Integrand size = 29, antiderivative size = 257 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=\frac {c^4 \left (c^3+4 a d^2\right )^4 x}{d^8}-\frac {8 c^5 \left (c^3+4 a d^2\right )^3 (c+d x)^3}{3 d^9}+\frac {4 c^3 \left (c^3+4 a d^2\right )^2 \left (7 c^3+4 a d^2\right ) (c+d x)^5}{5 d^9}-\frac {8 c^4 \left (c^3+4 a d^2\right ) \left (7 c^3+12 a d^2\right ) (c+d x)^7}{7 d^9}+\frac {2 c^2 \left (35 c^6+120 a c^3 d^2+48 a^2 d^4\right ) (c+d x)^9}{9 d^9}-\frac {8 c^3 \left (7 c^3+12 a d^2\right ) (c+d x)^{11}}{11 d^9}+\frac {4 c \left (7 c^3+4 a d^2\right ) (c+d x)^{13}}{13 d^9}-\frac {8 c^2 (c+d x)^{15}}{15 d^9}+\frac {(c+d x)^{17}}{17 d^9} \] Output:
c^4*(4*a*d^2+c^3)^4*x/d^8-8/3*c^5*(4*a*d^2+c^3)^3*(d*x+c)^3/d^9+4/5*c^3*(4 *a*d^2+c^3)^2*(4*a*d^2+7*c^3)*(d*x+c)^5/d^9-8/7*c^4*(4*a*d^2+c^3)*(12*a*d^ 2+7*c^3)*(d*x+c)^7/d^9+2/9*c^2*(48*a^2*d^4+120*a*c^3*d^2+35*c^6)*(d*x+c)^9 /d^9-8/11*c^3*(12*a*d^2+7*c^3)*(d*x+c)^11/d^9+4/13*c*(4*a*d^2+7*c^3)*(d*x+ c)^13/d^9-8/15*c^2*(d*x+c)^15/d^9+1/17*(d*x+c)^17/d^9
Time = 0.05 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.11 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=256 a^4 c^4 x+\frac {1024}{3} a^3 c^5 x^3+256 a^3 c^4 d x^4+\frac {256}{5} a^2 c^3 \left (6 c^3+a d^2\right ) x^5+512 a^2 c^5 d x^6+\frac {256}{7} a c^4 \left (4 c^3+9 a d^2\right ) x^7+96 a c^3 d \left (4 c^3+a d^2\right ) x^8+\frac {32}{9} c^2 \left (8 c^6+120 a c^3 d^2+3 a^2 d^4\right ) x^9+\frac {256}{5} c^4 d \left (2 c^3+5 a d^2\right ) x^{10}+\frac {64}{11} c^3 d^2 \left (28 c^3+15 a d^2\right ) x^{11}+\frac {16}{3} c^2 d^3 \left (28 c^3+3 a d^2\right ) x^{12}+\frac {16}{13} c d^4 \left (70 c^3+a d^2\right ) x^{13}+32 c^3 d^5 x^{14}+\frac {112}{15} c^2 d^6 x^{15}+c d^7 x^{16}+\frac {d^8 x^{17}}{17} \] Input:
Integrate[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^4,x]
Output:
256*a^4*c^4*x + (1024*a^3*c^5*x^3)/3 + 256*a^3*c^4*d*x^4 + (256*a^2*c^3*(6 *c^3 + a*d^2)*x^5)/5 + 512*a^2*c^5*d*x^6 + (256*a*c^4*(4*c^3 + 9*a*d^2)*x^ 7)/7 + 96*a*c^3*d*(4*c^3 + a*d^2)*x^8 + (32*c^2*(8*c^6 + 120*a*c^3*d^2 + 3 *a^2*d^4)*x^9)/9 + (256*c^4*d*(2*c^3 + 5*a*d^2)*x^10)/5 + (64*c^3*d^2*(28* c^3 + 15*a*d^2)*x^11)/11 + (16*c^2*d^3*(28*c^3 + 3*a*d^2)*x^12)/3 + (16*c* d^4*(70*c^3 + a*d^2)*x^13)/13 + 32*c^3*d^5*x^14 + (112*c^2*d^6*x^15)/15 + c*d^7*x^16 + (d^8*x^17)/17
Time = 0.59 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2458, 1403, 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.
