Integrand size = 24, antiderivative size = 83 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^3 \, dx=\frac {a^3 e^3 (c+d x)^4}{4 d}+\frac {3 a^2 b e^3 (c+d x)^7}{7 d}+\frac {3 a b^2 e^3 (c+d x)^{10}}{10 d}+\frac {b^3 e^3 (c+d x)^{13}}{13 d} \]
1/4*a^3*e^3*(d*x+c)^4/d+3/7*a^2*b*e^3*(d*x+c)^7/d+3/10*a*b^2*e^3*(d*x+c)^1 0/d+1/13*b^3*e^3*(d*x+c)^13/d
Leaf count is larger than twice the leaf count of optimal. \(327\) vs. \(2(83)=166\).
Time = 0.02 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.94 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^3 \, dx=e^3 \left (c^3 \left (a+b c^3\right )^3 x+\frac {3}{2} c^2 \left (a+b c^3\right )^2 \left (a+4 b c^3\right ) d x^2+c \left (a^3+15 a^2 b c^3+36 a b^2 c^6+22 b^3 c^9\right ) d^2 x^3+\frac {1}{4} \left (a^3+60 a^2 b c^3+252 a b^2 c^6+220 b^3 c^9\right ) d^3 x^4+\frac {9}{5} b c^2 \left (5 a^2+42 a b c^3+55 b^2 c^6\right ) d^4 x^5+3 b c \left (a^2+21 a b c^3+44 b^2 c^6\right ) d^5 x^6+\frac {3}{7} b \left (a^2+84 a b c^3+308 b^2 c^6\right ) d^6 x^7+\frac {9}{2} b^2 c^2 \left (3 a+22 b c^3\right ) d^7 x^8+b^2 c \left (3 a+55 b c^3\right ) d^8 x^9+\frac {1}{10} b^2 \left (3 a+220 b c^3\right ) d^9 x^{10}+6 b^3 c^2 d^{10} x^{11}+b^3 c d^{11} x^{12}+\frac {1}{13} b^3 d^{12} x^{13}\right ) \]
e^3*(c^3*(a + b*c^3)^3*x + (3*c^2*(a + b*c^3)^2*(a + 4*b*c^3)*d*x^2)/2 + c *(a^3 + 15*a^2*b*c^3 + 36*a*b^2*c^6 + 22*b^3*c^9)*d^2*x^3 + ((a^3 + 60*a^2 *b*c^3 + 252*a*b^2*c^6 + 220*b^3*c^9)*d^3*x^4)/4 + (9*b*c^2*(5*a^2 + 42*a* b*c^3 + 55*b^2*c^6)*d^4*x^5)/5 + 3*b*c*(a^2 + 21*a*b*c^3 + 44*b^2*c^6)*d^5 *x^6 + (3*b*(a^2 + 84*a*b*c^3 + 308*b^2*c^6)*d^6*x^7)/7 + (9*b^2*c^2*(3*a + 22*b*c^3)*d^7*x^8)/2 + b^2*c*(3*a + 55*b*c^3)*d^8*x^9 + (b^2*(3*a + 220* b*c^3)*d^9*x^10)/10 + 6*b^3*c^2*d^10*x^11 + b^3*c*d^11*x^12 + (b^3*d^12*x^ 13)/13)
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {895, 802, 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.
| \(\displaystyle \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^3 \, dx\) |
| \(\Big \downarrow \) 895 |
| \(\displaystyle \frac {e^3 \int (c+d x)^3 \left (b (c+d x)^3+a\right )^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 802 |
| \(\displaystyle \frac {e^3 \int \left (b^3 (c+d x)^{12}+3 a b^2 (c+d x)^9+3 a^2 b (c+d x)^6+a^3 (c+d x)^3\right )d(c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} a^3 (c+d x)^4+\frac {3}{7} a^2 b (c+d x)^7+\frac {3}{10} a b^2 (c+d x)^{10}+\frac {1}{13} b^3 (c+d x)^{13}\right )}{d}\) |
(e^3*((a^3*(c + d*x)^4)/4 + (3*a^2*b*(c + d*x)^7)/7 + (3*a*b^2*(c + d*x)^1 0)/10 + (b^3*(c + d*x)^13)/13))/d
3.29.58.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff icient[v, x, 1]*v^m) Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[ {a, b, m, n, p}, x] && LinearPairQ[u, v, x]
Leaf count of result is larger than twice the leaf count of optimal. \(440\) vs. \(2(75)=150\).
