\(\int (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4)^3 \, dx\) [26]

Optimal result

Integrand size = 34, antiderivative size = 320 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^3 \, dx=a^3 b^6 x+\frac {3}{2} a^2 b^7 x^2+a b^4 \left (b^4+a c\right ) x^3+\frac {1}{4} b^5 \left (b^4+6 a c+3 a^2 b d\right ) x^4+\frac {3}{5} b^2 \left (b^4 c+a c^2+2 a b^5 d+a^3 b^2 d^2\right ) x^5+\frac {1}{2} b^3 \left (c^2+2 a b c d+b^2 d \left (b^3+2 a^2 d\right )\right ) x^6+\frac {1}{7} \left (c^3+6 a b^6 d^2+6 b^2 c d \left (b^3+a^2 d\right )\right ) x^7+\frac {3}{8} b^2 d \left (c^2+2 a b c d+b^2 d \left (b^3+2 a^2 d\right )\right ) x^8+\frac {1}{3} d^2 \left (b^4 c+a c^2+2 a b^5 d+a^3 b^2 d^2\right ) x^9+\frac {1}{10} b^2 d^3 \left (b^4+6 a c+3 a^2 b d\right ) x^{10}+\frac {3}{11} a \left (b^4+a c\right ) d^4 x^{11}+\frac {1}{4} a^2 b^2 d^5 x^{12}+\frac {1}{13} a^3 d^6 x^{13} \] Output:

a^3*b^6*x+3/2*a^2*b^7*x^2+a*b^4*(b^4+a*c)*x^3+1/4*b^5*(3*a^2*b*d+b^4+6*a*c 
)*x^4+3/5*b^2*(a^3*b^2*d^2+2*a*b^5*d+b^4*c+a*c^2)*x^5+1/2*b^3*(c^2+2*a*b*c 
*d+b^2*d*(2*a^2*d+b^3))*x^6+1/7*(c^3+6*a*b^6*d^2+6*b^2*c*d*(a^2*d+b^3))*x^ 
7+3/8*b^2*d*(c^2+2*a*b*c*d+b^2*d*(2*a^2*d+b^3))*x^8+1/3*d^2*(a^3*b^2*d^2+2 
*a*b^5*d+b^4*c+a*c^2)*x^9+1/10*b^2*d^3*(3*a^2*b*d+b^4+6*a*c)*x^10+3/11*a*( 
b^4+a*c)*d^4*x^11+1/4*a^2*b^2*d^5*x^12+1/13*a^3*d^6*x^13
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^3 \, dx=a^3 b^6 x+\frac {3}{2} a^2 b^7 x^2+a b^4 \left (b^4+a c\right ) x^3+\frac {1}{4} b^5 \left (b^4+6 a c+3 a^2 b d\right ) x^4+\frac {3}{5} b^2 \left (b^4 c+a c^2+2 a b^5 d+a^3 b^2 d^2\right ) x^5+\frac {1}{2} b^3 \left (c^2+2 a b c d+b^2 d \left (b^3+2 a^2 d\right )\right ) x^6+\frac {1}{7} \left (c^3+6 a b^6 d^2+6 b^2 c d \left (b^3+a^2 d\right )\right ) x^7+\frac {3}{8} b^2 d \left (c^2+2 a b c d+b^2 d \left (b^3+2 a^2 d\right )\right ) x^8+\frac {1}{3} d^2 \left (b^4 c+a c^2+2 a b^5 d+a^3 b^2 d^2\right ) x^9+\frac {1}{10} b^2 d^3 \left (b^4+6 a c+3 a^2 b d\right ) x^{10}+\frac {3}{11} a \left (b^4+a c\right ) d^4 x^{11}+\frac {1}{4} a^2 b^2 d^5 x^{12}+\frac {1}{13} a^3 d^6 x^{13} \] Input:

Integrate[(a*b^2 + b^3*x + c*x^2 + b^2*d*x^3 + a*d^2*x^4)^3,x]
 

Output:

a^3*b^6*x + (3*a^2*b^7*x^2)/2 + a*b^4*(b^4 + a*c)*x^3 + (b^5*(b^4 + 6*a*c 
+ 3*a^2*b*d)*x^4)/4 + (3*b^2*(b^4*c + a*c^2 + 2*a*b^5*d + a^3*b^2*d^2)*x^5 
)/5 + (b^3*(c^2 + 2*a*b*c*d + b^2*d*(b^3 + 2*a^2*d))*x^6)/2 + ((c^3 + 6*a* 
b^6*d^2 + 6*b^2*c*d*(b^3 + a^2*d))*x^7)/7 + (3*b^2*d*(c^2 + 2*a*b*c*d + b^ 
2*d*(b^3 + 2*a^2*d))*x^8)/8 + (d^2*(b^4*c + a*c^2 + 2*a*b^5*d + a^3*b^2*d^ 
2)*x^9)/3 + (b^2*d^3*(b^4 + 6*a*c + 3*a^2*b*d)*x^10)/10 + (3*a*(b^4 + a*c) 
*d^4*x^11)/11 + (a^2*b^2*d^5*x^12)/4 + (a^3*d^6*x^13)/13
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2465, 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.

