\(\int \frac {(a+b x)^3 (A+B x+C x^2+D x^3)}{(c+d x)^{4/3}} \, dx\) [167]

Optimal result

Integrand size = 32, antiderivative size = 436 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{4/3}} \, dx=\frac {3 (b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^7 \sqrt [3]{c+d x}}-\frac {3 (b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) (c+d x)^{2/3}}{2 d^7}-\frac {3 (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{5/3}}{5 d^7}+\frac {3 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{8/3}}{8 d^7}+\frac {3 b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{11/3}}{11 d^7}+\frac {3 b^2 (b C d-6 b c D+3 a d D) (c+d x)^{14/3}}{14 d^7}+\frac {3 b^3 D (c+d x)^{17/3}}{17 d^7} \] Output:

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)^(1/3)-3/2*(-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*x+c)^(2/3)/d^7-3/5*(-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)^(5/3 
)/d^7+3/8*(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)^(8/3)/d^7+3/11*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)^( 
11/3)/d^7+3/14*b^2*(C*b*d+3*D*a*d-6*D*b*c)*(d*x+c)^(14/3)/d^7+3/17*b^3*D*( 
d*x+c)^(17/3)/d^7
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{4/3}} \, dx=\frac {3 \left (1309 a^3 d^3 \left (81 c^3 D-9 c^2 d (8 C-3 D x)+3 c d^2 (20 B-x (8 C+3 D x))+d^3 \left (-40 A+x \left (20 B+8 C x+5 D x^2\right )\right )\right )+357 a^2 b d^2 \left (-972 c^4 D+81 c^3 d (11 C-4 D x)-9 c^2 d^2 (88 B-3 x (11 C+4 D x))+3 c d^3 \left (220 A-x \left (88 B+33 C x+20 D x^2\right )\right )+d^4 x \left (220 A+x \left (88 B+55 C x+40 D x^2\right )\right )\right )+b^3 \left (-131220 c^6 D+7290 c^5 d (17 C-6 D x)+5 d^6 x^3 \left (1309 A+952 B x+748 C x^2+616 D x^3\right )-486 c^4 d^2 (238 B-5 x (17 C+6 D x))+81 c^3 d^3 \left (1309 A-2 x \left (238 B+85 C x+50 D x^2\right )\right )-3 c d^5 x^2 \left (3927 A+20 x \left (119 B+85 C x+66 D x^2\right )\right )+9 c^2 d^4 x \left (3927 A+2 x \left (714 B+425 C x+300 D x^2\right )\right )\right )+51 a b^2 d \left (7290 c^5 D-486 c^4 d (14 C-5 D x)+81 c^3 d^2 (77 B-2 x (14 C+5 D x))+d^5 x^2 \left (616 A+5 x \left (77 B+56 C x+44 D x^2\right )\right )-9 c^2 d^3 \left (616 A-x \left (231 B+84 C x+50 D x^2\right )\right )-3 c d^4 x (616 A+x (231 B+20 x (7 C+5 D x)))\right )\right )}{52360 d^7 \sqrt [3]{c+d x}} \] Input:

Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(4/3),x]
 

Output:

(3*(1309*a^3*d^3*(81*c^3*D - 9*c^2*d*(8*C - 3*D*x) + 3*c*d^2*(20*B - x*(8* 
C + 3*D*x)) + d^3*(-40*A + x*(20*B + 8*C*x + 5*D*x^2))) + 357*a^2*b*d^2*(- 
972*c^4*D + 81*c^3*d*(11*C - 4*D*x) - 9*c^2*d^2*(88*B - 3*x*(11*C + 4*D*x) 
) + 3*c*d^3*(220*A - x*(88*B + 33*C*x + 20*D*x^2)) + d^4*x*(220*A + x*(88* 
B + 55*C*x + 40*D*x^2))) + b^3*(-131220*c^6*D + 7290*c^5*d*(17*C - 6*D*x) 
+ 5*d^6*x^3*(1309*A + 952*B*x + 748*C*x^2 + 616*D*x^3) - 486*c^4*d^2*(238* 
B - 5*x*(17*C + 6*D*x)) + 81*c^3*d^3*(1309*A - 2*x*(238*B + 85*C*x + 50*D* 
x^2)) - 3*c*d^5*x^2*(3927*A + 20*x*(119*B + 85*C*x + 66*D*x^2)) + 9*c^2*d^ 
4*x*(3927*A + 2*x*(714*B + 425*C*x + 300*D*x^2))) + 51*a*b^2*d*(7290*c^5*D 
 - 486*c^4*d*(14*C - 5*D*x) + 81*c^3*d^2*(77*B - 2*x*(14*C + 5*D*x)) + d^5 
*x^2*(616*A + 5*x*(77*B + 56*C*x + 44*D*x^2)) - 9*c^2*d^3*(616*A - x*(231* 
B + 84*C*x + 50*D*x^2)) - 3*c*d^4*x*(616*A + x*(231*B + 20*x*(7*C + 5*D*x) 
)))))/(52360*d^7*(c + d*x)^(1/3))
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 436, 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.