Time = 3.88 (sec) , antiderivative size = 441, normalized size of antiderivative = 5.31
| method | result | size |
| gosper | \(\frac {e^{3} x \left (140 d^{12} b^{3} x^{12}+1820 c \,d^{11} b^{3} x^{11}+10920 c^{2} d^{10} b^{3} x^{10}+40040 x^{9} c^{3} d^{9} b^{3}+100100 b^{3} c^{4} d^{8} x^{8}+180180 x^{7} c^{5} d^{7} b^{3}+546 x^{9} a \,b^{2} d^{9}+240240 x^{6} b^{3} c^{6} d^{6}+5460 a \,b^{2} c \,d^{8} x^{8}+240240 b^{3} c^{7} d^{5} x^{5}+24570 x^{7} a \,b^{2} c^{2} d^{7}+180180 x^{4} b^{3} c^{8} d^{4}+65520 x^{6} a \,b^{2} c^{3} d^{6}+100100 x^{3} b^{3} c^{9} d^{3}+114660 a \,b^{2} c^{4} d^{5} x^{5}+40040 b^{3} c^{10} d^{2} x^{2}+137592 x^{4} a \,b^{2} c^{5} d^{4}+10920 x \,b^{3} c^{11} d +780 x^{6} a^{2} b \,d^{6}+114660 x^{3} a \,b^{2} c^{6} d^{3}+1820 b^{3} c^{12}+5460 a^{2} b c \,d^{5} x^{5}+65520 a \,b^{2} c^{7} d^{2} x^{2}+16380 x^{4} a^{2} b \,c^{2} d^{4}+24570 x a \,b^{2} c^{8} d +27300 x^{3} a^{2} b \,c^{3} d^{3}+5460 a \,b^{2} c^{9}+27300 a^{2} b \,c^{4} d^{2} x^{2}+16380 x \,a^{2} b \,c^{5} d +455 x^{3} a^{3} d^{3}+5460 a^{2} b \,c^{6}+1820 a^{3} c \,d^{2} x^{2}+2730 a^{3} c^{2} d x +1820 c^{3} a^{3}\right )}{1820}\) | \(441\) |
| norman | \(\left (22 c^{3} e^{3} d^{9} b^{3}+\frac {3}{10} a \,b^{2} d^{9} e^{3}\right ) x^{10}+\left (99 c^{5} e^{3} d^{7} b^{3}+\frac {27}{2} a \,b^{2} c^{2} d^{7} e^{3}\right ) x^{8}+\left (132 b^{3} c^{6} d^{6} e^{3}+36 a \,b^{2} c^{3} d^{6} e^{3}+\frac {3}{7} a^{2} b \,d^{6} e^{3}\right ) x^{7}+\left (99 b^{3} c^{8} d^{4} e^{3}+\frac {378}{5} a \,b^{2} c^{5} d^{4} e^{3}+9 a^{2} b \,c^{2} d^{4} e^{3}\right ) x^{5}+\left (55 b^{3} c^{9} d^{3} e^{3}+63 a \,b^{2} c^{6} d^{3} e^{3}+15 a^{2} b \,c^{3} d^{3} e^{3}+\frac {1}{4} a^{3} d^{3} e^{3}\right ) x^{4}+\left (6 b^{3} c^{11} d \,e^{3}+\frac {27}{2} a \,b^{2} c^{8} d \,e^{3}+9 a^{2} b \,c^{5} d \,e^{3}+\frac {3}{2} a^{3} c^{2} d \,e^{3}\right ) x^{2}+\left (55 c^{4} e^{3} d^{8} b^{3}+3 a \,b^{2} c \,d^{8} e^{3}\right ) x^{9}+\left (132 b^{3} c^{7} d^{5} e^{3}+63 a \,b^{2} c^{4} d^{5} e^{3}+3 a^{2} b c \,d^{5} e^{3}\right ) x^{6}+\left (b^{3} c^{12} e^{3}+3 a \,b^{2} c^{9} e^{3}+3 a^{2} b \,c^{6} e^{3}+a^{3} c^{3} e^{3}\right ) x +\left (22 b^{3} c^{10} d^{2} e^{3}+36 a \,b^{2} c^{7} d^{2} e^{3}+15 a^{2} b \,c^{4} d^{2} e^{3}+a^{3} c \,d^{2} e^{3}\right ) x^{3}+c \,e^{3} d^{11} b^{3} x^{12}+\frac {d^{12} e^{3} b^{3} x^{13}}{13}+6 c^{2} e^{3} d^{10} b^{3} x^{11}\) | \(508\) |
