Integrand size = 21, antiderivative size = 23 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^3 \, dx=\frac {\left (a+b (c+d x)^4\right )^4}{16 b d} \]
Leaf count is larger than twice the leaf count of optimal. \(308\) vs. \(2(23)=46\).
Time = 0.07 (sec) , antiderivative size = 308, normalized size of antiderivative = 13.39 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^3 \, dx=\frac {1}{16} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \left (4 a^3+6 a^2 b \left (2 c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right )+4 a b^2 \left (3 c^8+12 c^7 d x+34 c^6 d^2 x^2+60 c^5 d^3 x^3+71 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8\right )+b^3 \left (4 c^{12}+24 c^{11} d x+100 c^{10} d^2 x^2+280 c^9 d^3 x^3+566 c^8 d^4 x^4+848 c^7 d^5 x^5+952 c^6 d^6 x^6+800 c^5 d^7 x^7+496 c^4 d^8 x^8+220 c^3 d^9 x^9+66 c^2 d^{10} x^{10}+12 c d^{11} x^{11}+d^{12} x^{12}\right )\right ) \]
(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*(4*a^3 + 6*a^2*b*(2*c^4 + 4 *c^3*d*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x^4) + 4*a*b^2*(3*c^8 + 12*c^ 7*d*x + 34*c^6*d^2*x^2 + 60*c^5*d^3*x^3 + 71*c^4*d^4*x^4 + 56*c^3*d^5*x^5 + 28*c^2*d^6*x^6 + 8*c*d^7*x^7 + d^8*x^8) + b^3*(4*c^12 + 24*c^11*d*x + 10 0*c^10*d^2*x^2 + 280*c^9*d^3*x^3 + 566*c^8*d^4*x^4 + 848*c^7*d^5*x^5 + 952 *c^6*d^6*x^6 + 800*c^5*d^7*x^7 + 496*c^4*d^8*x^8 + 220*c^3*d^9*x^9 + 66*c^ 2*d^10*x^10 + 12*c*d^11*x^11 + d^12*x^12)))/16
Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {895, 793}
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.
| \(\displaystyle \int (c+d x)^3 \left (a+b (c+d x)^4\right )^3 \, dx\) |
| \(\Big \downarrow \) 895 |
| \(\displaystyle \frac {\int (c+d x)^3 \left (b (c+d x)^4+a\right )^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 793 |
| \(\displaystyle \frac {\left (a+b (c+d x)^4\right )^4}{16 b d}\) |
3.30.13.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) ^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
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]
Time = 3.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(\frac {\left (a +b \left (d x +c \right )^{4}\right )^{4}}{16 b d}\) | \(22\) |