\(\displaystyle \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{4/3}} \, dx\)

\(\Big \downarrow \) 2123

\(\displaystyle \int \left (\frac {(c+d x)^{2/3} (b c-a d) \left (-a^2 d^2 (C d-3 c D)+a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )-\left (b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{d^6}+\frac {b (c+d x)^{8/3} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{d^6}+\frac {(c+d x)^{5/3} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{d^6}+\frac {(b c-a d)^2 \left (b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )-a d \left (-B d^2-3 c^2 D+2 c C d\right )\right )}{d^6 \sqrt [3]{c+d x}}+\frac {(a d-b c)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6 (c+d x)^{4/3}}+\frac {b^2 (c+d x)^{11/3} (3 a d D-6 b c D+b C d)}{d^6}+\frac {b^3 D (c+d x)^{14/3}}{d^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 (c+d x)^{5/3} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{5 d^7}+\frac {3 b (c+d x)^{11/3} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{11 d^7}+\frac {3 (c+d x)^{8/3} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{8 d^7}-\frac {3 (c+d x)^{2/3} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{2 d^7}+\frac {3 (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7 \sqrt [3]{c+d x}}+\frac {3 b^2 (c+d x)^{14/3} (3 a d D-6 b c D+b C d)}{14 d^7}+\frac {3 b^3 D (c+d x)^{17/3}}{17 d^7}\)

Input:

Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(4/3),x]
 

Output:

(3*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^7*(c + d*x)^(1/3) 
) - (3*(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)^(2/3))/(2*d^7) - (3*(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)^(5/3))/(5*d^7) + (3*(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)^(8/3))/(8*d^7 
) + (3*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)^(11/3))/(11*d^7) + (3*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c 
 + d*x)^(14/3))/(14*d^7) + (3*b^3*D*(c + d*x)^(17/3))/(17*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 = 0.60 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.01