| risch | \(15 e^{3} a^{2} b \,c^{4} d^{2} x^{3}+3 e^{3} a \,b^{2} c \,d^{8} x^{9}+\frac {27}{2} e^{3} x^{8} a \,b^{2} c^{2} d^{7}+36 e^{3} x^{7} a \,b^{2} c^{3} d^{6}+\frac {378}{5} e^{3} x^{5} a \,b^{2} c^{5} d^{4}+9 e^{3} x^{5} a^{2} b \,c^{2} d^{4}+63 e^{3} x^{4} a \,b^{2} c^{6} d^{3}+c^{3} a^{3} e^{3} x +c \,e^{3} d^{11} b^{3} x^{12}+3 e^{3} a^{2} b \,c^{6} x +132 e^{3} b^{3} c^{7} d^{5} x^{6}+22 e^{3} b^{3} c^{10} d^{2} x^{3}+3 e^{3} a \,b^{2} c^{9} x +\frac {3}{2} e^{3} d \,a^{3} c^{2} x^{2}+e^{3} d^{2} a^{3} c \,x^{3}+3 e^{3} a^{2} b c \,d^{5} x^{6}+36 e^{3} a \,b^{2} c^{7} d^{2} x^{3}+15 e^{3} x^{4} a^{2} b \,c^{3} d^{3}+\frac {27}{2} e^{3} x^{2} a \,b^{2} c^{8} d +9 e^{3} x^{2} a^{2} b \,c^{5} d +63 e^{3} a \,b^{2} c^{4} d^{5} x^{6}+\frac {1}{4} e^{3} a^{3} d^{3} x^{4}+e^{3} b^{3} c^{12} x +6 c^{2} e^{3} d^{10} b^{3} x^{11}+\frac {1}{13} d^{12} e^{3} b^{3} x^{13}+22 e^{3} x^{10} c^{3} d^{9} b^{3}+\frac {3}{10} e^{3} x^{10} a \,b^{2} d^{9}+99 e^{3} x^{8} c^{5} d^{7} b^{3}+132 e^{3} x^{7} b^{3} c^{6} d^{6}+\frac {3}{7} e^{3} x^{7} a^{2} b \,d^{6}+99 e^{3} x^{5} b^{3} c^{8} d^{4}+55 e^{3} x^{4} b^{3} c^{9} d^{3}+6 e^{3} x^{2} b^{3} c^{11} d +55 e^{3} b^{3} c^{4} d^{8} x^{9}\) | \(545\) |
| parallelrisch | \(15 e^{3} a^{2} b \,c^{4} d^{2} x^{3}+3 e^{3} a \,b^{2} c \,d^{8} x^{9}+\frac {27}{2} e^{3} x^{8} a \,b^{2} c^{2} d^{7}+36 e^{3} x^{7} a \,b^{2} c^{3} d^{6}+\frac {378}{5} e^{3} x^{5} a \,b^{2} c^{5} d^{4}+9 e^{3} x^{5} a^{2} b \,c^{2} d^{4}+63 e^{3} x^{4} a \,b^{2} c^{6} d^{3}+c^{3} a^{3} e^{3} x +c \,e^{3} d^{11} b^{3} x^{12}+3 e^{3} a^{2} b \,c^{6} x +132 e^{3} b^{3} c^{7} d^{5} x^{6}+22 e^{3} b^{3} c^{10} d^{2} x^{3}+3 e^{3} a \,b^{2} c^{9} x +\frac {3}{2} e^{3} d \,a^{3} c^{2} x^{2}+e^{3} d^{2} a^{3} c \,x^{3}+3 e^{3} a^{2} b c \,d^{5} x^{6}+36 e^{3} a \,b^{2} c^{7} d^{2} x^{3}+15 e^{3} x^{4} a^{2} b \,c^{3} d^{3}+\frac {27}{2} e^{3} x^{2} a \,b^{2} c^{8} d +9 e^{3} x^{2} a^{2} b \,c^{5} d +63 e^{3} a \,b^{2} c^{4} d^{5} x^{6}+\frac {1}{4} e^{3} a^{3} d^{3} x^{4}+e^{3} b^{3} c^{12} x +6 c^{2} e^{3} d^{10} b^{3} x^{11}+\frac {1}{13} d^{12} e^{3} b^{3} x^{13}+22 e^{3} x^{10} c^{3} d^{9} b^{3}+\frac {3}{10} e^{3} x^{10} a \,b^{2} d^{9}+99 e^{3} x^{8} c^{5} d^{7} b^{3}+132 e^{3} x^{7} b^{3} c^{6} d^{6}+\frac {3}{7} e^{3} x^{7} a^{2} b \,d^{6}+99 e^{3} x^{5} b^{3} c^{8} d^{4}+55 e^{3} x^{4} b^{3} c^{9} d^{3}+6 e^{3} x^{2} b^{3} c^{11} d +55 e^{3} b^{3} c^{4} d^{8} x^{9}\) | \(545\) |