\(\displaystyle \int \left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^3 \, dx\)

\(\Big \downarrow \) 2465

\(\displaystyle \int \left (a^3 b^6+3 d^2 x^8 \left (a^3 b^2 d^2+2 a b^5 d+a c^2+b^4 c\right )+3 b^2 x^4 \left (a^3 b^2 d^2+2 a b^5 d+a c^2+b^4 c\right )+a^3 d^6 x^{12}+3 a^2 b^7 x+3 a^2 b^2 d^5 x^{11}+b^5 x^3 \left (3 a^2 b d+6 a c+b^4\right )+b^2 d^3 x^9 \left (3 a^2 b d+6 a c+b^4\right )+3 b^2 d x^7 \left (b^2 d \left (2 a^2 d+b^3\right )+2 a b c d+c^2\right )+3 b^3 x^5 \left (b^2 d \left (2 a^2 d+b^3\right )+2 a b c d+c^2\right )+x^6 \left (6 b^2 c d \left (a^2 d+b^3\right )+6 a b^6 d^2+c^3\right )+3 a d^4 x^{10} \left (a c+b^4\right )+3 a b^4 x^2 \left (a c+b^4\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle a^3 b^6 x+\frac {1}{3} d^2 x^9 \left (a^3 b^2 d^2+2 a b^5 d+a c^2+b^4 c\right )+\frac {3}{5} b^2 x^5 \left (a^3 b^2 d^2+2 a b^5 d+a c^2+b^4 c\right )+\frac {1}{13} a^3 d^6 x^{13}+\frac {3}{2} a^2 b^7 x^2+\frac {1}{4} a^2 b^2 d^5 x^{12}+\frac {1}{4} b^5 x^4 \left (3 a^2 b d+6 a c+b^4\right )+\frac {1}{10} b^2 d^3 x^{10} \left (3 a^2 b d+6 a c+b^4\right )+\frac {3}{8} b^2 d x^8 \left (b^2 d \left (2 a^2 d+b^3\right )+2 a b c d+c^2\right )+\frac {1}{2} b^3 x^6 \left (b^2 d \left (2 a^2 d+b^3\right )+2 a b c d+c^2\right )+\frac {1}{7} x^7 \left (6 b^2 c d \left (a^2 d+b^3\right )+6 a b^6 d^2+c^3\right )+\frac {3}{11} a d^4 x^{11} \left (a c+b^4\right )+a b^4 x^3 \left (a c+b^4\right )\)

Input:

Int[(a*b^2 + b^3*x + c*x^2 + b^2*d*x^3 + a*d^2*x^4)^3,x]
 

Output:

a^3*b^6*x + (3*a^2*b^7*x^2)/2 + a*b^4*(b^4 + a*c)*x^3 + (b^5*(b^4 + 6*a*c 
+ 3*a^2*b*d)*x^4)/4 + (3*b^2*(b^4*c + a*c^2 + 2*a*b^5*d + a^3*b^2*d^2)*x^5 
)/5 + (b^3*(c^2 + 2*a*b*c*d + b^2*d*(b^3 + 2*a^2*d))*x^6)/2 + ((c^3 + 6*a* 
b^6*d^2 + 6*b^2*c*d*(b^3 + a^2*d))*x^7)/7 + (3*b^2*d*(c^2 + 2*a*b*c*d + b^ 
2*d*(b^3 + 2*a^2*d))*x^8)/8 + (d^2*(b^4*c + a*c^2 + 2*a*b^5*d + a^3*b^2*d^ 
2)*x^9)/3 + (b^2*d^3*(b^4 + 6*a*c + 3*a^2*b*d)*x^10)/10 + (3*a*(b^4 + a*c) 
*d^4*x^11)/11 + (a^2*b^2*d^5*x^12)/4 + (a^3*d^6*x^13)/13
 

Defintions of rubi rules used

rule 2009

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

rule 2465

Int[(u_.)*(Px_)^(p_), x_Symbol] :> Int[ExpandToSum[u, Px^p, x], x] /; PolyQ 
[Px, x] && GtQ[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x 
] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.08

