a \,b^{2} c \,d^{3} \sqrt {x d +c}-6 D a \,b^{2} c d \left (x d +c \right )^{\frac {5}{2}}-2 C \,b^{3} c d \left (x d +c \right )^{\frac {5}{2}}+\frac {6 D a^{2} b \,d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}+6 A a \,b^{2} d^{4} \sqrt {x d +c}-6 A \,b^{3} c \,d^{3} \sqrt {x d +c}+6 B \,a^{2} b \,d^{4} \sqrt {x d +c}+12 B \,b^{3} c^{2} d^{2} \sqrt {x d +c}-20 C \,b^{3} c^{3} d \sqrt {x d +c}-6 D a^{3} c \,d^{3} \sqrt {x d +c}+2 B a \,b^{2} d^{3} \left (x d +c \right )^{\frac {3}{2}}+2 C \,a^{2} b \,d^{3} \left (x d +c \right )^{\frac {3}{2}}-\frac {2 \left (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}\right )}{\sqrt {x d +c}}-\frac {2 \left (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}\right )}{3 \left (x d +c \right )^{\frac {3}{2}}}+\frac {2 b^{3} D \left (x d +c \right )^{\frac {9}{2}}}{9}}{d^{7}}\)	\(927\)
default	\(\frac {-18 C \,a^{2} b c \,d^{3} \sqrt {x d +c}+36 C a \,b^{2} c^{2} d^{2} \sqrt {x d +c}+30 D b^{3} c^{4} \sqrt {x d +c}+2 C \,a^{3} d^{4} \sqrt {x d +c}+\frac {2 C \,b^{3} d \left (x d +c \right )^{\frac {7}{2}}}{7}-\frac {12 D b^{3} c \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 B \,b^{3} d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}+6 D b^{3} c^{2} \left (x d +c \right )^{\frac {5}{2}}+\frac {2 A \,b^{3} d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {8 B \,b^{3} c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {20 C \,b^{3} c^{2} d \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {6 D a \,b^{2} d \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {6 C a \,b^{2} d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}-8 C a \,b^{2} c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}-8 D a^{2} b c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}+20 D a \,b^{2} c^{2} d \left (x d +c \right )^{\frac {3}{2}}+36 D a^{2} b \,c^{2} d^{2} \sqrt {x d +c}-60 D a \,b^{2} c^{3} d \sqrt {x d +c}+\frac {2 D a^{3} d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {40 D b^{3} c^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-18 B a \,b^{2} c \,d^{3} \sqrt {x d +c}-6 D a \,b^{2} c d \left (x d +c \right )^{\frac {5}{2}}-2 C \,b^{3} c d \left (x d +c \right )^{\frac {5}{2}}+\frac {6 D a^{2} b \,d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}+6 A a \,b^{2} d^{4} \sqrt {x d +c}-6 A \,b^{3} c \,d^{3} \sqrt {x d +c}+6 B \,a^{2} b \,d^{4} \sqrt {x d +c}+12 B \,b^{3} c^{2} d^{2} \sqrt {x d +c}-20 C \,b^{3} c^{3} d \sqrt {x d +c}-6 D a^{3} c \,d^{3} \sqrt {x d +c}+2 B a \,b^{2} d^{3} \left (x d +c \right )^{\frac {3}{2}}+2 C \,a^{2} b \,d^{3} \left (x d +c \right )^{\frac {3}{2}}-\frac {2 \left (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}\right )}{\sqrt {x d +c}}-\frac {2 \left (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}\right )}{3 \left (x d +c \right )^{\frac {3}{2}}}+\frac {2 b^{3} D \left (x d +c \right )^{\frac {9}{2}}}{9}}{d^{7}}\)	\(927\)