| norman | \(\left (b^{3} c^{15}+3 a \,b^{2} c^{11}+3 a^{2} b \,c^{7}+c^{3} a^{3}\right ) x +\left (\frac {15}{2} b^{3} c^{14} d +\frac {33}{2} a \,b^{2} c^{10} d +\frac {21}{2} a^{2} b \,c^{6} d +\frac {3}{2} a^{3} c^{2} d \right ) x^{2}+\left (35 b^{3} c^{13} d^{2}+55 a \,b^{2} c^{9} d^{2}+21 a^{2} b \,c^{5} d^{2}+a^{3} c \,d^{2}\right ) x^{3}+\left (\frac {455}{4} b^{3} c^{12} d^{3}+\frac {495}{4} a \,b^{2} c^{8} d^{3}+\frac {105}{4} a^{2} b \,c^{4} d^{3}+\frac {1}{4} a^{3} d^{3}\right ) x^{4}+\left (273 b^{3} c^{11} d^{4}+198 a \,b^{2} c^{7} d^{4}+21 a^{2} b \,c^{3} d^{4}\right ) x^{5}+\left (\frac {1001}{2} b^{3} c^{10} d^{5}+231 a \,b^{2} c^{6} d^{5}+\frac {21}{2} a^{2} b \,c^{2} d^{5}\right ) x^{6}+\left (715 b^{3} c^{9} d^{6}+198 a \,b^{2} c^{5} d^{6}+3 a^{2} b c \,d^{6}\right ) x^{7}+\left (\frac {6435}{8} b^{3} c^{8} d^{7}+\frac {495}{4} a \,b^{2} c^{4} d^{7}+\frac {3}{8} a^{2} b \,d^{7}\right ) x^{8}+\left (715 b^{3} c^{7} d^{8}+55 a \,b^{2} c^{3} d^{8}\right ) x^{9}+\left (\frac {1001}{2} c^{6} d^{9} b^{3}+\frac {33}{2} a \,b^{2} c^{2} d^{9}\right ) x^{10}+\left (273 c^{5} d^{10} b^{3}+3 a \,b^{2} c \,d^{10}\right ) x^{11}+\left (\frac {455}{4} c^{4} b^{3} d^{11}+\frac {1}{4} a \,b^{2} d^{11}\right ) x^{12}+35 c^{3} b^{3} d^{12} x^{13}+\frac {15 c^{2} d^{13} b^{3} x^{14}}{2}+c \,d^{14} b^{3} x^{15}+\frac {d^{15} b^{3} x^{16}}{16}\) | \(488\) |
| gosper | \(\frac {x \left (d^{15} b^{3} x^{15}+16 c \,d^{14} b^{3} x^{14}+120 c^{2} d^{13} b^{3} x^{13}+560 c^{3} b^{3} d^{12} x^{12}+1820 x^{11} c^{4} b^{3} d^{11}+4368 b^{3} c^{5} d^{10} x^{10}+8008 x^{9} c^{6} d^{9} b^{3}+11440 b^{3} c^{7} d^{8} x^{8}+4 x^{11} a \,b^{2} d^{11}+12870 x^{7} b^{3} c^{8} d^{7}+48 a \,b^{2} c \,d^{10} x^{10}+11440 b^{3} c^{9} d^{6} x^{6}+264 x^{9} a \,b^{2} c^{2} d^{9}+8008 x^{5} b^{3} c^{10} d^{5}+880 a \,b^{2} c^{3} d^{8} x^{8}+4368 b^{3} c^{11} d^{4} x^{4}+1980 x^{7} a \,b^{2} c^{4} d^{7}+1820 x^{3} b^{3} c^{12} d^{3}+3168 a \,b^{2} c^{5} d^{6} x^{6}+560 b^{3} c^{13} d^{2} x^{2}+3696 x^{5} a \,b^{2} c^{6} d^{5}+120 x \,b^{3} c^{14} d +3168 a \,b^{2} c^{7} d^{4} x^{4}+16 b^{3} c^{15}+6 x^{7} a^{2} b \,d^{7}+1980 x^{3} a \,b^{2} c^{8} d^{3}+48 a^{2} b c \,d^{6} x^{6}+880 a \,b^{2} c^{9} d^{2} x^{2}+168 x^{5} a^{2} b \,c^{2} d^{5}+264 x a \,b^{2} c^{10} d +336 a^{2} b \,c^{3} d^{4} x^{4}+48 a \,b^{2} c^{11}+420 x^{3} a^{2} b \,c^{4} d^{3}+336 a^{2} b \,c^{5} d^{2} x^{2}+168 x \,a^{2} b \,c^{6} d +48 a^{2} b \,c^{7}+4 x^{3} a^{3} d^{3}+16 a^{3} c \,d^{2} x^{2}+24 a^{3} c^{2} d x +16 c^{3} a^{3}\right )}{16}\) | \(524\) |