method	result	size
norman	\(\frac {a^{3} d^{6} x^{13}}{13}+\frac {a^{2} b^{2} d^{5} x^{12}}{4}+\left (\frac {3}{11} a \,b^{4} d^{4}+\frac {3}{11} a^{2} c \,d^{4}\right ) x^{11}+\left (\frac {3}{10} b^{3} a^{2} d^{4}+\frac {1}{10} b^{6} d^{3}+\frac {3}{5} a \,b^{2} c \,d^{3}\right ) x^{10}+\left (\frac {1}{3} a^{3} b^{2} d^{4}+\frac {2}{3} a \,b^{5} d^{3}+\frac {1}{3} b^{4} c \,d^{2}+\frac {1}{3} a \,c^{2} d^{2}\right ) x^{9}+\left (\frac {3}{4} b^{4} a^{2} d^{3}+\frac {3}{8} b^{7} d^{2}+\frac {3}{4} a \,b^{3} c \,d^{2}+\frac {3}{8} b^{2} c^{2} d \right ) x^{8}+\left (\frac {6}{7} a \,b^{6} d^{2}+\frac {6}{7} a^{2} b^{2} c \,d^{2}+\frac {6}{7} b^{5} c d +\frac {1}{7} c^{3}\right ) x^{7}+\left (a^{2} d^{2} b^{5}+\frac {1}{2} b^{8} d +a \,b^{4} c d +\frac {1}{2} b^{3} c^{2}\right ) x^{6}+\left (\frac {3}{5} a^{3} d^{2} b^{4}+\frac {6}{5} d \,b^{7} a +\frac {3}{5} b^{6} c +\frac {3}{5} a \,b^{2} c^{2}\right ) x^{5}+\left (\frac {3}{4} d \,b^{6} a^{2}+\frac {1}{4} b^{9}+\frac {3}{2} a \,b^{5} c \right ) x^{4}+\left (a \,b^{8}+a^{2} b^{4} c \right ) x^{3}+\frac {3 a^{2} b^{7} x^{2}}{2}+a^{3} b^{6} x\)	\(345\)
gosper	\(a^{3} b^{6} x +\frac {6}{7} x^{7} b^{5} c d +x^{6} a^{2} d^{2} b^{5}+\frac {3}{5} x^{5} a^{3} d^{2} b^{4}+\frac {6}{5} x^{5} d \,b^{7} a +\frac {3}{5} x^{5} a \,b^{2} c^{2}+\frac {3}{4} x^{4} d \,b^{6} a^{2}+\frac {3}{2} x^{4} a \,b^{5} c +a^{2} b^{4} c \,x^{3}+\frac {3}{8} x^{8} b^{2} c^{2} d +\frac {6}{7} x^{7} a \,b^{6} d^{2}+\frac {3}{4} x^{8} b^{4} a^{2} d^{3}+\frac {1}{3} x^{9} a^{3} b^{2} d^{4}+\frac {2}{3} x^{9} a \,b^{5} d^{3}+\frac {1}{3} x^{9} b^{4} c \,d^{2}+\frac {1}{3} x^{9} a \,c^{2} d^{2}+\frac {3}{11} x^{11} a \,b^{4} d^{4}+\frac {3}{11} x^{11} a^{2} c \,d^{4}+\frac {3}{10} x^{10} b^{3} a^{2} d^{4}+\frac {3}{5} x^{10} a \,b^{2} c \,d^{3}+\frac {3}{4} x^{8} a \,b^{3} c \,d^{2}+\frac {6}{7} x^{7} a^{2} b^{2} c \,d^{2}+x^{6} a \,b^{4} c d +\frac {1}{4} a^{2} b^{2} d^{5} x^{12}+\frac {1}{7} x^{7} c^{3}+\frac {1}{2} x^{6} b^{3} c^{2}+\frac {3}{5} x^{5} b^{6} c +a \,b^{8} x^{3}+\frac {1}{4} x^{4} b^{9}+\frac {1}{13} a^{3} d^{6} x^{13}+\frac {1}{10} x^{10} b^{6} d^{3}+\frac {3}{8} x^{8} b^{7} d^{2}+\frac {1}{2} x^{6} b^{8} d +\frac {3}{2} a^{2} b^{7} x^{2}\)	\(390\)
risch	\(a^{3} b^{6} x +\frac {6}{7} x^{7} b^{5} c d +x^{6} a^{2} d^{2} b^{5}+\frac {3}{5} x^{5} a^{3} d^{2} b^{4}+\frac {6}{5} x^{5} d \,b^{7} a +\frac {3}{5} x^{5} a \,b^{2} c^{2}+\frac {3}{4} x^{4} d \,b^{6} a^{2}+\frac {3}{2} x^{4} a \,b^{5} c +a^{2} b^{4} c \,x^{3}+\frac {3}{8} x^{8} b^{2} c^{2} d +\frac {6}{7} x^{7} a \,b^{6} d^{2}+\frac {3}{4} x^{8} b^{4} a^{2} d^{3}+\frac {1}{3} x^{9} a^{3} b^{2} d^{4}+\frac {2}{3} x^{9} a \,b^{5} d^{3}+\frac {1}{3} x^{9} b^{4} c \,d^{2}+\frac {1}{3} x^{9} a \,c^{2} d^{2}+\frac {3}{11} x^{11} a \,b^{4} d^{4}+\frac {3}{11} x^{11} a^{2} c \,d^{4}+\frac {3}{10} x^{10} b^{3} a^{2} d^{4}+\frac {3}{5} x^{10} a \,b^{2} c \,d^{3}+\frac {3}{4} x^{8} a \,b^{3} c \,d^{2}+\frac {6}{7} x^{7} a^{2} b^{2} c \,d^{2}+x^{6} a \,b^{4} c d +\frac {1}{4} a^{2} b^{2} d^{5} x^{12}+\frac {1}{7} x^{7} c^{3}+\frac {1}{2} x^{6} b^{3} c^{2}+\frac {3}{5} x^{5} b^{6} c +a \,b^{8} x^{3}+\frac {1}{4} x^{4} b^{9}+\frac {1}{13} a^{3} d^{6} x^{13}+\frac {1}{10} x^{10} b^{6} d^{3}+\frac {3}{8} x^{8} b^{7} d^{2}+\frac {1}{2} x^{6} b^{8} d +\frac {3}{2} a^{2} b^{7} x^{2}\)	\(390\)