| \(\displaystyle \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )^4d\left (\frac {c}{d}+x\right )\) |
| \(\Big \downarrow \) 1403 |
| \(\displaystyle \int \left (16 c^8 \left (\frac {6 a^2 d^4}{c^6}+\frac {15 a d^2}{c^3}+\frac {35}{8}\right ) \left (\frac {c}{d}+x\right )^8+\frac {\left (4 a c d^2+c^4\right )^4}{d^8}-\frac {32 c^7 \left (4 a d^2+c^3\right ) \left (\frac {3 a d^2}{c^3}+\frac {7}{4}\right ) \left (\frac {c}{d}+x\right )^6}{d^2}-32 c^6 d^2 \left (\frac {3 a d^2}{c^3}+\frac {7}{4}\right ) \left (\frac {c}{d}+x\right )^{10}+\frac {24 c^6 \left (4 a d^2+c^3\right )^2 \left (\frac {2 a d^2}{3 c^3}+\frac {7}{6}\right ) \left (\frac {c}{d}+x\right )^4}{d^4}-\frac {8 c^5 \left (4 a d^2+c^3\right )^3 \left (\frac {c}{d}+x\right )^2}{d^6}+24 c^4 d^4 \left (\frac {2 a d^2}{3 c^3}+\frac {7}{6}\right ) \left (\frac {c}{d}+x\right )^{12}-8 c^2 d^6 \left (\frac {c}{d}+x\right )^{14}+d^8 \left (\frac {c}{d}+x\right )^{16}\right )d\left (\frac {c}{d}+x\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{9} c^2 \left (48 a^2 d^4+120 a c^3 d^2+35 c^6\right ) \left (\frac {c}{d}+x\right )^9-\frac {8}{11} c^3 d^2 \left (12 a d^2+7 c^3\right ) \left (\frac {c}{d}+x\right )^{11}+\frac {4}{13} c d^4 \left (4 a d^2+7 c^3\right ) \left (\frac {c}{d}+x\right )^{13}+\frac {4 c^3 \left (4 a d^2+c^3\right )^2 \left (4 a d^2+7 c^3\right ) \left (\frac {c}{d}+x\right )^5}{5 d^4}-\frac {8 c^5 \left (4 a d^2+c^3\right )^3 \left (\frac {c}{d}+x\right )^3}{3 d^6}-\frac {8 c^4 \left (4 a d^2+c^3\right ) \left (12 a d^2+7 c^3\right ) \left (\frac {c}{d}+x\right )^7}{7 d^2}+\frac {c^4 \left (4 a d^2+c^3\right )^4 \left (\frac {c}{d}+x\right )}{d^8}-\frac {8}{15} c^2 d^6 \left (\frac {c}{d}+x\right )^{15}+\frac {1}{17} d^8 \left (\frac {c}{d}+x\right )^{17}\) |
Input:
Int[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^4,x]
Output:
(c^4*(c^3 + 4*a*d^2)^4*(c/d + x))/d^8 - (8*c^5*(c^3 + 4*a*d^2)^3*(c/d + x) ^3)/(3*d^6) + (4*c^3*(c^3 + 4*a*d^2)^2*(7*c^3 + 4*a*d^2)*(c/d + x)^5)/(5*d ^4) - (8*c^4*(c^3 + 4*a*d^2)*(7*c^3 + 12*a*d^2)*(c/d + x)^7)/(7*d^2) + (2* c^2*(35*c^6 + 120*a*c^3*d^2 + 48*a^2*d^4)*(c/d + x)^9)/9 - (8*c^3*d^2*(7*c ^3 + 12*a*d^2)*(c/d + x)^11)/11 + (4*c*d^4*(7*c^3 + 4*a*d^2)*(c/d + x)^13) /13 - (8*c^2*d^6*(c/d + x)^15)/15 + (d^8*(c/d + x)^17)/17
Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandInte grand[(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a *c, 0] && IGtQ[p, 0]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Time = 0.08 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.04
| method | result | size |