| parallelrisch | \(\frac {455}{4} b^{3} d^{11} c^{4} x^{12}+273 b^{3} d^{10} c^{5} x^{11}+\frac {1001}{2} b^{3} d^{9} c^{6} x^{10}+715 b^{3} d^{8} c^{7} x^{9}+\frac {1}{4} b^{2} d^{11} a \,x^{12}+\frac {6435}{8} b^{3} d^{7} c^{8} x^{8}+715 b^{3} d^{6} c^{9} x^{7}+\frac {1001}{2} b^{3} d^{5} c^{10} x^{6}+273 b^{3} d^{4} c^{11} x^{5}+\frac {455}{4} b^{3} d^{3} c^{12} x^{4}+35 b^{3} d^{2} c^{13} x^{3}+\frac {15}{2} b^{3} d \,c^{14} x^{2}+b^{3} c^{15} x +\frac {3}{8} b \,d^{7} a^{2} x^{8}+\frac {1}{16} d^{15} b^{3} x^{16}+\frac {1}{4} a^{3} d^{3} x^{4}+a^{3} c^{3} x +35 c^{3} b^{3} d^{12} x^{13}+\frac {15}{2} c^{2} d^{13} b^{3} x^{14}+c \,d^{14} b^{3} x^{15}+3 b \,d^{6} a^{2} c \,x^{7}+55 b^{2} d^{2} a \,c^{9} x^{3}+\frac {21}{2} b \,d^{5} a^{2} c^{2} x^{6}+\frac {33}{2} b^{2} d a \,c^{10} x^{2}+21 b \,d^{4} a^{2} c^{3} x^{5}+\frac {3}{2} d \,a^{3} c^{2} x^{2}+d^{2} a^{3} c \,x^{3}+3 b^{2} a \,c^{11} x +\frac {105}{4} b \,d^{3} a^{2} c^{4} x^{4}+21 b \,d^{2} a^{2} c^{5} x^{3}+\frac {21}{2} b d \,a^{2} c^{6} x^{2}+3 b \,a^{2} c^{7} x +3 b^{2} d^{10} a c \,x^{11}+\frac {33}{2} b^{2} d^{9} a \,c^{2} x^{10}+55 b^{2} d^{8} a \,c^{3} x^{9}+\frac {495}{4} b^{2} d^{7} a \,c^{4} x^{8}+198 b^{2} d^{6} a \,c^{5} x^{7}+231 b^{2} d^{5} a \,c^{6} x^{6}+198 b^{2} d^{4} a \,c^{7} x^{5}+\frac {495}{4} b^{2} d^{3} a \,c^{8} x^{4}\) | \(530\) |
| risch | \(\frac {455 b^{3} d^{11} c^{4} x^{12}}{4}+273 b^{3} d^{10} c^{5} x^{11}+\frac {1001 b^{3} d^{9} c^{6} x^{10}}{2}+715 b^{3} d^{8} c^{7} x^{9}+\frac {b^{2} d^{11} a \,x^{12}}{4}+\frac {6435 b^{3} d^{7} c^{8} x^{8}}{8}+715 b^{3} d^{6} c^{9} x^{7}+\frac {1001 b^{3} d^{5} c^{10} x^{6}}{2}+273 b^{3} d^{4} c^{11} x^{5}+\frac {455 b^{3} d^{3} c^{12} x^{4}}{4}+35 b^{3} d^{2} c^{13} x^{3}+\frac {15 b^{3} d \,c^{14} x^{2}}{2}+b^{3} c^{15} x +\frac {3 b \,d^{7} a^{2} x^{8}}{8}+\frac {b^{2} a \,c^{12}}{4 d}+\frac {d^{15} b^{3} x^{16}}{16}+\frac {a^{3} d^{3} x^{4}}{4}+a^{3} c^{3} x +35 c^{3} b^{3} d^{12} x^{13}+\frac {15 c^{2} d^{13} b^{3} x^{14}}{2}+c \,d^{14} b^{3} x^{15}+3 b \,d^{6} a^{2} c \,x^{7}+55 b^{2} d^{2} a \,c^{9} x^{3}+\frac {21 b \,d^{5} a^{2} c^{2} x^{6}}{2}+\frac {33 b^{2} d a \,c^{10} x^{2}}{2}+21 b \,d^{4} a^{2} c^{3} x^{5}+\frac {3 b \,a^{2} c^{8}}{8 d}+\frac {a^{3} c^{4}}{4 d}+\frac {3 d \,a^{3} c^{2} x^{2}}{2}+d^{2} a^{3} c \,x^{3}+\frac {b^{3} c^{16}}{16 d}+\frac {a^{4}}{16 b d}+3 b^{2} a \,c^{11} x +\frac {105 b \,d^{3} a^{2} c^{4} x^{4}}{4}+21 b \,d^{2} a^{2} c^{5} x^{3}+\frac {21 b d \,a^{2} c^{6} x^{2}}{2}+3 b \,a^{2} c^{7} x +3 b^{2} d^{10} a c \,x^{11}+\frac {33 b^{2} d^{9} a \,c^{2} x^{10}}{2}+55 b^{2} d^{8} a \,c^{3} x^{9}+\frac {495 b^{2} d^{7} a \,c^{4} x^{8}}{4}+198 b^{2} d^{6} a \,c^{5} x^{7}+231 b^{2} d^{5} a \,c^{6} x^{6}+198 b^{2} d^{4} a \,c^{7} x^{5}+\frac {495 b^{2} d^{3} a \,c^{8} x^{4}}{4}\) | \(587\) |
Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 432, normalized size of antiderivative = 18.78 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^3 \, dx=\frac {1}{16} \, b^{3} d^{15} x^{16} + b^{3} c d^{14} x^{15} + \frac {15}{2} \, b^{3} c^{2} d^{13} x^{14} + 35 \, b^{3} c^{3} d^{12} x^{13} + \frac {1}{4} \, {\left (455 \, b^{3} c^{4} + a b^{2}\right )} d^{11} x^{12} + 3 \, {\left (91 \, b^{3} c^{5} + a b^{2} c\right )} d^{10} x^{11} + \frac {11}{2} \, {\left (91 \, b^{3} c^{6} + 3 \, a b^{2} c^{2}\right )} d^{9} x^{10} + 55 \, {\left (13 \, b^{3} c^{7} + a b^{2} c^{3}\right )} d^{8} x^{9} + \frac {3}{8} \, {\left (2145 \, b^{3} c^{8} + 330 \, a b^{2} c^{4} + a^{2} b\right )} d^{7} x^{8} + {\left (715 \, b^{3} c^{9} + 198 \, a b^{2} c^{5} + 3 \, a^{2} b c\right )} d^{6} x^{7} + \frac {7}{2} \, {\left (143 \, b^{3} c^{10} + 66 \, a b^{2} c^{6} + 3 \, a^{2} b c^{2}\right )} d^{5} x^{6} + 3 \, {\left (91 \, b^{3} c^{11} + 66 \, a b^{2} c^{7} + 7 \, a^{2} b c^{3}\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (455 \, b^{3} c^{12} + 495 \, a b^{2} c^{8} + 105 \, a^{2} b c^{4} + a^{3}\right )} d^{3} x^{4} + {\left (35 \, b^{3} c^{13} + 55 \, a b^{2} c^{9} + 21 \, a^{2} b c^{5} + a^{3} c\right )} d^{2} x^{3} + \frac {3}{2} \, {\left (5 \, b^{3} c^{14} + 11 \, a b^{2} c^{10} + 7 \, a^{2} b c^{6} + a^{3} c^{2}\right )} d x^{2} + {\left (b^{3} c^{15} + 3 \, a b^{2} c^{11} + 3 \, a^{2} b c^{7} + a^{3} c^{3}\right )} x \]
1/16*b^3*d^15*x^16 + b^3*c*d^14*x^15 + 15/2*b^3*c^2*d^13*x^14 + 35*b^3*c^3 *d^12*x^13 + 1/4*(455*b^3*c^4 + a*b^2)*d^11*x^12 + 3*(91*b^3*c^5 + a*b^2*c )*d^10*x^11 + 11/2*(91*b^3*c^6 + 3*a*b^2*c^2)*d^9*x^10 + 55*(13*b^3*c^7 + a*b^2*c^3)*d^8*x^9 + 3/8*(2145*b^3*c^8 + 330*a*b^2*c^4 + a^2*b)*d^7*x^8 + (715*b^3*c^9 + 198*a*b^2*c^5 + 3*a^2*b*c)*d^6*x^7 + 7/2*(143*b^3*c^10 + 66 *a*b^2*c^6 + 3*a^2*b*c^2)*d^5*x^6 + 3*(91*b^3*c^11 + 66*a*b^2*c^7 + 7*a^2* b*c^3)*d^4*x^5 + 1/4*(455*b^3*c^12 + 495*a*b^2*c^8 + 105*a^2*b*c^4 + a^3)* d^3*x^4 + (35*b^3*c^13 + 55*a*b^2*c^9 + 21*a^2*b*c^5 + a^3*c)*d^2*x^3 + 3/ 2*(5*b^3*c^14 + 11*a*b^2*c^10 + 7*a^2*b*c^6 + a^3*c^2)*d*x^2 + (b^3*c^15 + 3*a*b^2*c^11 + 3*a^2*b*c^7 + a^3*c^3)*x
Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (15) = 30\).