parallelrisch	\(a^{3} b^{6} x +\frac {6}{7} x^{7} b^{5} c d +x^{6} a^{2} d^{2} b^{5}+\frac {3}{5} x^{5} a^{3} d^{2} b^{4}+\frac {6}{5} x^{5} d \,b^{7} a +\frac {3}{5} x^{5} a \,b^{2} c^{2}+\frac {3}{4} x^{4} d \,b^{6} a^{2}+\frac {3}{2} x^{4} a \,b^{5} c +a^{2} b^{4} c \,x^{3}+\frac {3}{8} x^{8} b^{2} c^{2} d +\frac {6}{7} x^{7} a \,b^{6} d^{2}+\frac {3}{4} x^{8} b^{4} a^{2} d^{3}+\frac {1}{3} x^{9} a^{3} b^{2} d^{4}+\frac {2}{3} x^{9} a \,b^{5} d^{3}+\frac {1}{3} x^{9} b^{4} c \,d^{2}+\frac {1}{3} x^{9} a \,c^{2} d^{2}+\frac {3}{11} x^{11} a \,b^{4} d^{4}+\frac {3}{11} x^{11} a^{2} c \,d^{4}+\frac {3}{10} x^{10} b^{3} a^{2} d^{4}+\frac {3}{5} x^{10} a \,b^{2} c \,d^{3}+\frac {3}{4} x^{8} a \,b^{3} c \,d^{2}+\frac {6}{7} x^{7} a^{2} b^{2} c \,d^{2}+x^{6} a \,b^{4} c d +\frac {1}{4} a^{2} b^{2} d^{5} x^{12}+\frac {1}{7} x^{7} c^{3}+\frac {1}{2} x^{6} b^{3} c^{2}+\frac {3}{5} x^{5} b^{6} c +a \,b^{8} x^{3}+\frac {1}{4} x^{4} b^{9}+\frac {1}{13} a^{3} d^{6} x^{13}+\frac {1}{10} x^{10} b^{6} d^{3}+\frac {3}{8} x^{8} b^{7} d^{2}+\frac {1}{2} x^{6} b^{8} d +\frac {3}{2} a^{2} b^{7} x^{2}\)	\(390\)
orering	\(\frac {x \left (9240 a^{3} d^{6} x^{12}+30030 d^{5} b^{2} a^{2} x^{11}+32760 a \,b^{4} d^{4} x^{10}+36036 a^{2} b^{3} d^{4} x^{9}+12012 b^{6} d^{3} x^{9}+40040 a^{3} b^{2} d^{4} x^{8}+32760 a^{2} c \,d^{4} x^{10}+80080 a \,b^{5} d^{3} x^{8}+90090 a^{2} b^{4} d^{3} x^{7}+72072 a \,b^{2} c \,d^{3} x^{9}+45045 b^{7} d^{2} x^{7}+102960 a \,b^{6} d^{2} x^{6}+40040 b^{4} c \,d^{2} x^{8}+120120 a^{2} b^{5} d^{2} x^{5}+90090 a \,b^{3} c \,d^{2} x^{7}+60060 b^{8} d \,x^{5}+72072 a^{3} b^{4} d^{2} x^{4}+102960 a^{2} b^{2} c \,d^{2} x^{6}+144144 a \,b^{7} d \,x^{4}+40040 a \,c^{2} d^{2} x^{8}+102960 b^{5} c d \,x^{6}+90090 a^{2} b^{6} d \,x^{3}+120120 a \,b^{4} c d \,x^{5}+30030 b^{9} x^{3}+45045 b^{2} c^{2} d \,x^{7}+120120 a \,b^{8} x^{2}+72072 b^{6} c \,x^{4}+180180 a^{2} b^{7} x +180180 a \,b^{5} c \,x^{3}+60060 b^{3} c^{2} x^{5}+120120 a^{3} b^{6}+120120 a^{2} b^{4} c \,x^{2}+72072 a \,b^{2} c^{2} x^{4}+17160 c^{3} x^{6}\right )}{120120}\)	\(395\)
default	\(\frac {a^{3} d^{6} x^{13}}{13}+\frac {a^{2} b^{2} d^{5} x^{12}}{4}+\frac {\left (a^{2} c \,d^{4}+2 a \,b^{4} d^{4}+a \,d^{2} \left (b^{4} d^{2}+2 a c \,d^{2}\right )\right ) x^{11}}{11}+\frac {\left (b^{3} a^{2} d^{4}+2 a \,b^{2} c \,d^{3}+d \,b^{2} \left (b^{4} d^{2}+2 a c \,d^{2}\right )+a \,d^{2} \left (2 b^{3} a \,d^{2}+2 b^{2} c d \right )\right ) x^{10}}{10}+\frac {\left (a^{3} b^{2} d^{4}+2 a \,b^{5} d^{3}+c \left (b^{4} d^{2}+2 a c \,d^{2}\right )+d \,b^{2} \left (2 b^{3} a \,d^{2}+2 b^{2} c d \right )+a \,d^{2} \left (2 a^{2} b^{2} d^{2}+2 b^{5} d +c^{2}\right )\right ) x^{9}}{9}+\frac {\left (2 b^{4} a^{2} d^{3}+b^{3} \left (b^{4} d^{2}+2 a c \,d^{2}\right )+c \left (2 b^{3} a \,d^{2}+2 b^{2} c d \right )+d \,b^{2} \left (2 a^{2} b^{2} d^{2}+2 b^{5} d +c^{2}\right )+a \,d^{2} \left (2 a \,b^{4} d +2 b^{3} c \right )\right ) x^{8}}{8}+\frac {\left (b^{2} a \left (b^{4} d^{2}+2 a c \,d^{2}\right )+b^{3} \left (2 b^{3} a \,d^{2}+2 b^{2} c d \right )+c \left (2 a^{2} b^{2} d^{2}+2 b^{5} d +c^{2}\right )+d \,b^{2} \left (2 a \,b^{4} d +2 b^{3} c \right )+a \,d^{2} \left (b^{6}+2 a \,b^{2} c \right )\right ) x^{7}}{7}+\frac {\left (b^{2} a \left (2 b^{3} a \,d^{2}+2 b^{2} c d \right )+b^{3} \left (2 a^{2} b^{2} d^{2}+2 b^{5} d +c^{2}\right )+c \left (2 a \,b^{4} d +2 b^{3} c \right )+d \,b^{2} \left (b^{6}+2 a \,b^{2} c \right )+2 a^{2} d^{2} b^{5}\right ) x^{6}}{6}+\frac {\left (b^{2} a \left (2 a^{2} b^{2} d^{2}+2 b^{5} d +c^{2}\right )+b^{3} \left (2 a \,b^{4} d +2 b^{3} c \right )+c \left (b^{6}+2 a \,b^{2} c \right )+2 d \,b^{7} a +a^{3} d^{2} b^{4}\right ) x^{5}}{5}+\frac {\left (b^{2} a \left (2 a \,b^{4} d +2 b^{3} c \right )+b^{3} \left (b^{6}+2 a \,b^{2} c \right )+2 a \,b^{5} c +d \,b^{6} a^{2}\right ) x^{4}}{4}+\frac {\left (b^{2} a \left (b^{6}+2 a \,b^{2} c \right )+2 a \,b^{8}+a^{2} b^{4} c \right ) x^{3}}{3}+\frac {3 a^{2} b^{7} x^{2}}{2}+a^{3} b^{6} x\)	\(719\)