Input:

int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3*((-x^3*(1/3*D*x^3+3/7*C*x^2+3/5*B*x+A)*b^3-9*x^2*(1/5*C*x^2+1/3*B*x+1 
/7*D*x^3+A)*a*b^2+9*x*(-1/5*D*x^3-1/3*C*x^2-B*x+A)*a^2*b+a^3*(-D*x^3-3*C*x 
^2+3*B*x+A))*d^6+6*(x^2*(2/21*D*x^3+1/7*C*x^2+4/15*B*x+A)*b^3-6*x*(-1/14*D 
*x^3-2/15*C*x^2-1/2*B*x+A)*a*b^2+a^2*(4/5*D*x^3+3*C*x^2-6*B*x+A)*b+1/3*a^3 
*(3*D*x^2-6*C*x+B))*c*d^5-24*(-x*(-1/21*D*x^3-2/21*C*x^2-2/5*B*x+A)*b^3+a* 
(2/7*D*x^3+6/5*C*x^2-3*B*x+A)*b^2+a^2*(6/5*D*x^2-3*C*x+B)*b+1/3*a^3*(-3*D* 
x+C))*c^2*d^4+16*c^3*((4/21*D*x^3+6/7*C*x^2-12/5*B*x+A)*b^3+3*a*(6/7*D*x^2 
-12/5*C*x+B)*b^2+3*(-12/5*D*x+C)*a^2*b+a^3*D)*d^3-128/5*((5/7*D*x^2-15/7*C 
*x+B)*b^2+3*(-15/7*D*x+C)*a*b+3*D*a^2)*c^4*b*d^2+256/7*((-2*D*x+C)*b+3*D*a 
)*c^5*b^2*d-1024/21*D*b^3*c^6)/(d*x+c)^(3/2)/d^7
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.49 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (35 \, D b^{3} d^{6} x^{6} + 5120 \, D b^{3} c^{6} - 105 \, A a^{3} d^{6} - 3840 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d + 2688 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} - 1680 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} + 840 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} - 210 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} - 15 \, {\left (4 \, D b^{3} c d^{5} - 3 \, {\left (3 \, D a b^{2} + C b^{3}\right )} d^{6}\right )} x^{5} + 3 \, {\left (40 \, D b^{3} c^{2} d^{4} - 30 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c d^{5} + 21 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{6}\right )} x^{4} - {\left (320 \, D b^{3} c^{3} d^{3} - 240 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d^{4} + 168 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{5} - 105 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{6}\right )} x^{3} + 3 \, {\left (640 \, D b^{3} c^{4} d^{2} - 480 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d^{3} + 336 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{4} - 210 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{5} + 105 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{6}\right )} x^{2} + 3 \, {\left (2560 \, D b^{3} c^{5} d - 1920 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d^{2} + 1344 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{3} - 840 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{4} + 420 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{5} - 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{6}\right )} x\right )} \sqrt {d x + c}}{315 \, {\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} \] Input:

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm="fricas 
")
 

Output:

2/315*(35*D*b^3*d^6*x^6 + 5120*D*b^3*c^6 - 105*A*a^3*d^6 - 3840*(3*D*a*b^2 
 + C*b^3)*c^5*d + 2688*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 - 1680*(D*a 
^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 + 840*(C*a^3 + 3*B*a^2*b + 3*A 
*a*b^2)*c^2*d^4 - 210*(B*a^3 + 3*A*a^2*b)*c*d^5 - 15*(4*D*b^3*c*d^5 - 3*(3 
*D*a*b^2 + C*b^3)*d^6)*x^5 + 3*(40*D*b^3*c^2*d^4 - 30*(3*D*a*b^2 + C*b^3)* 
c*d^5 + 21*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^6)*x^4 - (320*D*b^3*c^3*d^3 - 
 240*(3*D*a*b^2 + C*b^3)*c^2*d^4 + 168*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d 
^5 - 105*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^6)*x^3 + 3*(640*D*b^3*c 
^4*d^2 - 480*(3*D*a*b^2 + C*b^3)*c^3*d^3 + 336*(3*D*a^2*b + 3*C*a*b^2 + B* 
b^3)*c^2*d^4 - 210*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c*d^5 + 105*(C* 
a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^6)*x^2 + 3*(2560*D*b^3*c^5*d - 1920*(3*D*a* 
b^2 + C*b^3)*c^4*d^2 + 1344*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^3*d^3 - 840* 
(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^4 + 420*(C*a^3 + 3*B*a^2*b + 
 3*A*a*b^2)*c*d^5 - 105*(B*a^3 + 3*A*a^2*b)*d^6)*x)*sqrt(d*x + c)/(d^9*x^2 
 + 2*c*d^8*x + c^2*d^7)
 

Sympy [A] (verification not implemented)