| norman | \(256 c^{4} a^{4} x +\frac {1024 a^{3} c^{5} x^{3}}{3}+256 a^{3} c^{4} d \,x^{4}+\left (\frac {256}{5} a^{3} c^{3} d^{2}+\frac {1536}{5} a^{2} c^{6}\right ) x^{5}+512 a^{2} c^{5} d \,x^{6}+\left (\frac {2304}{7} a^{2} c^{4} d^{2}+\frac {1024}{7} a \,c^{7}\right ) x^{7}+\left (96 a^{2} c^{3} d^{3}+384 a \,c^{6} d \right ) x^{8}+\left (\frac {32}{3} a^{2} c^{2} d^{4}+\frac {1280}{3} a \,c^{5} d^{2}+\frac {256}{9} c^{8}\right ) x^{9}+\left (256 a \,c^{4} d^{3}+\frac {512}{5} c^{7} d \right ) x^{10}+\left (\frac {960}{11} a \,c^{3} d^{4}+\frac {1792}{11} c^{6} d^{2}\right ) x^{11}+\left (16 a \,c^{2} d^{5}+\frac {448}{3} c^{5} d^{3}\right ) x^{12}+\left (\frac {16}{13} a c \,d^{6}+\frac {1120}{13} c^{4} d^{4}\right ) x^{13}+32 c^{3} d^{5} x^{14}+\frac {112 c^{2} d^{6} x^{15}}{15}+c \,d^{7} x^{16}+\frac {d^{8} x^{17}}{17}\) | \(267\) |
| gosper | \(\frac {16}{13} x^{13} a c \,d^{6}+96 a^{2} c^{3} d^{3} x^{8}+384 a \,c^{6} d \,x^{8}+\frac {256}{5} x^{5} a^{3} c^{3} d^{2}+\frac {2304}{7} x^{7} a^{2} c^{4} d^{2}+\frac {32}{3} x^{9} a^{2} c^{2} d^{4}+\frac {1280}{3} x^{9} a \,c^{5} d^{2}+256 x^{10} a \,c^{4} d^{3}+\frac {960}{11} x^{11} a \,c^{3} d^{4}+16 x^{12} a \,c^{2} d^{5}+256 a^{3} c^{4} d \,x^{4}+512 a^{2} c^{5} d \,x^{6}+\frac {1}{17} d^{8} x^{17}+\frac {1024}{3} a^{3} c^{5} x^{3}+\frac {1792}{11} x^{11} c^{6} d^{2}+\frac {256}{9} x^{9} c^{8}+\frac {512}{5} x^{10} c^{7} d +\frac {112}{15} c^{2} d^{6} x^{15}+\frac {1120}{13} x^{13} c^{4} d^{4}+32 c^{3} d^{5} x^{14}+\frac {448}{3} x^{12} c^{5} d^{3}+\frac {1536}{5} x^{5} a^{2} c^{6}+\frac {1024}{7} x^{7} a \,c^{7}+256 c^{4} a^{4} x +c \,d^{7} x^{16}\) | \(278\) |
| risch | \(\frac {16}{13} x^{13} a c \,d^{6}+96 a^{2} c^{3} d^{3} x^{8}+384 a \,c^{6} d \,x^{8}+\frac {256}{5} x^{5} a^{3} c^{3} d^{2}+\frac {2304}{7} x^{7} a^{2} c^{4} d^{2}+\frac {32}{3} x^{9} a^{2} c^{2} d^{4}+\frac {1280}{3} x^{9} a \,c^{5} d^{2}+256 x^{10} a \,c^{4} d^{3}+\frac {960}{11} x^{11} a \,c^{3} d^{4}+16 x^{12} a \,c^{2} d^{5}+256 a^{3} c^{4} d \,x^{4}+512 a^{2} c^{5} d \,x^{6}+\frac {1}{17} d^{8} x^{17}+\frac {1024}{3} a^{3} c^{5} x^{3}+\frac {1792}{11} x^{11} c^{6} d^{2}+\frac {256}{9} x^{9} c^{8}+\frac {512}{5} x^{10} c^{7} d +\frac {112}{15} c^{2} d^{6} x^{15}+\frac {1120}{13} x^{13} c^{4} d^{4}+32 c^{3} d^{5} x^{14}+\frac {448}{3} x^{12} c^{5} d^{3}+\frac {1536}{5} x^{5} a^{2} c^{6}+\frac {1024}{7} x^{7} a \,c^{7}+256 c^{4} a^{4} x +c \,d^{7} x^{16}\) | \(278\) |