| default | \(\text {Expression too large to display}\) | \(2050\) |
1/1820*e^3*x*(140*b^3*d^12*x^12+1820*b^3*c*d^11*x^11+10920*b^3*c^2*d^10*x^ 10+40040*b^3*c^3*d^9*x^9+100100*b^3*c^4*d^8*x^8+180180*b^3*c^5*d^7*x^7+546 *a*b^2*d^9*x^9+240240*b^3*c^6*d^6*x^6+5460*a*b^2*c*d^8*x^8+240240*b^3*c^7* d^5*x^5+24570*a*b^2*c^2*d^7*x^7+180180*b^3*c^8*d^4*x^4+65520*a*b^2*c^3*d^6 *x^6+100100*b^3*c^9*d^3*x^3+114660*a*b^2*c^4*d^5*x^5+40040*b^3*c^10*d^2*x^ 2+137592*a*b^2*c^5*d^4*x^4+10920*b^3*c^11*d*x+780*a^2*b*d^6*x^6+114660*a*b ^2*c^6*d^3*x^3+1820*b^3*c^12+5460*a^2*b*c*d^5*x^5+65520*a*b^2*c^7*d^2*x^2+ 16380*a^2*b*c^2*d^4*x^4+24570*a*b^2*c^8*d*x+27300*a^2*b*c^3*d^3*x^3+5460*a *b^2*c^9+27300*a^2*b*c^4*d^2*x^2+16380*a^2*b*c^5*d*x+455*a^3*d^3*x^3+5460* a^2*b*c^6+1820*a^3*c*d^2*x^2+2730*a^3*c^2*d*x+1820*a^3*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (75) = 150\).
Time = 0.27 (sec) , antiderivative size = 398, normalized size of antiderivative = 4.80 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^3 \, dx=\frac {1}{13} \, b^{3} d^{12} e^{3} x^{13} + b^{3} c d^{11} e^{3} x^{12} + 6 \, b^{3} c^{2} d^{10} e^{3} x^{11} + \frac {1}{10} \, {\left (220 \, b^{3} c^{3} + 3 \, a b^{2}\right )} d^{9} e^{3} x^{10} + {\left (55 \, b^{3} c^{4} + 3 \, a b^{2} c\right )} d^{8} e^{3} x^{9} + \frac {9}{2} \, {\left (22 \, b^{3} c^{5} + 3 \, a b^{2} c^{2}\right )} d^{7} e^{3} x^{8} + \frac {3}{7} \, {\left (308 \, b^{3} c^{6} + 84 \, a b^{2} c^{3} + a^{2} b\right )} d^{6} e^{3} x^{7} + 3 \, {\left (44 \, b^{3} c^{7} + 21 \, a b^{2} c^{4} + a^{2} b c\right )} d^{5} e^{3} x^{6} + \frac {9}{5} \, {\left (55 \, b^{3} c^{8} + 42 \, a b^{2} c^{5} + 5 \, a^{2} b c^{2}\right )} d^{4} e^{3} x^{5} + \frac {1}{4} \, {\left (220 \, b^{3} c^{9} + 252 \, a b^{2} c^{6} + 60 \, a^{2} b c^{3} + a^{3}\right )} d^{3} e^{3} x^{4} + {\left (22 \, b^{3} c^{10} + 36 \, a b^{2} c^{7} + 15 \, a^{2} b c^{4} + a^{3} c\right )} d^{2} e^{3} x^{3} + \frac {3}{2} \, {\left (4 \, b^{3} c^{11} + 9 \, a b^{2} c^{8} + 6 \, a^{2} b c^{5} + a^{3} c^{2}\right )} d e^{3} x^{2} + {\left (b^{3} c^{12} + 3 \, a b^{2} c^{9} + 3 \, a^{2} b c^{6} + a^{3} c^{3}\right )} e^{3} x \]