Time = 80.90 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.68 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {D b^{3} \left (c + d x\right )^{\frac {9}{2}}}{9 d^{6}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (C b^{3} d + 3 D a b^{2} d - 6 D b^{3} c\right )}{7 d^{6}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (B b^{3} d^{2} + 3 C a b^{2} d^{2} - 5 C b^{3} c d + 3 D a^{2} b d^{2} - 15 D a b^{2} c d + 15 D b^{3} c^{2}\right )}{5 d^{6}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (A b^{3} d^{3} + 3 B a b^{2} d^{3} - 4 B b^{3} c d^{2} + 3 C a^{2} b d^{3} - 12 C a b^{2} c d^{2} + 10 C b^{3} c^{2} d + D a^{3} d^{3} - 12 D a^{2} b c d^{2} + 30 D a b^{2} c^{2} d - 20 D b^{3} c^{3}\right )}{3 d^{6}} + \frac {\sqrt {c + d x} \left (3 A a b^{2} d^{4} - 3 A b^{3} c d^{3} + 3 B a^{2} b d^{4} - 9 B a b^{2} c d^{3} + 6 B b^{3} c^{2} d^{2} + C a^{3} d^{4} - 9 C a^{2} b c d^{3} + 18 C a b^{2} c^{2} d^{2} - 10 C b^{3} c^{3} d - 3 D a^{3} c d^{3} + 18 D a^{2} b c^{2} d^{2} - 30 D a b^{2} c^{3} d + 15 D b^{3} c^{4}\right )}{d^{6}} - \frac {\left (a d - b c\right )^{2} \cdot \left (3 A b d^{3} + B a d^{3} - 4 B b c d^{2} - 2 C a c d^{2} + 5 C b c^{2} d + 3 D a c^{2} d - 6 D b c^{3}\right )}{d^{6} \sqrt {c + d x}} + \frac {\left (a d - b c\right )^{3} \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{3 d^{6} \left (c + d x\right )^{\frac {3}{2}}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {A a^{3} x + \frac {D b^{3} x^{7}}{7} + \frac {x^{6} \left (C b^{3} + 3 D a b^{2}\right )}{6} + \frac {x^{5} \left (B b^{3} + 3 C a b^{2} + 3 D a^{2} b\right )}{5} + \frac {x^{4} \left (A b^{3} + 3 B a b^{2} + 3 C a^{2} b + D a^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 A a b^{2} + 3 B a^{2} b + C a^{3}\right )}{3} + \frac {x^{2} \cdot \left (3 A a^{2} b + B a^{3}\right )}{2}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:

integrate((b*x+a)**3*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2),x)
 

Output:

Piecewise((2*(D*b**3*(c + d*x)**(9/2)/(9*d**6) + (c + d*x)**(7/2)*(C*b**3* 
d + 3*D*a*b**2*d - 6*D*b**3*c)/(7*d**6) + (c + d*x)**(5/2)*(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)/(5*d**6) + (c + d*x)**(3/2)*(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)/(3 
*d**6) + sqrt(c + d*x)*(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)/d**6 - (a*d - b*c)**2 
*(3*A*b*d**3 + B*a*d**3 - 4*B*b*c*d**2 - 2*C*a*c*d**2 + 5*C*b*c**2*d + 3*D 
*a*c**2*d - 6*D*b*c**3)/(d**6*sqrt(c + d*x)) + (a*d - b*c)**3*(-A*d**3 + B 
*c*d**2 - C*c**2*d + D*c**3)/(3*d**6*(c + d*x)**(3/2)))/d, Ne(d, 0)), ((A* 
a**3*x + D*b**3*x**7/7 + x**6*(C*b**3 + 3*D*a*b**2)/6 + x**5*(B*b**3 + 3*C 
*a*b**2 + 3*D*a**2*b)/5 + x**4*(A*b**3 + 3*B*a*b**2 + 3*C*a**2*b + D*a**3) 
/4 + x**3*(3*A*a*b**2 + 3*B*a**2*b + C*a**3)/3 + x**2*(3*A*a**2*b + B*a**3 
)/2)/c**(5/2), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.44 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (d x + c\right )}^{\frac {9}{2}} D b^{3} - 45 \, {\left (6 \, D b^{3} c - {\left (3 \, D a b^{2} + C b^{3}\right )} d\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 63 \, {\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 {5}{2}} - 105 \, {\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 {3}{2}} + 315 \, {\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 )} \sqrt {d x + c}}{d^{6}} - \frac {105 \, {\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} - 3 \, {\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 )}\right )}}{{\left (d x + c\right )}^{\frac {3}{2}} d^{6}}\right )}}{315 \, d} \] Input:

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm="maxima 
")
 

Output:

2/315*((35*(d*x + c)^(9/2)*D*b^3 - 45*(6*D*b^3*c - (3*D*a*b^2 + C*b^3)*d)* 
(d*x + c)^(7/2) + 63*(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)^(5/2) - 105*(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)^(3/2) + 315*(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)*sqrt(d*x + c))/d^6 - 105*(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 - 3*(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))/((d*x + c)^(3/2)*d^6))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (412) = 824\).

Time = 0.16 (sec) , antiderivative size = 1030, normalized size of antiderivative = 2.37 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

2/3*(18*(d*x + c)*D*b^3*c^5 - D*b^3*c^6 - 45*(d*x + c)*D*a*b^2*c^4*d - 15* 
(d*x + c)*C*b^3*c^4*d + 3*D*a*b^2*c^5*d + C*b^3*c^5*d + 36*(d*x + c)*D*a^2 
*b*c^3*d^2 + 36*(d*x + c)*C*a*b^2*c^3*d^2 + 12*(d*x + c)*B*b^3*c^3*d^2 - 3 
*D*a^2*b*c^4*d^2 - 3*C*a*b^2*c^4*d^2 - B*b^3*c^4*d^2 - 9*(d*x + c)*D*a^3*c 
^2*d^3 - 27*(d*x + c)*C*a^2*b*c^2*d^3 - 27*(d*x + c)*B*a*b^2*c^2*d^3 - 9*( 
d*x + c)*A*b^3*c^2*d^3 + D*a^3*c^3*d^3 + 3*C*a^2*b*c^3*d^3 + 3*B*a*b^2*c^3 
*d^3 + A*b^3*c^3*d^3 + 6*(d*x + c)*C*a^3*c*d^4 + 18*(d*x + c)*B*a^2*b*c*d^ 
4 + 18*(d*x + c)*A*a*b^2*c*d^4 - C*a^3*c^2*d^4 - 3*B*a^2*b*c^2*d^4 - 3*A*a 
*b^2*c^2*d^4 - 3*(d*x + c)*B*a^3*d^5 - 9*(d*x + c)*A*a^2*b*d^5 + B*a^3*c*d 
^5 + 3*A*a^2*b*c*d^5 - A*a^3*d^6)/((d*x + c)^(3/2)*d^7) + 2/315*(35*(d*x + 
 c)^(9/2)*D*b^3*d^56 - 270*(d*x + c)^(7/2)*D*b^3*c*d^56 + 945*(d*x + c)^(5 
/2)*D*b^3*c^2*d^56 - 2100*(d*x + c)^(3/2)*D*b^3*c^3*d^56 + 4725*sqrt(d*x + 
 c)*D*b^3*c^4*d^56 + 135*(d*x + c)^(7/2)*D*a*b^2*d^57 + 45*(d*x + c)^(7/2) 
*C*b^3*d^57 - 945*(d*x + c)^(5/2)*D*a*b^2*c*d^57 - 315*(d*x + c)^(5/2)*C*b 
^3*c*d^57 + 3150*(d*x + c)^(3/2)*D*a*b^2*c^2*d^57 + 1050*(d*x + c)^(3/2)*C 
*b^3*c^2*d^57 - 9450*sqrt(d*x + c)*D*a*b^2*c^3*d^57 - 3150*sqrt(d*x + c)*C 
*b^3*c^3*d^57 + 189*(d*x + c)^(5/2)*D*a^2*b*d^58 + 189*(d*x + c)^(5/2)*C*a 
*b^2*d^58 + 63*(d*x + c)^(5/2)*B*b^3*d^58 - 1260*(d*x + c)^(3/2)*D*a^2*b*c 
*d^58 - 1260*(d*x + c)^(3/2)*C*a*b^2*c*d^58 - 420*(d*x + c)^(3/2)*B*b^3*c* 
d^58 + 5670*sqrt(d*x + c)*D*a^2*b*c^2*d^58 + 5670*sqrt(d*x + c)*C*a*b^2...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^3\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(5/2),x)
 