| parallelrisch | \(\frac {16}{13} x^{13} a c \,d^{6}+96 a^{2} c^{3} d^{3} x^{8}+384 a \,c^{6} d \,x^{8}+\frac {256}{5} x^{5} a^{3} c^{3} d^{2}+\frac {2304}{7} x^{7} a^{2} c^{4} d^{2}+\frac {32}{3} x^{9} a^{2} c^{2} d^{4}+\frac {1280}{3} x^{9} a \,c^{5} d^{2}+256 x^{10} a \,c^{4} d^{3}+\frac {960}{11} x^{11} a \,c^{3} d^{4}+16 x^{12} a \,c^{2} d^{5}+256 a^{3} c^{4} d \,x^{4}+512 a^{2} c^{5} d \,x^{6}+\frac {1}{17} d^{8} x^{17}+\frac {1024}{3} a^{3} c^{5} x^{3}+\frac {1792}{11} x^{11} c^{6} d^{2}+\frac {256}{9} x^{9} c^{8}+\frac {512}{5} x^{10} c^{7} d +\frac {112}{15} c^{2} d^{6} x^{15}+\frac {1120}{13} x^{13} c^{4} d^{4}+32 c^{3} d^{5} x^{14}+\frac {448}{3} x^{12} c^{5} d^{3}+\frac {1536}{5} x^{5} a^{2} c^{6}+\frac {1024}{7} x^{7} a \,c^{7}+256 c^{4} a^{4} x +c \,d^{7} x^{16}\) | \(278\) |
| orering | \(\frac {x \left (45045 d^{8} x^{16}+765765 c \,d^{7} x^{15}+5717712 c^{2} d^{6} x^{14}+24504480 c^{3} d^{5} x^{13}+942480 a c \,d^{6} x^{12}+65973600 c^{4} d^{4} x^{12}+12252240 a \,c^{2} d^{5} x^{11}+114354240 c^{5} d^{3} x^{11}+66830400 a \,c^{3} d^{4} x^{10}+124750080 c^{6} d^{2} x^{10}+196035840 a \,c^{4} d^{3} x^{9}+78414336 c^{7} d \,x^{9}+8168160 a^{2} c^{2} d^{4} x^{8}+326726400 a \,c^{5} d^{2} x^{8}+21781760 c^{8} x^{8}+73513440 a^{2} c^{3} d^{3} x^{7}+294053760 a \,c^{6} d \,x^{7}+252046080 a^{2} c^{4} d^{2} x^{6}+112020480 a \,c^{7} x^{6}+392071680 a^{2} c^{5} d \,x^{5}+39207168 a^{3} c^{3} d^{2} x^{4}+235243008 a^{2} c^{6} x^{4}+196035840 a^{3} c^{4} d \,x^{3}+261381120 a^{3} c^{5} x^{2}+196035840 c^{4} a^{4}\right )}{765765}\) | \(281\) |
| default | \(\frac {d^{8} x^{17}}{17}+c \,d^{7} x^{16}+\frac {112 c^{2} d^{6} x^{15}}{15}+32 c^{3} d^{5} x^{14}+\frac {\left (2 \left (8 a c \,d^{2}+16 c^{4}\right ) d^{4}+1088 c^{4} d^{4}\right ) x^{13}}{13}+\frac {\left (64 a \,c^{2} d^{5}+16 \left (8 a c \,d^{2}+16 c^{4}\right ) c \,d^{3}+1536 c^{5} d^{3}\right ) x^{12}}{12}+\frac {\left (576 a \,c^{3} d^{4}+48 \left (8 a c \,d^{2}+16 c^{4}\right ) c^{2} d^{2}+1024 c^{6} d^{2}\right ) x^{11}}{11}+\frac {\left (2048 a \,c^{4} d^{3}+64 \left (8 a c \,d^{2}+16 c^{4}\right ) c^{3} d \right ) x^{10}}{10}+\frac {\left (32 a^{2} c^{2} d^{4}+3584 a \,c^{5} d^{2}+\left (8 a c \,d^{2}+16 c^{4}\right )^{2}\right ) x^{9}}{9}+\frac {\left (256 a^{2} c^{3} d^{3}+2048 a \,c^{6} d +64 a \,c^{2} d \left (8 a c \,d^{2}+16 c^{4}\right )\right ) x^{8}}{8}+\frac {\left (1792 a^{2} c^{4} d^{2}+64 a \,c^{3} \left (8 a c \,d^{2}+16 c^{4}\right )\right ) x^{7}}{7}+512 a^{2} c^{5} d \,x^{6}+\frac {\left (32 a^{2} c^{2} \left (8 a c \,d^{2}+16 c^{4}\right )+1024 a^{2} c^{6}\right ) x^{5}}{5}+256 a^{3} c^{4} d \,x^{4}+\frac {1024 a^{3} c^{5} x^{3}}{3}+256 c^{4} a^{4} x\) | \(392\) |
Input:
int((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^4,x,method=_RETURNVERBOSE)
Output:
256*c^4*a^4*x+1024/3*a^3*c^5*x^3+256*a^3*c^4*d*x^4+(256/5*a^3*c^3*d^2+1536 /5*a^2*c^6)*x^5+512*a^2*c^5*d*x^6+(2304/7*a^2*c^4*d^2+1024/7*a*c^7)*x^7+(9 6*a^2*c^3*d^3+384*a*c^6*d)*x^8+(32/3*a^2*c^2*d^4+1280/3*a*c^5*d^2+256/9*c^ 8)*x^9+(256*a*c^4*d^3+512/5*c^7*d)*x^10+(960/11*a*c^3*d^4+1792/11*c^6*d^2) *x^11+(16*a*c^2*d^5+448/3*c^5*d^3)*x^12+(16/13*a*c*d^6+1120/13*c^4*d^4)*x^ 13+32*c^3*d^5*x^14+112/15*c^2*d^6*x^15+c*d^7*x^16+1/17*d^8*x^17
Time = 0.09 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.05 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=\frac {1}{17} \, d^{8} x^{17} + c d^{7} x^{16} + \frac {112}{15} \, c^{2} d^{6} x^{15} + 32 \, c^{3} d^{5} x^{14} + 512 \, a^{2} c^{5} d x^{6} + \frac {16}{13} \, {\left (70 \, c^{4} d^{4} + a c d^{6}\right )} x^{13} + \frac {16}{3} \, {\left (28 \, c^{5} d^{3} + 3 \, a c^{2} d^{5}\right )} x^{12} + 256 \, a^{3} c^{4} d x^{4} + \frac {64}{11} \, {\left (28 \, c^{6} d^{2} + 15 \, a c^{3} d^{4}\right )} x^{11} + \frac {1024}{3} \, a^{3} c^{5} x^{3} + \frac {256}{5} \, {\left (2 \, c^{7} d + 5 \, a c^{4} d^{3}\right )} x^{10} + \frac {32}{9} \, {\left (8 \, c^{8} + 120 \, a c^{5} d^{2} + 3 \, a^{2} c^{2} d^{4}\right )} x^{9} + 256 \, a^{4} c^{4} x + 96 \, {\left (4 \, a c^{6} d + a^{2} c^{3} d^{3}\right )} x^{8} + \frac {256}{7} \, {\left (4 \, a c^{7} + 9 \, a^{2} c^{4} d^{2}\right )} x^{7} + \frac {256}{5} \, {\left (6 \, a^{2} c^{6} + a^{3} c^{3} d^{2}\right )} x^{5} \] Input:
integrate((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^4,x, algorithm="fricas")
Output:
1/17*d^8*x^17 + c*d^7*x^16 + 112/15*c^2*d^6*x^15 + 32*c^3*d^5*x^14 + 512*a ^2*c^5*d*x^6 + 16/13*(70*c^4*d^4 + a*c*d^6)*x^13 + 16/3*(28*c^5*d^3 + 3*a* c^2*d^5)*x^12 + 256*a^3*c^4*d*x^4 + 64/11*(28*c^6*d^2 + 15*a*c^3*d^4)*x^11 + 1024/3*a^3*c^5*x^3 + 256/5*(2*c^7*d + 5*a*c^4*d^3)*x^10 + 32/9*(8*c^8 + 120*a*c^5*d^2 + 3*a^2*c^2*d^4)*x^9 + 256*a^4*c^4*x + 96*(4*a*c^6*d + a^2* c^3*d^3)*x^8 + 256/7*(4*a*c^7 + 9*a^2*c^4*d^2)*x^7 + 256/5*(6*a^2*c^6 + a^ 3*c^3*d^2)*x^5
Time = 0.06 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.16 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=256 a^{4} c^{4} x + \frac {1024 a^{3} c^{5} x^{3}}{3} + 256 a^{3} c^{4} d x^{4} + 512 a^{2} c^{5} d x^{6} + 32 c^{3} d^{5} x^{14} + \frac {112 c^{2} d^{6} x^{15}}{15} + c d^{7} x^{16} + \frac {d^{8} x^{17}}{17} + x^{13} \cdot \left (\frac {16 a c d^{6}}{13} + \frac {1120 c^{4} d^{4}}{13}\right ) + x^{12} \cdot \left (16 a c^{2} d^{5} + \frac {448 c^{5} d^{3}}{3}\right ) + x^{11} \cdot \left (\frac {960 a c^{3} d^{4}}{11} + \frac {1792 c^{6} d^{2}}{11}\right ) + x^{10} \cdot \left (256 