1/13*b^3*d^12*e^3*x^13 + b^3*c*d^11*e^3*x^12 + 6*b^3*c^2*d^10*e^3*x^11 + 1 /10*(220*b^3*c^3 + 3*a*b^2)*d^9*e^3*x^10 + (55*b^3*c^4 + 3*a*b^2*c)*d^8*e^ 3*x^9 + 9/2*(22*b^3*c^5 + 3*a*b^2*c^2)*d^7*e^3*x^8 + 3/7*(308*b^3*c^6 + 84 *a*b^2*c^3 + a^2*b)*d^6*e^3*x^7 + 3*(44*b^3*c^7 + 21*a*b^2*c^4 + a^2*b*c)* d^5*e^3*x^6 + 9/5*(55*b^3*c^8 + 42*a*b^2*c^5 + 5*a^2*b*c^2)*d^4*e^3*x^5 + 1/4*(220*b^3*c^9 + 252*a*b^2*c^6 + 60*a^2*b*c^3 + a^3)*d^3*e^3*x^4 + (22*b ^3*c^10 + 36*a*b^2*c^7 + 15*a^2*b*c^4 + a^3*c)*d^2*e^3*x^3 + 3/2*(4*b^3*c^ 11 + 9*a*b^2*c^8 + 6*a^2*b*c^5 + a^3*c^2)*d*e^3*x^2 + (b^3*c^12 + 3*a*b^2* c^9 + 3*a^2*b*c^6 + a^3*c^3)*e^3*x
Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (73) = 146\).
Time = 0.08 (sec) , antiderivative size = 552, normalized size of antiderivative = 6.65 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^3 \, dx=6 b^{3} c^{2} d^{10} e^{3} x^{11} + b^{3} c d^{11} e^{3} x^{12} + \frac {b^{3} d^{12} e^{3} x^{13}}{13} + x^{10} \cdot \left (\frac {3 a b^{2} d^{9} e^{3}}{10} + 22 b^{3} c^{3} d^{9} e^{3}\right ) + x^{9} \cdot \left (3 a b^{2} c d^{8} e^{3} + 55 b^{3} c^{4} d^{8} e^{3}\right ) + x^{8} \cdot \left (\frac {27 a b^{2} c^{2} d^{7} e^{3}}{2} + 99 b^{3} c^{5} d^{7} e^{3}\right ) + x^{7} \cdot \left (\frac {3 a^{2} b d^{6} e^{3}}{7} + 36 a b^{2} c^{3} d^{6} e^{3} + 132 b^{3} c^{6} d^{6} e^{3}\right ) + x^{6} \cdot \left (3 a^{2} b c d^{5} e^{3} + 63 a b^{2} c^{4} d^{5} e^{3} + 132 b^{3} c^{7} d^{5} e^{3}\right ) + x^{5} \cdot \left (9 a^{2} b c^{2} d^{4} e^{3} + \frac {378 a b^{2} c^{5} d^{4} e^{3}}{5} + 99 b^{3} c^{8} d^{4} e^{3}\right ) + x^{4} \left (\frac {a^{3} d^{3} e^{3}}{4} + 15 a^{2} b c^{3} d^{3} e^{3} + 63 a b^{2} c^{6} d^{3} e^{3} + 55 b^{3} c^{9} d^{3} e^{3}\right ) + x^{3} \left (a^{3} c d^{2} e^{3} + 15 a^{2} b c^{4} d^{2} e^{3} + 36 a b^{2} c^{7} d^{2} e^{3} + 22 b^{3} c^{10} d^{2} e^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{3} c^{2} d e^{3}}{2} + 9 a^{2} b c^{5} d e^{3} + \frac {27 a b^{2} c^{8} d e^{3}}{2} + 6 b^{3} c^{11} d e^{3}\right ) + x \left (a^{3} c^{3} e^{3} + 3 a^{2} b c^{6} e^{3} + 3 a b^{2} c^{9} e^{3} + b^{3} c^{12} e^{3}\right ) \]
6*b**3*c**2*d**10*e**3*x**11 + b**3*c*d**11*e**3*x**12 + b**3*d**12*e**3*x **13/13 + x**10*(3*a*b**2*d**9*e**3/10 + 22*b**3*c**3*d**9*e**3) + x**9*(3 *a*b**2*c*d**8*e**3 + 55*b**3*c**4*d**8*e**3) + x**8*(27*a*b**2*c**2*d**7* e**3/2 + 99*b**3*c**5*d**7*e**3) + x**7*(3*a**2*b*d**6*e**3/7 + 36*a*b**2* c**3*d**6*e**3 + 132*b**3*c**6*d**6*e**3) + x**6*(3*a**2*b*c*d**5*e**3 + 6 3*a*b**2*c**4*d**5*e**3 + 132*b**3*c**7*d**5*e**3) + x**5*(9*a**2*b*c**2*d **4*e**3 + 378*a*b**2*c**5*d**4*e**3/5 + 99*b**3*c**8*d**4*e**3) + x**4*(a **3*d**3*e**3/4 + 