Output:

int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {-64 a \,b^{3} c^{2} d^{3} x +\frac {96}{5} a^{2} b \,c^{4} d^{2}-\frac {2}{3} a^{4} d^{5}+\frac {512}{63} b^{3} c^{6}-\frac {16}{3} a^{3} b c \,d^{4}-8 a^{3} b \,d^{5} x -8 a^{3} c^{2} d^{4} x -2 a^{3} c \,d^{5} x^{2}+32 a^{2} b^{2} c^{2} d^{3}+12 a^{2} b^{2} d^{5} x^{2}+\frac {6}{5} a^{2} b \,d^{6} x^{4}-\frac {128}{3} a \,b^{3} c^{3} d^{2}+\frac {8}{3} a \,b^{3} d^{5} x^{3}-\frac {768}{35} a \,b^{2} c^{5} d +\frac {6}{7} a \,b^{2} d^{6} x^{5}+\frac {128}{5} b^{4} c^{3} d^{2} x +\frac {32}{5} b^{4} c^{2} d^{3} x^{2}-\frac {16}{15} b^{4} c \,d^{4} x^{3}+\frac {256}{21} b^{3} c^{5} d x +\frac {64}{21} b^{3} c^{4} d^{2} x^{2}-\frac {32}{63} b^{3} c^{3} d^{3} x^{3}+\frac {4}{21} b^{3} c^{2} d^{4} x^{4}-\frac {2}{21} b^{3} c \,d^{5} x^{5}-\frac {16}{3} a^{3} c^{3} d^{3}+\frac {2}{3} a^{3} d^{6} x^{3}+\frac {256}{15} b^{4} c^{4} d +\frac {2}{5} b^{4} d^{5} x^{4}+\frac {2}{9} b^{3} d^{6} x^{6}+48 a^{2} b^{2} c \,d^{4} x +\frac {144}{5} a^{2} b \,c^{3} d^{3} x +\frac {36}{5} a^{2} b \,c^{2} d^{4} x^{2}-\frac {6}{5} a^{2} b c \,d^{5} x^{3}-16 a \,b^{3} c \,d^{4} x^{2}-\frac {1152}{35} a \,b^{2} c^{4} d^{2} x -\frac {288}{35} a \,b^{2} c^{3} d^{3} x^{2}+\frac {48}{35} a \,b^{2} c^{2} d^{4} x^{3}-\frac {18}{35} a \,b^{2} c \,d^{5} x^{4}}{\sqrt {d x +c}\, d^{6} \left (d x +c \right )} \] Input:

int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x)
 

Output:

(2*( - 105*a**4*d**5 - 840*a**3*b*c*d**4 - 1260*a**3*b*d**5*x - 840*a**3*c 
**3*d**3 - 1260*a**3*c**2*d**4*x - 315*a**3*c*d**5*x**2 + 105*a**3*d**6*x* 
*3 + 5040*a**2*b**2*c**2*d**3 + 7560*a**2*b**2*c*d**4*x + 1890*a**2*b**2*d 
**5*x**2 + 3024*a**2*b*c**4*d**2 + 4536*a**2*b*c**3*d**3*x + 1134*a**2*b*c 
**2*d**4*x**2 - 189*a**2*b*c*d**5*x**3 + 189*a**2*b*d**6*x**4 - 6720*a*b** 
3*c**3*d**2 - 10080*a*b**3*c**2*d**3*x - 2520*a*b**3*c*d**4*x**2 + 420*a*b 
**3*d**5*x**3 - 3456*a*b**2*c**5*d - 5184*a*b**2*c**4*d**2*x - 1296*a*b**2 
*c**3*d**3*x**2 + 216*a*b**2*c**2*d**4*x**3 - 81*a*b**2*c*d**5*x**4 + 135* 
a*b**2*d**6*x**5 + 2688*b**4*c**4*d + 4032*b**4*c**3*d**2*x + 1008*b**4*c* 
*2*d**3*x**2 - 168*b**4*c*d**4*x**3 + 63*b**4*d**5*x**4 + 1280*b**3*c**6 + 
 1920*b**3*c**5*d*x + 480*b**3*c**4*d**2*x**2 - 80*b**3*c**3*d**3*x**3 + 3 
0*b**3*c**2*d**4*x**4 - 15*b**3*c*d**5*x**5 + 35*b**3*d**6*x**6))/(315*sqr 
t(c + d*x)*d**6*(c + d*x))