method	result	size
pseudoelliptic	\(-\frac {3 \left (\left (-\frac {\left (\frac {8}{17} D x^{3}+\frac {4}{7} C \,x^{2}+\frac {8}{11} B x +A \right ) x^{3} b^{3}}{8}-\frac {3 x^{2} \left (\frac {5}{14} D x^{3}+\frac {5}{11} C \,x^{2}+\frac {5}{8} B x +A \right ) a \,b^{2}}{5}-\frac {3 \left (\frac {2}{11} D x^{3}+\frac {1}{4} C \,x^{2}+\frac {2}{5} B x +A \right ) x \,a^{2} b}{2}+a^{3} \left (-\frac {1}{8} D x^{3}-\frac {1}{5} C \,x^{2}-\frac {1}{2} B x +A \right )\right ) d^{6}-\frac {9 \left (-\frac {x^{2} \left (\frac {40}{119} D x^{3}+\frac {100}{231} C \,x^{2}+\frac {20}{33} B x +A \right ) b^{3}}{20}-\frac {2 \left (\frac {25}{154} D x^{3}+\frac {5}{22} C \,x^{2}+\frac {3}{8} B x +A \right ) x a \,b^{2}}{5}+a^{2} \left (-\frac {1}{11} D x^{3}-\frac {3}{20} C \,x^{2}-\frac {2}{5} B x +A \right ) b +\frac {\left (-\frac {3}{20} D x^{2}-\frac {2}{5} C x +B \right ) a^{3}}{3}\right ) c \,d^{5}}{2}+\frac {27 \left (-\frac {x \left (\frac {200}{1309} D x^{3}+\frac {50}{231} C \,x^{2}+\frac {4}{11} B x +A \right ) b^{3}}{8}+a \left (-\frac {25}{308} D x^{3}-\frac {3}{22} C \,x^{2}-\frac {3}{8} B x +A \right ) b^{2}+a^{2} \left (-\frac {3}{22} D x^{2}-\frac {3}{8} C x +B \right ) b +\frac {\left (-\frac {3 D x}{8}+C \right ) a^{3}}{3}\right ) c^{2} d^{4}}{5}-\frac {81 c^{3} \left (\left (-\frac {100}{1309} D x^{3}-\frac {10}{77} C \,x^{2}-\frac {4}{11} B x +A \right ) b^{3}+3 \left (-\frac {10}{77} D x^{2}-\frac {4}{11} C x +B \right ) a \,b^{2}+3 \left (-\frac {4 D x}{11}+C \right ) a^{2} b +a^{3} D\right ) d^{3}}{40}+\frac {243 c^{4} b \left (\left (-\frac {15}{119} D x^{2}-\frac {5}{14} C x +B \right ) b^{2}+3 \left (-\frac {5 D x}{14}+C \right ) a b +3 D a^{2}\right ) d^{2}}{110}-\frac {729 \left (\left (-\frac {6 D x}{17}+C \right ) b +3 D a \right ) c^{5} b^{2} d}{308}+\frac {6561 D b^{3} c^{6}}{2618}\right )}{\left (x d +c \right )^{\frac {1}{3}} d^{7}}\)	\(440\)
gosper	\(-\frac {3 \left (-3080 D x^{6} b^{3} d^{6}-3740 C \,b^{3} d^{6} x^{5}-11220 D x^{5} a \,b^{2} d^{6}+3960 D x^{5} b^{3} c \,d^{5}-4760 B \,x^{4} b^{3} d^{6}-14280 C \,x^{4} a \,b^{2} d^{6}+5100 C \,x^{4} b^{3} c \,d^{5}-14280 D a^{2} b \,d^{6} x^{4}+15300 D x^{4} a \,b^{2} c \,d^{5}-5400 D x^{4} b^{3} c^{2} d^{4}-6545 A \,x^{3} b^{3} d^{6}-19635 B a \,b^{2} d^{6} x^{3}+7140 B \,b^{3} c \,d^{5} x^{3}-19635 C \,a^{2} b \,d^{6} x^{3}+21420 C \,x^{3} a \,b^{2} c \,d^{5}-7650 C \,x^{3} b^{3} c^{2} d^{4}-6545 D x^{3} a^{3} d^{6}+21420 D x^{3} a^{2} b c \,d^{5}-22950 D a \,b^{2} c^{2} d^{4} x^{3}+8100 D x^{3} b^{3} c^{3} d^{3}-31416 A \,x^{2} a \,b^{2} d^{6}+11781 A \,x^{2} b^{3} c \,d^{5}-31416 B \,x^{2} a^{2} b \,d^{6}+35343 B \,x^{2} a \,b^{2} c \,d^{5}-12852 B \,b^{3} c^{2} d^{4} x^{2}-10472 C \,x^{2} a^{3} d^{6}+35343 C \,x^{2} a^{2} b c \,d^{5}-38556 C \,x^{2} a \,b^{2} c^{2} d^{4}+13770 C \,b^{3} c^{3} d^{3} x^{2}+11781 D x^{2} a^{3} c \,d^{5}-38556 D x^{2} a^{2} b \,c^{2} d^{4}+41310 D x^{2} a \,b^{2} c^{3} d^{3}-14580 D x^{2} b^{3} c^{4} d^{2}-78540 A x \,a^{2} b \,d^{6}+94248 A x a \,b^{2} c \,d^{5}-35343 A x \,b^{3} c^{2} d^{4}-26180 B x \,a^{3} d^{6}+94248 B x \,a^{2} b c \,d^{5}-106029 B x a \,b^{2} c^{2} d^{4}+38556 B \,b^{3} c^{3} d^{3} x +31416 C x \,a^{3} c \,d^{5}-106029 C x \,a^{2} b \,c^{2} d^{4}+115668 C x a \,b^{2} c^{3} d^{3}-41310 C x \,b^{3} c^{4} d^{2}-35343 D x \,a^{3} c^{2} d^{4}+115668 D x \,a^{2} b \,c^{3} d^{3}-123930 D x a \,b^{2} c^{4} d^{2}+43740 D x \,b^{3} c^{5} d +52360 A \,a^{3} d^{6}-235620 A \,a^{2} b c \,d^{5}+282744 A a \,b^{2} c^{2} d^{4}-106029 A \,b^{3} c^{3} d^{3}-78540 B \,a^{3} c \,d^{5}+282744 B \,a^{2} b \,c^{2} d^{4}-318087 B a \,b^{2} c^{3} d^{3}+115668 B \,b^{3} c^{4} d^{2}+94248 C \,a^{3} c^{2} d^{4}-318087 C \,a^{2} b \,c^{3} d^{3}+347004 C a \,b^{2} c^{4} d^{2}-123930 C \,b^{3} c^{5} d -106029 D a^{3} c^{3} d^{3}+347004 D a^{2} b \,c^{4} d^{2}-371790 D a \,b^{2} c^{5} d +131220 D b^{3} c^{6}\right )}{52360 \left (x d +c \right )^{\frac {1}{3}} d^{7}}\)	\(841\)