Input:

int((a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^3,x,method=_RETURNVERBOSE)
 

Output:

1/13*a^3*d^6*x^13+1/4*a^2*b^2*d^5*x^12+(3/11*a*b^4*d^4+3/11*a^2*c*d^4)*x^1 
1+(3/10*b^3*a^2*d^4+1/10*b^6*d^3+3/5*a*b^2*c*d^3)*x^10+(1/3*a^3*b^2*d^4+2/ 
3*a*b^5*d^3+1/3*b^4*c*d^2+1/3*a*c^2*d^2)*x^9+(3/4*b^4*a^2*d^3+3/8*b^7*d^2+ 
3/4*a*b^3*c*d^2+3/8*b^2*c^2*d)*x^8+(6/7*a*b^6*d^2+6/7*a^2*b^2*c*d^2+6/7*b^ 
5*c*d+1/7*c^3)*x^7+(a^2*d^2*b^5+1/2*b^8*d+a*b^4*c*d+1/2*b^3*c^2)*x^6+(3/5* 
a^3*d^2*b^4+6/5*d*b^7*a+3/5*b^6*c+3/5*a*b^2*c^2)*x^5+(3/4*d*b^6*a^2+1/4*b^ 
9+3/2*a*b^5*c)*x^4+(a*b^8+a^2*b^4*c)*x^3+3/2*a^2*b^7*x^2+a^3*b^6*x
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.02 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^3 \, dx=\frac {1}{13} \, a^{3} d^{6} x^{13} + \frac {1}{4} \, a^{2} b^{2} d^{5} x^{12} + \frac {3}{11} \, {\left (a b^{4} + a^{2} c\right )} d^{4} x^{11} + \frac {3}{2} \, a^{2} b^{7} x^{2} + \frac {1}{10} \, {\left (3 \, a^{2} b^{3} d^{4} + {\left (b^{6} + 6 \, a b^{2} c\right )} d^{3}\right )} x^{10} + a^{3} b^{6} x + \frac {1}{3} \, {\left (2 \, a b^{5} d^{3} + a^{3} b^{2} d^{4} + {\left (b^{4} c + a c^{2}\right )} d^{2}\right )} x^{9} + \frac {3}{8} \, {\left (2 \, a^{2} b^{4} d^{3} + b^{2} c^{2} d + {\left (b^{7} + 2 \, a b^{3} c\right )} d^{2}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, b^{5} c d + c^{3} + 6 \, {\left (a b^{6} + a^{2} b^{2} c\right )} d^{2}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a^{2} b^{5} d^{2} + b^{3} c^{2} + {\left (b^{8} + 2 \, a b^{4} c\right )} d\right )} x^{6} + \frac {3}{5} \, {\left (2 \, a b^{7} d + a^{3} b^{4} d^{2} + b^{6} c + a b^{2} c^{2}\right )} x^{5} + \frac {1}{4} \, {\left (b^{9} + 3 \, a^{2} b^{6} d + 6 \, a b^{5} c\right )} x^{4} + {\left (a b^{8} + a^{2} b^{4} c\right )} x^{3} \] Input:

integrate((a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^3,x, algorithm="fricas")
 

Output:

1/13*a^3*d^6*x^13 + 1/4*a^2*b^2*d^5*x^12 + 3/11*(a*b^4 + a^2*c)*d^4*x^11 + 
 3/2*a^2*b^7*x^2 + 1/10*(3*a^2*b^3*d^4 + (b^6 + 6*a*b^2*c)*d^3)*x^10 + a^3 
*b^6*x + 1/3*(2*a*b^5*d^3 + a^3*b^2*d^4 + (b^4*c + a*c^2)*d^2)*x^9 + 3/8*( 
2*a^2*b^4*d^3 + b^2*c^2*d + (b^7 + 2*a*b^3*c)*d^2)*x^8 + 1/7*(6*b^5*c*d + 
c^3 + 6*(a*b^6 + a^2*b^2*c)*d^2)*x^7 + 1/2*(2*a^2*b^5*d^2 + b^3*c^2 + (b^8 
 + 2*a*b^4*c)*d)*x^6 + 3/5*(2*a*b^7*d + a^3*b^4*d^2 + b^6*c + a*b^2*c^2)*x 
^5 + 1/4*(b^9 + 3*a^2*b^6*d + 6*a*b^5*c)*x^4 + (a*b^8 + a^2*b^4*c)*x^3
 