a c^{4} d^{3} + \frac {512 c^{7} d}{5}\right ) + x^{9} \cdot \left (\frac {32 a^{2} c^{2} d^{4}}{3} + \frac {1280 a c^{5} d^{2}}{3} + \frac {256 c^{8}}{9}\right ) + x^{8} \cdot \left (96 a^{2} c^{3} d^{3} + 384 a c^{6} d\right ) + x^{7} \cdot \left (\frac {2304 a^{2} c^{4} d^{2}}{7} + \frac {1024 a c^{7}}{7}\right ) + x^{5} \cdot \left (\frac {256 a^{3} c^{3} d^{2}}{5} + \frac {1536 a^{2} c^{6}}{5}\right ) \] Input:
integrate((d**2*x**4+4*c*d*x**3+4*c**2*x**2+4*a*c)**4,x)
Output:
256*a**4*c**4*x + 1024*a**3*c**5*x**3/3 + 256*a**3*c**4*d*x**4 + 512*a**2* c**5*d*x**6 + 32*c**3*d**5*x**14 + 112*c**2*d**6*x**15/15 + c*d**7*x**16 + d**8*x**17/17 + x**13*(16*a*c*d**6/13 + 1120*c**4*d**4/13) + x**12*(16*a* c**2*d**5 + 448*c**5*d**3/3) + x**11*(960*a*c**3*d**4/11 + 1792*c**6*d**2/ 11) + x**10*(256*a*c**4*d**3 + 512*c**7*d/5) + x**9*(32*a**2*c**2*d**4/3 + 1280*a*c**5*d**2/3 + 256*c**8/9) + x**8*(96*a**2*c**3*d**3 + 384*a*c**6*d ) + x**7*(2304*a**2*c**4*d**2/7 + 1024*a*c**7/7) + x**5*(256*a**3*c**3*d** 2/5 + 1536*a**2*c**6/5)
Time = 0.04 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.45 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=\frac {1}{17} \, d^{8} x^{17} + c d^{7} x^{16} + \frac {32}{5} \, c^{2} d^{6} x^{15} + \frac {128}{7} \, c^{3} d^{5} x^{14} + \frac {256}{13} \, c^{4} d^{4} x^{13} + \frac {256}{9} \, c^{8} x^{9} + 256 \, a^{4} c^{4} x + \frac {256}{15} \, {\left (3 \, d^{2} x^{5} + 15 \, c d x^{4} + 20 \, c^{2} x^{3}\right )} a^{3} c^{3} + \frac {256}{55} \, {\left (5 \, d^{2} x^{11} + 22 \, c d x^{10}\right )} c^{6} + \frac {32}{105} \, {\left (35 \, d^{4} x^{9} + 315 \, c d^{3} x^{8} + 720 \, c^{2} d^{2} x^{7} + 1008 \, c^{4} x^{5} + 120 \, {\left (3 \, d^{2} x^{7} + 14 \, c d x^{6}\right )} c^{2}\right )} a^{2} c^{2} + \frac {32}{143} \, {\left (33 \, d^{4} x^{13} + 286 \, c d^{3} x^{12} + 624 \, c^{2} d^{2} x^{11}\right )} c^{4} + \frac {16}{15015} \, {\left (1155 \, d^{6} x^{13} + 15015 \, c d^{5} x^{12} + 65520 \, c^{2} d^{4} x^{11} + 96096 \, c^{3} d^{3} x^{10} + 137280 \, c^{6} x^{7} + 40040 \, {\left (2 \, d^{2} x^{9} + 9 \, c d x^{8}\right )} c^{4} + 364 \, {\left (45 \, d^{4} x^{11} + 396 \, c d^{3} x^{10} + 880 \, c^{2} d^{2} x^{9}\right )} c^{2}\right )} a c + \frac {16}{1365} \, {\left (91 \, d^{6} x^{15} + 1170 \, c d^{5} x^{14} + 5040 \, c^{2} d^{4} x^{13} + 7280 \, c^{3} d^{3} x^{12}\right )} c^{2} \] Input:
integrate((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^4,x, algorithm="maxima")
Output:
1/17*d^8*x^17 + c*d^7*x^16 + 32/5*c^2*d^6*x^15 + 128/7*c^3*d^5*x^14 + 256/ 13*c^4*d^4*x^13 + 256/9*c^8*x^9 + 256*a^4*c^4*x + 256/15*(3*d^2*x^5 + 15*c *d*x^4 + 20*c^2*x^3)*a^3*c^3 + 256/55*(5*d^2*x^11 + 22*c*d*x^10)*c^6 + 32/ 105*(35*d^4*x^9 + 315*c*d^3*x^8 + 720*c^2*d^2*x^7 + 1008*c^4*x^5 + 120*(3* d^2*x^7 + 14*c*d*x^6)*c^2)*a^2*c^2 + 32/143*(33*d^4*x^13 + 286*c*d^3*x^12 + 624*c^2*d^2*x^11)*c^4 + 16/15015*(1155*d^6*x^13 + 15015*c*d^5*x^12 + 655 20*c^2*d^4*x^11 + 96096*c^3*d^3*x^10 + 137280*c^6*x^7 + 40040*(2*d^2*x^9 + 9*c*d*x^8)*c^4 + 364*(45*d^4*x^11 + 396*c*d^3*x^10 + 880*c^2*d^2*x^9)*c^2 )*a*c + 16/1365*(91*d^6*x^15 + 1170*c*d^5*x^14 + 5040*c^2*d^4*x^13 + 7280* c^3*d^3*x^12)*c^2
Time = 0.11 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.08 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=\frac {1}{17} \, d^{8} x^{17} + c d^{7} x^{16} + \frac {112}{15} \, c^{2} d^{6} x^{15} + 32 \, c^{3} d^{5} x^{14} + \frac {1120}{13} \, c^{4} d^{4} x^{13} + \frac {16}{13} \, a c d^{6} x^{13} + \frac {448}{3} \, c^{5} d^{3} x^{12} + 16 \, a c^{2} d^{5} x^{12} + \frac {1792}{11} \, c^{6} d^{2} x^{11} + \frac {960}{11} \, a c^{3} d^{4} x^{11} + \frac {512}{5} \, c^{7} d x^{10} + 256 \, a c^{4} d^{3} x^{10} + \frac {256}{9} \, c^{8} x^{9} + \frac {1280}{3} \, a c^{5} d^{2} x^{9} + \frac {32}{3} \, a^{2} c^{2} d^{4} x^{9} + 384 \, a c^{6} d x^{8} + 96 \, a^{2} c^{3} d^{3} x^{8} + \frac {1024}{7} \, a c^{7} x^{7} + \frac {2304}{7} \, a^{2} c^{4} d^{2} x^{7} + 512 \, a^{2} c^{5} d x^{6} + \frac {1536}{5} \, a^{2} c^{6} x^{5} + \frac {256}{5} \, a^{3} c^{3} d^{2} x^{5} + 256 \, a^{3} c^{4} d x^{4} + \frac {1024}{3} \, a^{3} c^{5} x^{3} + 256 \, a^{4} c^{4} x \] Input:
integrate((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^4,x, algorithm="giac")
Output:
1/17*d^8*x^17 + c*d^7*x^16 + 112/15*c^2*d^6*x^15 + 32*c^3*d^5*x^14 + 1120/ 13*c^4*d^4*x^13 + 16/13*a*c*d^6*x^13 + 448/3*c^5*d^3*x^12 + 16*a*c^2*d^5*x ^12 + 1792/11*c^6*d^2*x^11 + 960/11*a*c^3*d^4*x^11 + 512/5*c^7*d*x^10 + 25 6*a*c^4*d^3*x^10 + 256/9*c^8*x^9 + 1280/3*a*c^5*d^2*x^9 + 32/3*a^2*c^2*d^4 *x^9 + 384*a*c^6*d*x^8 + 96*a^2*c^3*d^3*x^8 + 1024/7*a*c^7*x^7 + 2304/7*a^ 2*c^4*d^2*x^7 + 512*a^2*c^5*d*x^6 + 1536/5*a^2*c^6*x^5 + 256/5*a^3*c^3*d^2 *x^5 + 256*a^3*c^4*d*x^4 + 1024/3*a^3*c^5*x^3 + 256*a^4*c^4*x
Time = 0.25 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.02 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=x^{10}\,\left (\frac {512\,c^7\,d}{5}+256\,a\,c^4\,d^3\right )+x^{13}\,\left (\frac {1120\,c^4\,d^4}{13}+\frac {16\,a\,c\,d^6}{13}\right )+x^9\,\left (\frac {32\,a^2\,c^2\,d^4}{3}+\frac {1280\,a\,c^5\,d^2}{3}+\frac {256\,c^8}{9}\right )+x^{12}\,\left (\frac {448\,c^5\,d^3}{3}+16\,a\,c^2\,d^5\right )+x^{11}\,\left (\frac {1792\,c^6\,d^2}{11}+\frac {960\,a\,c^3\,d^4}{11}\right )+\frac {d^8\,x^{17}}{17}+256\,a^4\,c^4\,x+c\,d^7\,x^{16}+\frac {1024\,a^3\,c^5\,x^3}{3}+32\,c^3\,d^5\,x^{14}+\frac {112\,c^2\,d^6\,x^{15}}{15}+256\,a^3\,c^4\,d\,x^4+512\,a^2\,c^5\,d\,x^6+\frac {256\,a\,c^4\,x^7\,\left (4\,c^3+9\,a\,d^2\right )}{7}+\frac {256\,a^2\,c^3\,x^5\,\left (6\,c^3+a\,d^2\right )}{5}+96\,a\,c^3\,d\,x^8\,\left (4\,c^3+a\,d^2\right ) \] Input:
int((4*a*c + 4*c^2*x^2 + d^2*x^4 + 4*c*d*x^3)^4,x)
Output:
x^10*((512*c^7*d)/5 + 256*a*c^4*d^3) + x^13*((1120*c^4*d^4)/13 + (16*a*c*d ^6)/13) + x^9*((256*c^8)/9 + (1280*a*c^5*d^2)/3 + (32*a^2*c^2*d^4)/3) + x^ 12*((448*c^5*d^3)/3 + 16*a*c^2*d^5) + x^11*((1792*c^6*d^2)/11 + (960*a*c^3 *d^4)/11) + (d^8*x^17)/17 + 256*a^4*c^4*x + c*d^7*x^16 + (1024*a^3*c^5*x^3 )/3 + 32*c^3*d^5*x^14 + (112*c^2*d^6*x^15)/15 + 256*a^3*c^4*d*x^4 + 512*a^ 2*c^5*d*x^6 + (256*a*c^4*x^7*(9*a*d^2 + 4*c^3))/7 + (256*a^2*c^3*x^5*(a*d^ 2 + 6*c^3))/5 + 96*a*c^3*d*x^8*(a*d^2 + 4*c^3)
Time = 0.14 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.09 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=\frac {x \left (45045 d^{8} x^{16}+765765 c \,d^{7} x^{15}+5717712 c^{2} d^{6} x^{14}+24504480 c^{3} d^{5} x^{13}+942480 a c \,d^{6} x^{12}+65973600 c^{4} d^{4} x^{12}+12252240 a \,c^{2} d^{5} x^{11}+114354240 c^{5} d^{3} x^{11}+66830400 a \,c^{3} d^{4} x^{10}+124750080 c^{6} d^{2} x^{10}+196035840 a \,c^{4} d^{3} x^{9}+78414336 c^{7} d \,x^{9}+8168160 a^{2} c^{2} d^{4} x^{8}+326726400 a \,c^{5} d^{2} x^{8}+21781760 c^{8} x^{8}+73513440 a^{2} c^{3} d^{3} x^{7}+294053760 a \,c^{6} d \,x^{7}+252046080 a^{2} c^{4} d^{2} x^{6}+112020480 a \,c^{7} x^{6}+392071680 a^{2} c^{5} d \,x^{5}+39207168 a^{3} c^{3} d^{2} x^{4}+235243008 a^{2} c^{6} x^{4}+196035840 a^{3} c^{4} d \,x^{3}+261381120 a^{3} c^{5} x^{2}+196035840 a^{4} c^{4}\right )}{765765} \] Input:
int((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^4,x)
Output:
(x*(196035840*a**4*c**4 + 261381120*a**3*c**5*x**2 + 196035840*a**3*c**4*d *x**3 + 39207168*a**3*c**3*d**2*x**4 + 235243008*a**2*c**6*x**4 + 39207168 0*a**2*c**5*d*x**5 + 252046080*a**2*c**4*d**2*x**6 + 73513440*a**2*c**3*d* *3*x**7 + 8168160*a**2*c**2*d**4*x**8 + 112020480*a*c**7*x**6 + 294053760* a*c**6*d*x**7 + 326726400*a*c**5*d**2*x**8 + 196035840*a*c**4*d**3*x**9 + 66830400*a*c**3*d**4*x**10 + 12252240*a*c**2*d**5*x**11 + 942480*a*c*d**6* x**12 + 21781760*c**8*x**8 + 78414336*c**7*d*x**9 + 124750080*c**6*d**2*x* *10 + 114354240*c**5*d**3*x**11 + 65973600*c**4*d**4*x**12 + 24504480*c**3 *d**5*x**13 + 5717712*c**2*d**6*x**14 + 765765*c*d**7*x**15 + 45045*d**8*x **16))/765765