trager	\(-\frac {3 \left (-3080 D x^{6} b^{3} d^{6}-3740 C \,b^{3} d^{6} x^{5}-11220 D x^{5} a \,b^{2} d^{6}+3960 D x^{5} b^{3} c \,d^{5}-4760 B \,x^{4} b^{3} d^{6}-14280 C \,x^{4} a \,b^{2} d^{6}+5100 C \,x^{4} b^{3} c \,d^{5}-14280 D a^{2} b \,d^{6} x^{4}+15300 D x^{4} a \,b^{2} c \,d^{5}-5400 D x^{4} b^{3} c^{2} d^{4}-6545 A \,x^{3} b^{3} d^{6}-19635 B a \,b^{2} d^{6} x^{3}+7140 B \,b^{3} c \,d^{5} x^{3}-19635 C \,a^{2} b \,d^{6} x^{3}+21420 C \,x^{3} a \,b^{2} c \,d^{5}-7650 C \,x^{3} b^{3} c^{2} d^{4}-6545 D x^{3} a^{3} d^{6}+21420 D x^{3} a^{2} b c \,d^{5}-22950 D a \,b^{2} c^{2} d^{4} x^{3}+8100 D x^{3} b^{3} c^{3} d^{3}-31416 A \,x^{2} a \,b^{2} d^{6}+11781 A \,x^{2} b^{3} c \,d^{5}-31416 B \,x^{2} a^{2} b \,d^{6}+35343 B \,x^{2} a \,b^{2} c \,d^{5}-12852 B \,b^{3} c^{2} d^{4} x^{2}-10472 C \,x^{2} a^{3} d^{6}+35343 C \,x^{2} a^{2} b c \,d^{5}-38556 C \,x^{2} a \,b^{2} c^{2} d^{4}+13770 C \,b^{3} c^{3} d^{3} x^{2}+11781 D x^{2} a^{3} c \,d^{5}-38556 D x^{2} a^{2} b \,c^{2} d^{4}+41310 D x^{2} a \,b^{2} c^{3} d^{3}-14580 D x^{2} b^{3} c^{4} d^{2}-78540 A x \,a^{2} b \,d^{6}+94248 A x a \,b^{2} c \,d^{5}-35343 A x \,b^{3} c^{2} d^{4}-26180 B x \,a^{3} d^{6}+94248 B x \,a^{2} b c \,d^{5}-106029 B x a \,b^{2} c^{2} d^{4}+38556 B \,b^{3} c^{3} d^{3} x +31416 C x \,a^{3} c \,d^{5}-106029 C x \,a^{2} b \,c^{2} d^{4}+115668 C x a \,b^{2} c^{3} d^{3}-41310 C x \,b^{3} c^{4} d^{2}-35343 D x \,a^{3} c^{2} d^{4}+115668 D x \,a^{2} b \,c^{3} d^{3}-123930 D x a \,b^{2} c^{4} d^{2}+43740 D x \,b^{3} c^{5} d +52360 A \,a^{3} d^{6}-235620 A \,a^{2} b c \,d^{5}+282744 A a \,b^{2} c^{2} d^{4}-106029 A \,b^{3} c^{3} d^{3}-78540 B \,a^{3} c \,d^{5}+282744 B \,a^{2} b \,c^{2} d^{4}-318087 B a \,b^{2} c^{3} d^{3}+115668 B \,b^{3} c^{4} d^{2}+94248 C \,a^{3} c^{2} d^{4}-318087 C \,a^{2} b \,c^{3} d^{3}+347004 C a \,b^{2} c^{4} d^{2}-123930 C \,b^{3} c^{5} d -106029 D a^{3} c^{3} d^{3}+347004 D a^{2} b \,c^{4} d^{2}-371790 D a \,b^{2} c^{5} d +131220 D b^{3} c^{6}\right )}{52360 \left (x d +c \right )^{\frac {1}{3}} d^{7}}\)	\(841\)
orering	\(-\frac {3 \left (-3080 D x^{6} b^{3} d^{6}-3740 C \,b^{3} d^{6} x^{5}-11220 D x^{5} a \,b^{2} d^{6}+3960 D x^{5} b^{3} c \,d^{5}-4760 B \,x^{4} b^{3} d^{6}-14280 C \,x^{4} a \,b^{2} d^{6}+5100 C \,x^{4} b^{3} c \,d^{5}-14280 D a^{2} b \,d^{6} x^{4}+15300 D x^{4} a \,b^{2} c \,d^{5}-5400 D x^{4} b^{3} c^{2} d^{4}-6545 A \,x^{3} b^{3} d^{6}-19635 B a \,b^{2} d^{6} x^{3}+7140 B \,b^{3} c \,d^{5} x^{3}-19635 C \,a^{2} b \,d^{6} x^{3}+21420 C \,x^{3} a \,b^{2} c \,d^{5}-7650 C \,x^{3} b^{3} c^{2} d^{4}-6545 D x^{3} a^{3} d^{6}+21420 D x^{3} a^{2} b c \,d^{5}-22950 D a \,b^{2} c^{2} d^{4} x^{3}+8100 D x^{3} b^{3} c^{3} d^{3}-31416 A \,x^{2} a \,b^{2} d^{6}+11781 A \,x^{2} b^{3} c \,d^{5}-31416 B \,x^{2} a^{2} b \,d^{6}+35343 B \,x^{2} a \,b^{2} c \,d^{5}-12852 B \,b^{3} c^{2} d^{4} x^{2}-10472 C \,x^{2} a^{3} d^{6}+35343 C \,x^{2} a^{2} b c \,d^{5}-38556 C \,x^{2} a \,b^{2} c^{2} d^{4}+13770 C \,b^{3} c^{3} d^{3} x^{2}+11781 D x^{2} a^{3} c \,d^{5}-38556 D x^{2} a^{2} b \,c^{2} d^{4}+41310 D x^{2} a \,b^{2} c^{3} d^{3}-14580 D x^{2} b^{3} c^{4} d^{2}-78540 A x \,a^{2} b \,d^{6}+94248 A x a \,b^{2} c \,d^{5}-35343 A x \,b^{3} c^{2} d^{4}-26180 B x \,a^{3} d^{6}+94248 B x \,a^{2} b c \,d^{5}-106029 B x a \,b^{2} c^{2} d^{4}+38556 B \,b^{3} c^{3} d^{3} x +31416 C x \,a^{3} c \,d^{5}-106029 C x \,a^{2} b \,c^{2} d^{4}+115668 C x a \,b^{2} c^{3} d^{3}-41310 C x \,b^{3} c^{4} d^{2}-35343 D x \,a^{3} c^{2} d^{4}+115668 D x \,a^{2} b \,c^{3} d^{3}-123930 D x a \,b^{2} c^{4} d^{2}+43740 D x \,b^{3} c^{5} d +52360 A \,a^{3} d^{6}-235620 A \,a^{2} b c \,d^{5}+282744 A a \,b^{2} c^{2} d^{4}-106029 A \,b^{3} c^{3} d^{3}-78540 B \,a^{3} c \,d^{5}+282744 B \,a^{2} b \,c^{2} d^{4}-318087 B a \,b^{2} c^{3} d^{3}+115668 B \,b^{3} c^{4} d^{2}+94248 C \,a^{3} c^{2} d^{4}-318087 C \,a^{2} b \,c^{3} d^{3}+347004 C a \,b^{2} c^{4} d^{2}-123930 C \,b^{3} c^{5} d -106029 D a^{3} c^{3} d^{3}+347004 D a^{2} b \,c^{4} d^{2}-371790 D a \,b^{2} c^{5} d +131220 D b^{3} c^{6}\right )}{52360 \left (x d +c \right )^{\frac {1}{3}} d^{7}}\)	\(841\)
derivativedivides	\(\text {Expression too large to display}\)	\(1026\)
default	\(\text {Expression too large to display}\)	\(1026\)