15*a**2*b*c**3*d**3*e**3 + 63*a*b**2*c**6*d**3*e**3 + 55 *b**3*c**9*d**3*e**3) + x**3*(a**3*c*d**2*e**3 + 15*a**2*b*c**4*d**2*e**3 + 36*a*b**2*c**7*d**2*e**3 + 22*b**3*c**10*d**2*e**3) + x**2*(3*a**3*c**2* d*e**3/2 + 9*a**2*b*c**5*d*e**3 + 27*a*b**2*c**8*d*e**3/2 + 6*b**3*c**11*d *e**3) + x*(a**3*c**3*e**3 + 3*a**2*b*c**6*e**3 + 3*a*b**2*c**9*e**3 + b** 3*c**12*e**3)
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (75) = 150\).
Time = 0.20 (sec) , antiderivative size = 398, normalized size of antiderivative = 4.80 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^3 \, dx=\frac {1}{13} \, b^{3} d^{12} e^{3} x^{13} + b^{3} c d^{11} e^{3} x^{12} + 6 \, b^{3} c^{2} d^{10} e^{3} x^{11} + \frac {1}{10} \, {\left (220 \, b^{3} c^{3} + 3 \, a b^{2}\right )} d^{9} e^{3} x^{10} + {\left (55 \, b^{3} c^{4} + 3 \, a b^{2} c\right )} d^{8} e^{3} x^{9} + \frac {9}{2} \, {\left (22 \, b^{3} c^{5} + 3 \, a b^{2} c^{2}\right )} d^{7} e^{3} x^{8} + \frac {3}{7} \, {\left (308 \, b^{3} c^{6} + 84 \, a b^{2} c^{3} + a^{2} b\right )} d^{6} e^{3} x^{7} + 3 \, {\left (44 \, b^{3} c^{7} + 21 \, a b^{2} c^{4} + a^{2} b c\right )} d^{5} e^{3} x^{6} + \frac {9}{5} \, {\left (55 \, b^{3} c^{8} + 42 \, a b^{2} c^{5} + 5 \, a^{2} b c^{2}\right )} d^{4} e^{3} x^{5} + \frac {1}{4} \, {\left (220 \, b^{3} c^{9} + 252 \, a b^{2} c^{6} + 60 \, a^{2} b c^{3} + a^{3}\right )} d^{3} e^{3} x^{4} + {\left (22 \, b^{3} c^{10} + 36 \, a b^{2} c^{7} + 15 \, a^{2} b c^{4} + a^{3} c\right )} d^{2} e^{3} x^{3} + \frac {3}{2} \, {\left (4 \, b^{3} c^{11} + 9 \, a b^{2} c^{8} + 6 \, a^{2} b c^{5} + a^{3} c^{2}\right )} d e^{3} x^{2} + {\left (b^{3} c^{12} + 3 \, a b^{2} c^{9} + 3 \, a^{2} b c^{6} + a^{3} c^{3}\right )} e^{3} x \]
1/13*b^3*d^12*e^3*x^13 + b^3*c*d^11*e^3*x^12 + 6*b^3*c^2*d^10*e^3*x^11 + 1 /10*(220*b^3*c^3 + 3*a*b^2)*d^9*e^3*x^10 + (55*b^3*c^4 + 3*a*b^2*c)*d^8*e^ 3*x^9 + 9/2*(22*b^3*c^5 + 3*a*b^2*c^2)*d^7*e^3*x^8 + 3/7*(308*b^3*c^6 + 84 *a*b^2*c^3 + a^2*b)*d^6*e^3*x^7 + 3*(44*b^3*c^7 + 21*a*b^2*c^4 + a^2*b*c)* d^5*e^3*x^6 + 9/5*(55*b^3*c^8 + 42*a*b^2*c^5 + 5*a^2*b*c^2)*d^4*e^3*x^5 + 1/4*(220*b^3*c^9 + 252*a*b^2*c^6 + 60*a^2*b*c^3 + a^3)*d^3*e^3*x^4 + (22*b ^3*c^10 + 36*a*b^2*c^7 + 15*a^2*b*c^4 + a^3*c)*d^2*e^3*x^3 + 3/2*(4*b^3*c^ 11 + 9*a*b^2*c^8 + 6*a^2*b*c^5 + a^3*c^2)*d*e^3*x^2 + (b^3*c^12 + 3*a*b^2* c^9 + 3*a^2*b*c^6 + a^3*c^3)*e^3*x
Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (75) = 150\).