Time = 0.07 (sec) , antiderivative size = 541, normalized size of antiderivative = 23.52 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^3 \, dx=35 b^{3} c^{3} d^{12} x^{13} + \frac {15 b^{3} c^{2} d^{13} x^{14}}{2} + b^{3} c d^{14} x^{15} + \frac {b^{3} d^{15} x^{16}}{16} + x^{12} \left (\frac {a b^{2} d^{11}}{4} + \frac {455 b^{3} c^{4} d^{11}}{4}\right ) + x^{11} \cdot \left (3 a b^{2} c d^{10} + 273 b^{3} c^{5} d^{10}\right ) + x^{10} \cdot \left (\frac {33 a b^{2} c^{2} d^{9}}{2} + \frac {1001 b^{3} c^{6} d^{9}}{2}\right ) + x^{9} \cdot \left (55 a b^{2} c^{3} d^{8} + 715 b^{3} c^{7} d^{8}\right ) + x^{8} \cdot \left (\frac {3 a^{2} b d^{7}}{8} + \frac {495 a b^{2} c^{4} d^{7}}{4} + \frac {6435 b^{3} c^{8} d^{7}}{8}\right ) + x^{7} \cdot \left (3 a^{2} b c d^{6} + 198 a b^{2} c^{5} d^{6} + 715 b^{3} c^{9} d^{6}\right ) + x^{6} \cdot \left (\frac {21 a^{2} b c^{2} d^{5}}{2} + 231 a b^{2} c^{6} d^{5} + \frac {1001 b^{3} c^{10} d^{5}}{2}\right ) + x^{5} \cdot \left (21 a^{2} b c^{3} d^{4} + 198 a b^{2} c^{7} d^{4} + 273 b^{3} c^{11} d^{4}\right ) + x^{4} \left (\frac {a^{3} d^{3}}{4} + \frac {105 a^{2} b c^{4} d^{3}}{4} + \frac {495 a b^{2} c^{8} d^{3}}{4} + \frac {455 b^{3} c^{12} d^{3}}{4}\right ) + x^{3} \left (a^{3} c d^{2} + 21 a^{2} b c^{5} d^{2} + 55 a b^{2} c^{9} d^{2} + 35 b^{3} c^{13} d^{2}\right ) + x^{2} \cdot \left (\frac {3 a^{3} c^{2} d}{2} + \frac {21 a^{2} b c^{6} d}{2} + \frac {33 a b^{2} c^{10} d}{2} + \frac {15 b^{3} c^{14} d}{2}\right ) + x \left (a^{3} c^{3} + 3 a^{2} b c^{7} + 3 a b^{2} c^{11} + b^{3} c^{15}\right ) \]
35*b**3*c**3*d**12*x**13 + 15*b**3*c**2*d**13*x**14/2 + b**3*c*d**14*x**15 + b**3*d**15*x**16/16 + x**12*(a*b**2*d**11/4 + 455*b**3*c**4*d**11/4) + x**11*(3*a*b**2*c*d**10 + 273*b**3*c**5*d**10) + x**10*(33*a*b**2*c**2*d** 9/2 + 1001*b**3*c**6*d**9/2) + x**9*(55*a*b**2*c**3*d**8 + 715*b**3*c**7*d **8) + x**8*(3*a**2*b*d**7/8 + 495*a*b**2*c**4*d**7/4 + 6435*b**3*c**8*d** 7/8) + x**7*(3*a**2*b*c*d**6 + 198*a*b**2*c**5*d**6 + 715*b**3*c**9*d**6) + x**6*(21*a**2*b*c**2*d**5/2 + 231*a*b**2*c**6*d**5 + 1001*b**3*c**10*d** 5/2) + x**5*(21*a**2*b*c**3*d**4 + 198*a*b**2*c**7*d**4 + 273*b**3*c**11*d **4) + x**4*(a**3*d**3/4 + 105*a**2*b*c**4*d**3/4 + 495*a*b**2*c**8*d**3/4 + 455*b**3*c**12*d**3/4) + x**3*(a**3*c*d**2 + 21*a**2*b*c**5*d**2 + 55*a *b**2*c**9*d**2 + 35*b**3*c**13*d**2) + x**2*(3*a**3*c**2*d/2 + 21*a**2*b* c**6*d/2 + 33*a*b**2*c**10*d/2 + 15*b**3*c**14*d/2) + x*(a**3*c**3 + 3*a** 2*b*c**7 + 3*a*b**2*c**11 + b**3*c**15)
Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^3 \, dx=\frac {{\left ({\left (d x + c\right )}^{4} b + a\right )}^{4}}{16 \, b d} \]
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^3 \, dx=\frac {{\left ({\left (d x + c\right )}^{4} b + a\right )}^{4}}{16 \, b d} \]
Time = 0.26 (sec) , antiderivative size = 376, normalized size of antiderivative = 16.35 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^3 \, dx=c^3\,x\,{\left (b\,c^4+a\right )}^3+\frac {d^3\,x^4\,\left (a^3+105\,a^2\,b\,c^4+495\,a\,b^2\,c^8+455\,b^3\,c^{12}\right )}{4}+\frac {b^3\,d^{15}\,x^{16}}{16}+\frac {b^2\,d^{11}\,x^{12}\,\left (455\,b\,c^4+a\right )}{4}+b^3\,c\,d^{14}\,x^{15}+\frac {3\,b\,d^7\,x^8\,\left (a^2+330\,a\,b\,c^4+2145\,b^2\,c^8\right )}{8}+35\,b^3\,c^3\,d^{12}\,x^{13}+\frac {15\,b^3\,c^2\,d^{13}\,x^{14}}{2}+c\,d^2\,x^3\,\left (a^3+21\,a^2\,b\,c^4+55\,a\,b^2\,c^8+35\,b^3\,c^{12}\right )+b\,c\,d^6\,x^7\,\left (3\,a^2+198\,a\,b\,c^4+715\,b^2\,c^8\right )+\frac {11\,b^2\,c^2\,d^9\,x^{10}\,\left (91\,b\,c^4+3\,a\right )}{2}+\frac {3\,c^2\,d\,x^2\,{\left (b\,c^4+a\right )}^2\,\left (5\,b\,c^4+a\right )}{2}+3\,b^2\,c\,d^{10}\,x^{11}\,\left (91\,b\,c^4+a\right )+3\,b\,c^3\,d^4\,x^5\,\left (7\,a^2+66\,a\,b\,c^4+91\,b^2\,c^8\right )+\frac {7\,b\,c^2\,d^5\,x^6\,\left (3\,a^2+66\,a\,b\,c^4+143\,b^2\,c^8\right )}{2}+55\,b^2\,c^3\,d^8\,x^9\,\left (13\,b\,c^4+a\right ) \]
c^3*x*(a + b*c^4)^3 + (d^3*x^4*(a^3 + 455*b^3*c^12 + 105*a^2*b*c^4 + 495*a *b^2*c^8))/4 + (b^3*d^15*x^16)/16 + (b^2*d^11*x^12*(a + 455*b*c^4))/4 + b^ 3*c*d^14*x^15 + (3*b*d^7*x^8*(a^2 + 2145*b^2*c^8 + 330*a*b*c^4))/8 + 35*b^ 3*c^3*d^12*x^13 + (15*b^3*c^2*d^13*x^14)/2 + c*d^2*x^3*(a^3 + 35*b^3*c^12 + 21*a^2*b*c^4 + 55*a*b^2*c^8) + b*c*d^6*x^7*(3*a^2 + 715*b^2*c^8 + 198*a* b*c^4) + (11*b^2*c^2*d^9*x^10*(3*a + 91*b*c^4))/2 + (3*c^2*d*x^2*(a + b*c^ 4)^2*(a + 5*b*c^4))/2 + 3*b^2*c*d^10*x^11*(a + 91*b*c^4) + 3*b*c^3*d^4*x^5 *(7*a^2 + 91*b^2*c^8 + 66*a*b*c^4) + (7*b*c^2*d^5*x^6*(3*a^2 + 143*b^2*c^8 + 66*a*b*c^4))/2 + 55*b^2*c^3*d^8*x^9*(a + 13*b*c^4)