Input:

int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(4/3),x,method=_RETURNVERBOSE)
 

Output:

-3/(d*x+c)^(1/3)*((-1/8*(8/17*D*x^3+4/7*C*x^2+8/11*B*x+A)*x^3*b^3-3/5*x^2* 
(5/14*D*x^3+5/11*C*x^2+5/8*B*x+A)*a*b^2-3/2*(2/11*D*x^3+1/4*C*x^2+2/5*B*x+ 
A)*x*a^2*b+a^3*(-1/8*D*x^3-1/5*C*x^2-1/2*B*x+A))*d^6-9/2*(-1/20*x^2*(40/11 
9*D*x^3+100/231*C*x^2+20/33*B*x+A)*b^3-2/5*(25/154*D*x^3+5/22*C*x^2+3/8*B* 
x+A)*x*a*b^2+a^2*(-1/11*D*x^3-3/20*C*x^2-2/5*B*x+A)*b+1/3*(-3/20*D*x^2-2/5 
*C*x+B)*a^3)*c*d^5+27/5*(-1/8*x*(200/1309*D*x^3+50/231*C*x^2+4/11*B*x+A)*b 
^3+a*(-25/308*D*x^3-3/22*C*x^2-3/8*B*x+A)*b^2+a^2*(-3/22*D*x^2-3/8*C*x+B)* 
b+1/3*(-3/8*D*x+C)*a^3)*c^2*d^4-81/40*c^3*((-100/1309*D*x^3-10/77*C*x^2-4/ 
11*B*x+A)*b^3+3*(-10/77*D*x^2-4/11*C*x+B)*a*b^2+3*(-4/11*D*x+C)*a^2*b+a^3* 
D)*d^3+243/110*c^4*b*((-15/119*D*x^2-5/14*C*x+B)*b^2+3*(-5/14*D*x+C)*a*b+3 
*D*a^2)*d^2-729/308*((-6/17*D*x+C)*b+3*D*a)*c^5*b^2*d+6561/2618*D*b^3*c^6) 
/d^7
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{4/3}} \, dx=\frac {3 \, {\left (3080 \, D b^{3} d^{6} x^{6} - 131220 \, D b^{3} c^{6} - 52360 \, A a^{3} d^{6} + 123930 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d - 115668 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} + 106029 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} - 94248 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} + 78540 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} - 220 \, {\left (18 \, D b^{3} c d^{5} - 17 \, {\left (3 \, D a b^{2} + C b^{3}\right )} d^{6}\right )} x^{5} + 20 \, {\left (270 \, D b^{3} c^{2} d^{4} - 255 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c d^{5} + 238 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{6}\right )} x^{4} - 5 \, {\left (1620 \, D b^{3} c^{3} d^{3} - 1530 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d^{4} + 1428 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{5} - 1309 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{6}\right )} x^{3} + {\left (14580 \, D b^{3} c^{4} d^{2} - 13770 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d^{3} + 12852 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{4} - 11781 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{5} + 10472 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{6}\right )} x^{2} - {\left (43740 \, D b^{3} c^{5} d - 41310 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d^{2} + 38556 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{3} - 35343 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{4} + 31416 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{5} - 26180 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{6}\right )} x\right )} {\left (d x + c\right )}^{\frac {2}{3}}}{52360 \, {\left (d^{8} x + c d^{7}\right )}} \] Input:

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(4/3),x, algorithm="fricas 
")
 

Output:

3/52360*(3080*D*b^3*d^6*x^6 - 131220*D*b^3*c^6 - 52360*A*a^3*d^6 + 123930* 
(3*D*a*b^2 + C*b^3)*c^5*d - 115668*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 
 + 106029*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 - 94248*(C*a^3 + 
 3*B*a^2*b + 3*A*a*b^2)*c^2*d^4 + 78540*(B*a^3 + 3*A*a^2*b)*c*d^5 - 220*(1 
8*D*b^3*c*d^5 - 17*(3*D*a*b^2 + C*b^3)*d^6)*x^5 + 20*(270*D*b^3*c^2*d^4 - 
255*(3*D*a*b^2 + C*b^3)*c*d^5 + 238*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^6)*x 
^4 - 5*(1620*D*b^3*c^3*d^3 - 1530*(3*D*a*b^2 + C*b^3)*c^2*d^4 + 1428*(3*D* 
a^2*b + 3*C*a*b^2 + B*b^3)*c*d^5 - 1309*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A 
*b^3)*d^6)*x^3 + (14580*D*b^3*c^4*d^2 - 13770*(3*D*a*b^2 + C*b^3)*c^3*d^3 
+ 12852*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^2*d^4 - 11781*(D*a^3 + 3*C*a^2*b 
 + 3*B*a*b^2 + A*b^3)*c*d^5 + 10472*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^6)*x 
^2 - (43740*D*b^3*c^5*d - 41310*(3*D*a*b^2 + C*b^3)*c^4*d^2 + 38556*(3*D*a 
^2*b + 3*C*a*b^2 + B*b^3)*c^3*d^3 - 35343*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + 
 A*b^3)*c^2*d^4 + 31416*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^5 - 26180*(B*a 
^3 + 3*A*a^2*b)*d^6)*x)*(d*x + c)^(2/3)/(d^8*x + c*d^7)
 

Sympy [A] (verification not implemented)

Time = 19.54 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.96 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{4/3}} \, dx=\begin {cases} \frac {3 \left (\frac {D b^{3} \left (c + d x\right )^{\frac {17}{3}}}{17 d^{6}} + \frac {\left (c + d x\right )^{\frac {14}{3}} \left (C b^{3} d + 3 D a b^{2} d - 6 D b^{3} c\right )}{14 d^{6}} + \frac {\left (c + d x\right )^{\frac {11}{3}} \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 )}{11 d^{6}} + \frac {\left (c + d x\right )^{\frac {8}{3}} \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 )}{8 d^{6}} + \frac {\left (c + d x\right )^{\frac {5}{3}} \cdot \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 )}{5 d^{6}} + \frac {\left (c + d x\right )^{\frac {2}{3}} \cdot \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 )}{2 d^{6}} + \frac {\left (a d - b c\right )^{3} \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{6} \sqrt [3]{c + d x}}\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 {4}{3}}} & \text {otherwise} \end {cases} \] Input:

integrate((b*x+a)**3*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(4/3),x)
 

Output:

Piecewise((3*(D*b**3*(c + d*x)**(17/3)/(17*d**6) + (c + d*x)**(14/3)*(C*b* 
*3*d + 3*D*a*b**2*d - 6*D*b**3*c)/(14*d**6) + (c + d*x)**(11/3)*(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)**(8/3)*(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)/(8*d**6) + (c + d*x)**(5/3)*(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)/(5*d**6) + ( 
c + d*x)**(2/3)*(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)/(2*d**6) + (a*d - b*c)**3*(-A*d**3 + B*c*d**2 - C*c* 
*2*d + D*c**3)/(d**6*(c + d*x)**(1/3)))/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**(4/3), T 
rue))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 629, 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)^{4/3}} \, dx=\frac {3 \, {\left (\frac {3080 \, {\left (d x + c\right )}^{\frac {17}{3}} D b^{3} - 3740 \, {\left (6 \, D b^{3} c - {\left (3 \, D a b^{2} + C b^{3}\right )} d\right )} {\left (d x + c\right )}^{\frac {14}{3}} + 4760 \, {\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}{3}} - 6545 \, {\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 {8}{3}} + 10472 \, {\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 {5}{3}} - 26180 \, {\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 {2}{3}}}{d^{6}} - \frac {52360 \, {\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 {1}{3}} d^{6}}\right )}}{52360 \, d} \] Input:

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(4/3),x, algorithm="maxima 
")
 

Output:

3/52360*((3080*(d*x + c)^(17/3)*D*b^3 - 3740*(6*D*b^3*c - (3*D*a*b^2 + C*b 
^3)*d)*(d*x + c)^(14/3) + 4760*(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)^(11/3) - 6545*(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)^(8/3) + 10472*(1 
5*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)^(5/3) - 26180*(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)^(2/3))/d^6 - 52360*(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)^(1/3 
)*d^6))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1067 vs. \(2 (412) = 824\).

Time = 0.15 (sec) , antiderivative size = 1067, normalized size of antiderivative = 2.45 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{4/3}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(4/3),x, algorithm="giac")
 

Output:

-3*(D*b^3*c^6 - 3*D*a*b^2*c^5*d - C*b^3*c^5*d + 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 - 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 + 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 - B*a^3*c*d^5 - 3*A*a^2*b*c*d^5 + A*a^3*d^6)/((d*x + c)^(1/3)*d^7) 
 + 3/52360*(3080*(d*x + c)^(17/3)*D*b^3*d^112 - 22440*(d*x + c)^(14/3)*D*b 
^3*c*d^112 + 71400*(d*x + c)^(11/3)*D*b^3*c^2*d^112 - 130900*(d*x + c)^(8/ 
3)*D*b^3*c^3*d^112 + 157080*(d*x + c)^(5/3)*D*b^3*c^4*d^112 - 157080*(d*x 
+ c)^(2/3)*D*b^3*c^5*d^112 + 11220*(d*x + c)^(14/3)*D*a*b^2*d^113 + 3740*( 
d*x + c)^(14/3)*C*b^3*d^113 - 71400*(d*x + c)^(11/3)*D*a*b^2*c*d^113 - 238 
00*(d*x + c)^(11/3)*C*b^3*c*d^113 + 196350*(d*x + c)^(8/3)*D*a*b^2*c^2*d^1 
13 + 65450*(d*x + c)^(8/3)*C*b^3*c^2*d^113 - 314160*(d*x + c)^(5/3)*D*a*b^ 
2*c^3*d^113 - 104720*(d*x + c)^(5/3)*C*b^3*c^3*d^113 + 392700*(d*x + c)^(2 
/3)*D*a*b^2*c^4*d^113 + 130900*(d*x + c)^(2/3)*C*b^3*c^4*d^113 + 14280*(d* 
x + c)^(11/3)*D*a^2*b*d^114 + 14280*(d*x + c)^(11/3)*C*a*b^2*d^114 + 4760* 
(d*x + c)^(11/3)*B*b^3*d^114 - 78540*(d*x + c)^(8/3)*D*a^2*b*c*d^114 - 785 
40*(d*x + c)^(8/3)*C*a*b^2*c*d^114 - 26180*(d*x + c)^(8/3)*B*b^3*c*d^114 + 
 188496*(d*x + c)^(5/3)*D*a^2*b*c^2*d^114 + 188496*(d*x + c)^(5/3)*C*a*b^2 
*c^2*d^114 + 62832*(d*x + c)^(5/3)*B*b^3*c^2*d^114 - 314160*(d*x + c)^(2/3 
)*D*a^2*b*c^3*d^114 - 314160*(d*x + c)^(2/3)*C*a*b^2*c^3*d^114 - 104720*(d 
*x + c)^(2/3)*B*b^3*c^3*d^114 + 6545*(d*x + c)^(8/3)*D*a^3*d^115 + 1963...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{4/3}} \, dx=\int \frac {{\left (a+b\,x\right )}^3\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{4/3}} \,d x \] Input:

int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(4/3),x)
 