Time = 0.29 (sec) , antiderivative size = 544, normalized size of antiderivative = 6.55 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^3 \, dx=\frac {1}{13} \, b^{3} d^{12} e^{3} x^{13} + b^{3} c d^{11} e^{3} x^{12} + 6 \, b^{3} c^{2} d^{10} e^{3} x^{11} + 22 \, b^{3} c^{3} d^{9} e^{3} x^{10} + 55 \, b^{3} c^{4} d^{8} e^{3} x^{9} + 99 \, b^{3} c^{5} d^{7} e^{3} x^{8} + 132 \, b^{3} c^{6} d^{6} e^{3} x^{7} + \frac {3}{10} \, a b^{2} d^{9} e^{3} x^{10} + 132 \, b^{3} c^{7} d^{5} e^{3} x^{6} + 3 \, a b^{2} c d^{8} e^{3} x^{9} + 99 \, b^{3} c^{8} d^{4} e^{3} x^{5} + \frac {27}{2} \, a b^{2} c^{2} d^{7} e^{3} x^{8} + 55 \, b^{3} c^{9} d^{3} e^{3} x^{4} + 36 \, a b^{2} c^{3} d^{6} e^{3} x^{7} + 22 \, b^{3} c^{10} d^{2} e^{3} x^{3} + 63 \, a b^{2} c^{4} d^{5} e^{3} x^{6} + 6 \, b^{3} c^{11} d e^{3} x^{2} + \frac {378}{5} \, a b^{2} c^{5} d^{4} e^{3} x^{5} + b^{3} c^{12} e^{3} x + 63 \, a b^{2} c^{6} d^{3} e^{3} x^{4} + \frac {3}{7} \, a^{2} b d^{6} e^{3} x^{7} + 36 \, a b^{2} c^{7} d^{2} e^{3} x^{3} + 3 \, a^{2} b c d^{5} e^{3} x^{6} + \frac {27}{2} \, a b^{2} c^{8} d e^{3} x^{2} + 9 \, a^{2} b c^{2} d^{4} e^{3} x^{5} + 3 \, a b^{2} c^{9} e^{3} x + 15 \, a^{2} b c^{3} d^{3} e^{3} x^{4} + 15 \, a^{2} b c^{4} d^{2} e^{3} x^{3} + 9 \, a^{2} b c^{5} d e^{3} x^{2} + 3 \, a^{2} b c^{6} e^{3} x + \frac {1}{4} \, a^{3} d^{3} e^{3} x^{4} + a^{3} c d^{2} e^{3} x^{3} + \frac {3}{2} \, a^{3} c^{2} d e^{3} x^{2} + a^{3} c^{3} e^{3} x \]
1/13*b^3*d^12*e^3*x^13 + b^3*c*d^11*e^3*x^12 + 6*b^3*c^2*d^10*e^3*x^11 + 2 2*b^3*c^3*d^9*e^3*x^10 + 55*b^3*c^4*d^8*e^3*x^9 + 99*b^3*c^5*d^7*e^3*x^8 + 132*b^3*c^6*d^6*e^3*x^7 + 3/10*a*b^2*d^9*e^3*x^10 + 132*b^3*c^7*d^5*e^3*x ^6 + 3*a*b^2*c*d^8*e^3*x^9 + 99*b^3*c^8*d^4*e^3*x^5 + 27/2*a*b^2*c^2*d^7*e ^3*x^8 + 55*b^3*c^9*d^3*e^3*x^4 + 36*a*b^2*c^3*d^6*e^3*x^7 + 22*b^3*c^10*d ^2*e^3*x^3 + 63*a*b^2*c^4*d^5*e^3*x^6 + 6*b^3*c^11*d*e^3*x^2 + 378/5*a*b^2 *c^5*d^4*e^3*x^5 + b^3*c^12*e^3*x + 63*a*b^2*c^6*d^3*e^3*x^4 + 3/7*a^2*b*d ^6*e^3*x^7 + 36*a*b^2*c^7*d^2*e^3*x^3 + 3*a^2*b*c*d^5*e^3*x^6 + 27/2*a*b^2 *c^8*d*e^3*x^2 + 9*a^2*b*c^2*d^4*e^3*x^5 + 3*a*b^2*c^9*e^3*x + 15*a^2*b*c^ 3*d^3*e^3*x^4 + 15*a^2*b*c^4*d^2*e^3*x^3 + 9*a^2*b*c^5*d*e^3*x^2 + 3*a^2*b *c^6*e^3*x + 1/4*a^3*d^3*e^3*x^4 + a^3*c*d^2*e^3*x^3 + 3/2*a^3*c^2*d*e^3*x ^2 + a^3*c^3*e^3*x
Time = 5.61 (sec) , antiderivative size = 348, normalized size of antiderivative = 4.19 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^3 \, dx=c^3\,e^3\,x\,{\left (b\,c^3+a\right )}^3+\frac {d^3\,e^3\,x^4\,\left (a^3+60\,a^2\,b\,c^3+252\,a\,b^2\,c^6+220\,b^3\,c^9\right )}{4}+\frac {b^3\,d^{12}\,e^3\,x^{13}}{13}+\frac {3\,b\,d^6\,e^3\,x^7\,\left (a^2+84\,a\,b\,c^3+308\,b^2\,c^6\right )}{7}+\frac {b^2\,d^9\,e^3\,x^{10}\,\left (220\,b\,c^3+3\,a\right )}{10}+6\,b^3\,c^2\,d^{10}\,e^3\,x^{11}+c\,d^2\,e^3\,x^3\,\left (a^3+15\,a^2\,b\,c^3+36\,a\,b^2\,c^6+22\,b^3\,c^9\right )+b^3\,c\,d^{11}\,e^3\,x^{12}+\frac {3\,c^2\,d\,e^3\,x^2\,{\left (b\,c^3+a\right )}^2\,\left (4\,b\,c^3+a\right )}{2}+\frac {9\,b\,c^2\,d^4\,e^3\,x^5\,\left (5\,a^2+42\,a\,b\,c^3+55\,b^2\,c^6\right )}{5}+3\,b\,c\,d^5\,e^3\,x^6\,\left (a^2+21\,a\,b\,c^3+44\,b^2\,c^6\right )+b^2\,c\,d^8\,e^3\,x^9\,\left (55\,b\,c^3+3\,a\right )+\frac {9\,b^2\,c^2\,d^7\,e^3\,x^8\,\left (22\,b\,c^3+3\,a\right )}{2} \]
c^3*e^3*x*(a + b*c^3)^3 + (d^3*e^3*x^4*(a^3 + 220*b^3*c^9 + 60*a^2*b*c^3 + 252*a*b^2*c^6))/4 + (b^3*d^12*e^3*x^13)/13 + (3*b*d^6*e^3*x^7*(a^2 + 308* b^2*c^6 + 84*a*b*c^3))/7 + (b^2*d^9*e^3*x^10*(3*a + 220*b*c^3))/10 + 6*b^3 *c^2*d^10*e^3*x^11 + c*d^2*e^3*x^3*(a^3 + 22*b^3*c^9 + 15*a^2*b*c^3 + 36*a *b^2*c^6) + b^3*c*d^11*e^3*x^12 + (3*c^2*d*e^3*x^2*(a + b*c^3)^2*(a + 4*b* c^3))/2 + (9*b*c^2*d^4*e^3*x^5*(5*a^2 + 55*b^2*c^6 + 42*a*b*c^3))/5 + 3*b* c*d^5*e^3*x^6*(a^2 + 44*b^2*c^6 + 21*a*b*c^3) + b^2*c*d^8*e^3*x^9*(3*a + 5 5*b*c^3) + (9*b^2*c^2*d^7*e^3*x^8*(3*a + 22*b*c^3))/2