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.23 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^3 \, dx=a^{3} b^{6} x + \frac {a^{3} d^{6} x^{13}}{13} + \frac {3 a^{2} b^{7} x^{2}}{2} + \frac {a^{2} b^{2} d^{5} x^{12}}{4} + x^{11} \cdot \left (\frac {3 a^{2} c d^{4}}{11} + \frac {3 a b^{4} d^{4}}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{2} b^{3} d^{4}}{10} + \frac {3 a b^{2} c d^{3}}{5} + \frac {b^{6} d^{3}}{10}\right ) + x^{9} \left (\frac {a^{3} b^{2} d^{4}}{3} + \frac {2 a b^{5} d^{3}}{3} + \frac {a c^{2} d^{2}}{3} + \frac {b^{4} c d^{2}}{3}\right ) + x^{8} \cdot \left (\frac {3 a^{2} b^{4} d^{3}}{4} + \frac {3 a b^{3} c d^{2}}{4} + \frac {3 b^{7} d^{2}}{8} + \frac {3 b^{2} c^{2} d}{8}\right ) + x^{7} \cdot \left (\frac {6 a^{2} b^{2} c d^{2}}{7} + \frac {6 a b^{6} d^{2}}{7} + \frac {6 b^{5} c d}{7} + \frac {c^{3}}{7}\right ) + x^{6} \left (a^{2} b^{5} d^{2} + a b^{4} c d + \frac {b^{8} d}{2} + \frac {b^{3} c^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{3} b^{4} d^{2}}{5} + \frac {6 a b^{7} d}{5} + \frac {3 a b^{2} c^{2}}{5} + \frac {3 b^{6} c}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} b^{6} d}{4} + \frac {3 a b^{5} c}{2} + \frac {b^{9}}{4}\right ) + x^{3} \left (a^{2} b^{4} c + a b^{8}\right ) \] Input:

integrate((a*d**2*x**4+b**2*d*x**3+b**3*x+a*b**2+c*x**2)**3,x)
 

Output:

a**3*b**6*x + a**3*d**6*x**13/13 + 3*a**2*b**7*x**2/2 + a**2*b**2*d**5*x** 
12/4 + x**11*(3*a**2*c*d**4/11 + 3*a*b**4*d**4/11) + x**10*(3*a**2*b**3*d* 
*4/10 + 3*a*b**2*c*d**3/5 + b**6*d**3/10) + x**9*(a**3*b**2*d**4/3 + 2*a*b 
**5*d**3/3 + a*c**2*d**2/3 + b**4*c*d**2/3) + x**8*(3*a**2*b**4*d**3/4 + 3 
*a*b**3*c*d**2/4 + 3*b**7*d**2/8 + 3*b**2*c**2*d/8) + x**7*(6*a**2*b**2*c* 
d**2/7 + 6*a*b**6*d**2/7 + 6*b**5*c*d/7 + c**3/7) + x**6*(a**2*b**5*d**2 + 
 a*b**4*c*d + b**8*d/2 + b**3*c**2/2) + x**5*(3*a**3*b**4*d**2/5 + 6*a*b** 
7*d/5 + 3*a*b**2*c**2/5 + 3*b**6*c/5) + x**4*(3*a**2*b**6*d/4 + 3*a*b**5*c 
/2 + b**9/4) + x**3*(a**2*b**4*c + a*b**8)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.18 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^3 \, dx=\frac {1}{13} \, a^{3} d^{6} x^{13} + \frac {1}{4} \, a^{2} b^{2} d^{5} x^{12} + \frac {3}{11} \, a b^{4} d^{4} x^{11} + \frac {1}{10} \, b^{6} d^{3} x^{10} + \frac {1}{4} \, b^{9} x^{4} + a^{3} b^{6} x + \frac {1}{7} \, c^{3} x^{7} + \frac {1}{20} \, {\left (12 \, a d^{2} x^{5} + 15 \, b^{2} d x^{4} + 30 \, b^{3} x^{2} + 20 \, c x^{3}\right )} a^{2} b^{4} + \frac {1}{70} \, {\left (30 \, a d^{2} x^{7} + 35 \, b^{2} d x^{6} + 42 \, c x^{5}\right )} b^{6} + \frac {1}{420} \, {\left (140 \, a^{2} d^{4} x^{9} + 315 \, a b^{2} d^{3} x^{8} + 180 \, b^{4} d^{2} x^{7} + 420 \, b^{6} x^{3} + 252 \, c^{2} x^{5} + 42 \, {\left (10 \, a d^{2} x^{6} + 12 \, b^{2} d x^{5} + 15 \, c x^{4}\right )} b^{3} + 60 \, {\left (6 \, a d^{2} x^{7} + 7 \, b^{2} d x^{6}\right )} c\right )} a b^{2} + \frac {1}{840} \, {\left (252 \, a^{2} d^{4} x^{10} + 560 \, a b^{2} d^{3} x^{9} + 720 \, b^{2} c d x^{7} + 315 \, {\left (b^{4} + 2 \, a c\right )} d^{2} x^{8} + 420 \, c^{2} x^{6}\right )} b^{3} + \frac {1}{24} \, {\left (8 \, a d^{2} x^{9} + 9 \, b^{2} d x^{8}\right )} c^{2} + \frac {1}{165} \, {\left (45 \, a^{2} d^{4} x^{11} + 99 \, a b^{2} d^{3} x^{10} + 55 \, b^{4} d^{2} x^{9}\right )} c \] Input:

integrate((a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^3,x, algorithm="maxima")
 

Output:

1/13*a^3*d^6*x^13 + 1/4*a^2*b^2*d^5*x^12 + 3/11*a*b^4*d^4*x^11 + 1/10*b^6* 
d^3*x^10 + 1/4*b^9*x^4 + a^3*b^6*x + 1/7*c^3*x^7 + 1/20*(12*a*d^2*x^5 + 15 
*b^2*d*x^4 + 30*b^3*x^2 + 20*c*x^3)*a^2*b^4 + 1/70*(30*a*d^2*x^7 + 35*b^2* 
d*x^6 + 42*c*x^5)*b^6 + 1/420*(140*a^2*d^4*x^9 + 315*a*b^2*d^3*x^8 + 180*b 
^4*d^2*x^7 + 420*b^6*x^3 + 252*c^2*x^5 + 42*(10*a*d^2*x^6 + 12*b^2*d*x^5 + 
 15*c*x^4)*b^3 + 60*(6*a*d^2*x^7 + 7*b^2*d*x^6)*c)*a*b^2 + 1/840*(252*a^2* 
d^4*x^10 + 560*a*b^2*d^3*x^9 + 720*b^2*c*d*x^7 + 315*(b^4 + 2*a*c)*d^2*x^8 
 + 420*c^2*x^6)*b^3 + 1/24*(8*a*d^2*x^9 + 9*b^2*d*x^8)*c^2 + 1/165*(45*a^2 
*d^4*x^11 + 99*a*b^2*d^3*x^10 + 55*b^4*d^2*x^9)*c
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.22 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^3 \, dx=\frac {1}{13} \, a^{3} d^{6} x^{13} + \frac {1}{4} \, a^{2} b^{2} d^{5} x^{12} + \frac {3}{11} \, a b^{4} d^{4} x^{11} + \frac {1}{10} \, b^{6} d^{3} x^{10} + \frac {3}{10} \, a^{2} b^{3} d^{4} x^{10} + \frac {2}{3} \, a b^{5} d^{3} x^{9} + \frac {1}{3} \, a^{3} b^{2} d^{4} x^{9} + \frac {3}{11} \, a^{2} c d^{4} x^{11} + \frac {3}{8} \, b^{7} d^{2} x^{8} + \frac {3}{4} \, a^{2} b^{4} d^{3} x^{8} + \frac {3}{5} \, a b^{2} c d^{3} x^{10} + \frac {6}{7} \, a b^{6} d^{2} x^{7} + \frac {1}{3} \, b^{4} c d^{2} x^{9} + \frac {1}{2} \, b^{8} d x^{6} + a^{2} b^{5} d^{2} x^{6} + \frac {3}{4} \, a b^{3} c d^{2} x^{8} + \frac {6}{5} \, a b^{7} d x^{5} + \frac {3}{5} \, a^{3} b^{4} d^{2} x^{5} + \frac {6}{7} \, b^{5} c d x^{7} + \frac {6}{7} \, a^{2} b^{2} c d^{2} x^{7} + \frac {1}{3} \, a c^{2} d^{2} x^{9} + \frac {1}{4} \, b^{9} x^{4} + \frac {3}{4} \, a^{2} b^{6} d x^{4} + a b^{4} c d x^{6} + \frac {3}{8} \, b^{2} c^{2} d x^{8} + a b^{8} x^{3} + \frac {3}{5} \, b^{6} c x^{5} + \frac {3}{2} \, a^{2} b^{7} x^{2} + \frac {3}{2} \, a b^{5} c x^{4} + \frac {1}{2} \, b^{3} c^{2} x^{6} + a^{3} b^{6} x + a^{2} b^{4} c x^{3} + \frac {3}{5} \, a b^{2} c^{2} x^{5} + \frac {1}{7} \, c^{3} x^{7} \] Input:

integrate((a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/13*a^3*d^6*x^13 + 1/4*a^2*b^2*d^5*x^12 + 3/11*a*b^4*d^4*x^11 + 1/10*b^6* 
d^3*x^10 + 3/10*a^2*b^3*d^4*x^10 + 2/3*a*b^5*d^3*x^9 + 1/3*a^3*b^2*d^4*x^9 
 + 3/11*a^2*c*d^4*x^11 + 3/8*b^7*d^2*x^8 + 3/4*a^2*b^4*d^3*x^8 + 3/5*a*b^2 
*c*d^3*x^10 + 6/7*a*b^6*d^2*x^7 + 1/3*b^4*c*d^2*x^9 + 1/2*b^8*d*x^6 + a^2* 
b^5*d^2*x^6 + 3/4*a*b^3*c*d^2*x^8 + 6/5*a*b^7*d*x^5 + 3/5*a^3*b^4*d^2*x^5 
+ 6/7*b^5*c*d*x^7 + 6/7*a^2*b^2*c*d^2*x^7 + 1/3*a*c^2*d^2*x^9 + 1/4*b^9*x^ 
4 + 3/4*a^2*b^6*d*x^4 + a*b^4*c*d*x^6 + 3/8*b^2*c^2*d*x^8 + a*b^8*x^3 + 3/ 
5*b^6*c*x^5 + 3/2*a^2*b^7*x^2 + 3/2*a*b^5*c*x^4 + 1/2*b^3*c^2*x^6 + a^3*b^ 
6*x + a^2*b^4*c*x^3 + 3/5*a*b^2*c^2*x^5 + 1/7*c^3*x^7
 

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.97 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^3 \, dx=x^4\,\left (\frac {3\,d\,a^2\,b^6}{4}+\frac {3\,c\,a\,b^5}{2}+\frac {b^9}{4}\right )+x^5\,\left (\frac {3\,a^3\,b^4\,d^2}{5}+\frac {6\,a\,b^7\,d}{5}+\frac {3\,a\,b^2\,c^2}{5}+\frac {3\,b^6\,c}{5}\right )+x^6\,\left (a^2\,b^5\,d^2+a\,b^4\,c\,d+\frac {b^8\,d}{2}+\frac {b^3\,c^2}{2}\right )+x^7\,\left (\frac {6\,a^2\,b^2\,c\,d^2}{7}+\frac {6\,a\,b^6\,d^2}{7}+\frac {6\,b^5\,c\,d}{7}+\frac {c^3}{7}\right )+a^3\,b^6\,x+\frac {3\,a^2\,b^7\,x^2}{2}+\frac {a^3\,d^6\,x^{13}}{13}+\frac {d^2\,x^9\,\left (a^3\,b^2\,d^2+2\,a\,b^5\,d+a\,c^2+b^4\,c\right )}{3}+\frac {a^2\,b^2\,d^5\,x^{12}}{4}+\frac {3\,b^2\,d\,x^8\,\left (2\,a^2\,b^2\,d^2+2\,a\,b\,c\,d+b^5\,d+c^2\right )}{8}+a\,b^4\,x^3\,\left (b^4+a\,c\right )+\frac {3\,a\,d^4\,x^{11}\,\left (b^4+a\,c\right )}{11}+\frac {b^2\,d^3\,x^{10}\,\left (3\,d\,a^2\,b+6\,c\,a+b^4\right )}{10} \] Input:

int((a*b^2 + b^3*x + c*x^2 + a*d^2*x^4 + b^2*d*x^3)^3,x)
 

Output:

x^4*(b^9/4 + (3*a^2*b^6*d)/4 + (3*a*b^5*c)/2) + x^5*((3*b^6*c)/5 + (3*a*b^ 
2*c^2)/5 + (3*a^3*b^4*d^2)/5 + (6*a*b^7*d)/5) + x^6*((b^8*d)/2 + (b^3*c^2) 
/2 + a^2*b^5*d^2 + a*b^4*c*d) + x^7*(c^3/7 + (6*a*b^6*d^2)/7 + (6*b^5*c*d) 
/7 + (6*a^2*b^2*c*d^2)/7) + a^3*b^6*x + (3*a^2*b^7*x^2)/2 + (a^3*d^6*x^13) 
/13 + (d^2*x^9*(a*c^2 + b^4*c + a^3*b^2*d^2 + 2*a*b^5*d))/3 + (a^2*b^2*d^5 
*x^12)/4 + (3*b^2*d*x^8*(b^5*d + c^2 + 2*a^2*b^2*d^2 + 2*a*b*c*d))/8 + a*b 
^4*x^3*(a*c + b^4) + (3*a*d^4*x^11*(a*c + b^4))/11 + (b^2*d^3*x^10*(6*a*c 
+ b^4 + 3*a^2*b*d))/10
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.23 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^3 \, dx=\frac {x \left (9240 a^{3} d^{6} x^{12}+30030 a^{2} b^{2} d^{5} x^{11}+32760 a \,b^{4} d^{4} x^{10}+36036 a^{2} b^{3} d^{4} x^{9}+12012 b^{6} d^{3} x^{9}+40040 a^{3} b^{2} d^{4} x^{8}+32760 a^{2} c \,d^{4} x^{10}+80080 a \,b^{5} d^{3} x^{8}+90090 a^{2} b^{4} d^{3} x^{7}+72072 a \,b^{2} c \,d^{3} x^{9}+45045 b^{7} d^{2} x^{7}+102960 a \,b^{6} d^{2} x^{6}+40040 b^{4} c \,d^{2} x^{8}+120120 a^{2} b^{5} d^{2} x^{5}+90090 a \,b^{3} c \,d^{2} x^{7}+60060 b^{8} d \,x^{5}+72072 a^{3} b^{4} d^{2} x^{4}+102960 a^{2} b^{2} c \,d^{2} x^{6}+144144 a \,b^{7} d \,x^{4}+40040 a \,c^{2} d^{2} x^{8}+102960 b^{5} c d \,x^{6}+90090 a^{2} b^{6} d \,x^{3}+120120 a \,b^{4} c d \,x^{5}+30030 b^{9} x^{3}+45045 b^{2} c^{2} d \,x^{7}+120120 a \,b^{8} x^{2}+72072 b^{6} c \,x^{4}+180180 a^{2} b^{7} x +180180 a \,b^{5} c \,x^{3}+60060 b^{3} c^{2} x^{5}+120120 a^{3} b^{6}+120120 a^{2} b^{4} c \,x^{2}+72072 a \,b^{2} c^{2} x^{4}+17160 c^{3} x^{6}\right )}{120120} \] Input:

int((a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^3,x)
 

Output:

(x*(120120*a**3*b**6 + 72072*a**3*b**4*d**2*x**4 + 40040*a**3*b**2*d**4*x* 
*8 + 9240*a**3*d**6*x**12 + 180180*a**2*b**7*x + 90090*a**2*b**6*d*x**3 + 
120120*a**2*b**5*d**2*x**5 + 120120*a**2*b**4*c*x**2 + 90090*a**2*b**4*d** 
3*x**7 + 36036*a**2*b**3*d**4*x**9 + 102960*a**2*b**2*c*d**2*x**6 + 30030* 
a**2*b**2*d**5*x**11 + 32760*a**2*c*d**4*x**10 + 120120*a*b**8*x**2 + 1441 
44*a*b**7*d*x**4 + 102960*a*b**6*d**2*x**6 + 180180*a*b**5*c*x**3 + 80080* 
a*b**5*d**3*x**8 + 120120*a*b**4*c*d*x**5 + 32760*a*b**4*d**4*x**10 + 9009 
0*a*b**3*c*d**2*x**7 + 72072*a*b**2*c**2*x**4 + 72072*a*b**2*c*d**3*x**9 + 
 40040*a*c**2*d**2*x**8 + 30030*b**9*x**3 + 60060*b**8*d*x**5 + 45045*b**7 
*d**2*x**7 + 72072*b**6*c*x**4 + 12012*b**6*d**3*x**9 + 102960*b**5*c*d*x* 
*6 + 40040*b**4*c*d**2*x**8 + 60060*b**3*c**2*x**5 + 45045*b**2*c**2*d*x** 
7 + 17160*c**3*x**6))/120120