Output:

int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(4/3), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{4/3}} \, dx=\frac {\frac {81}{10} a \,b^{3} c^{2} d^{3} x -\frac {729}{440} a^{2} b \,c^{4} d^{2}-3 a^{4} d^{5}-\frac {2187}{5236} b^{3} c^{6}+18 a^{3} b c \,d^{4}+6 a^{3} b \,d^{5} x +\frac {9}{40} a^{3} c^{2} d^{4} x -\frac {3}{40} a^{3} c \,d^{5} x^{2}-\frac {162}{5} a^{2} b^{2} c^{2} d^{3}+\frac {18}{5} a^{2} b^{2} d^{5} x^{2}+\frac {9}{11} a^{2} b \,d^{6} x^{4}+\frac {243}{10} a \,b^{3} c^{3} d^{2}+\frac {3}{2} a \,b^{3} d^{5} x^{3}+\frac {2187}{1540} a \,b^{2} c^{5} d +\frac {9}{14} a \,b^{2} d^{6} x^{5}-\frac {243}{110} b^{4} c^{3} d^{2} x +\frac {81}{110} b^{4} c^{2} d^{3} x^{2}-\frac {9}{22} b^{4} c \,d^{4} x^{3}-\frac {729}{5236} b^{3} c^{5} d x +\frac {243}{5236} b^{3} c^{4} d^{2} x^{2}-\frac {135}{5236} b^{3} c^{3} d^{3} x^{3}+\frac {45}{2618} b^{3} c^{2} d^{4} x^{4}-\frac {3}{238} b^{3} c \,d^{5} x^{5}+\frac {27}{40} a^{3} c^{3} d^{3}+\frac {3}{8} a^{3} d^{6} x^{3}-\frac {729}{110} b^{4} c^{4} d +\frac {3}{11} b^{4} d^{5} x^{4}+\frac {3}{17} b^{3} d^{6} x^{6}-\frac {54}{5} a^{2} b^{2} c \,d^{4} x -\frac {243}{440} a^{2} b \,c^{3} d^{3} x +\frac {81}{440} a^{2} b \,c^{2} d^{4} x^{2}-\frac {9}{88} a^{2} b c \,d^{5} x^{3}-\frac {27}{10} a \,b^{3} c \,d^{4} x^{2}+\frac {729}{1540} a \,b^{2} c^{4} d^{2} x -\frac {243}{1540} a \,b^{2} c^{3} d^{3} x^{2}+\frac {27}{308} a \,b^{2} c^{2} d^{4} x^{3}-\frac {9}{154} a \,b^{2} c \,d^{5} x^{4}}{\left (d x +c \right )^{\frac {1}{3}} d^{6}} \] Input:

int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(4/3),x)
 

Output:

(3*( - 52360*a**4*d**5 + 314160*a**3*b*c*d**4 + 104720*a**3*b*d**5*x + 117 
81*a**3*c**3*d**3 + 3927*a**3*c**2*d**4*x - 1309*a**3*c*d**5*x**2 + 6545*a 
**3*d**6*x**3 - 565488*a**2*b**2*c**2*d**3 - 188496*a**2*b**2*c*d**4*x + 6 
2832*a**2*b**2*d**5*x**2 - 28917*a**2*b*c**4*d**2 - 9639*a**2*b*c**3*d**3* 
x + 3213*a**2*b*c**2*d**4*x**2 - 1785*a**2*b*c*d**5*x**3 + 14280*a**2*b*d* 
*6*x**4 + 424116*a*b**3*c**3*d**2 + 141372*a*b**3*c**2*d**3*x - 47124*a*b* 
*3*c*d**4*x**2 + 26180*a*b**3*d**5*x**3 + 24786*a*b**2*c**5*d + 8262*a*b** 
2*c**4*d**2*x - 2754*a*b**2*c**3*d**3*x**2 + 1530*a*b**2*c**2*d**4*x**3 - 
1020*a*b**2*c*d**5*x**4 + 11220*a*b**2*d**6*x**5 - 115668*b**4*c**4*d - 38 
556*b**4*c**3*d**2*x + 12852*b**4*c**2*d**3*x**2 - 7140*b**4*c*d**4*x**3 + 
 4760*b**4*d**5*x**4 - 7290*b**3*c**6 - 2430*b**3*c**5*d*x + 810*b**3*c**4 
*d**2*x**2 - 450*b**3*c**3*d**3*x**3 + 300*b**3*c**2*d**4*x**4 - 220*b**3* 
c*d**5*x**5 + 3080*b**3*d**6*x**6))/(52360*(c + d*x)**(1/3)*d**6)