3.33 \(\int (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4)^4 \, dx\)

Optimal. Leaf size=270 \[ \frac{2}{9} c^2 \left (48 a^2 d^4+120 a c^3 d^2+35 c^6\right ) \left (\frac{c}{d}+x\right )^9+\frac{4}{13} c d^4 \left (4 a d^2+7 c^3\right ) \left (\frac{c}{d}+x\right )^{13}-\frac{8}{11} c^3 d^2 \left (12 a d^2+7 c^3\right ) \left (\frac{c}{d}+x\right )^{11}-\frac{8 c^4 \left (4 a d^2+c^3\right ) \left (12 a d^2+7 c^3\right ) \left (\frac{c}{d}+x\right )^7}{7 d^2}+\frac{4 c^3 \left (4 a d^2+c^3\right )^2 \left (4 a d^2+7 c^3\right ) \left (\frac{c}{d}+x\right )^5}{5 d^4}-\frac{8 c^5 \left (4 a d^2+c^3\right )^3 \left (\frac{c}{d}+x\right )^3}{3 d^6}+\frac{c^4 x \left (4 a d^2+c^3\right )^4}{d^8}-\frac{8}{15} c^2 d^6 \left (\frac{c}{d}+x\right )^{15}+\frac{1}{17} d^8 \left (\frac{c}{d}+x\right )^{17} \]

[Out]

(c^4*(c^3 + 4*a*d^2)^4*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*(3
5*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

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Rubi [A]  time = 0.538616, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {1106, 1090} \[ \frac{2}{9} c^2 \left (48 a^2 d^4+120 a c^3 d^2+35 c^6\right ) \left (\frac{c}{d}+x\right )^9+\frac{4}{13} c d^4 \left (4 a d^2+7 c^3\right ) \left (\frac{c}{d}+x\right )^{13}-\frac{8}{11} c^3 d^2 \left (12 a d^2+7 c^3\right ) \left (\frac{c}{d}+x\right )^{11}-\frac{8 c^4 \left (4 a d^2+c^3\right ) \left (12 a d^2+7 c^3\right ) \left (\frac{c}{d}+x\right )^7}{7 d^2}+\frac{4 c^3 \left (4 a d^2+c^3\right )^2 \left (4 a d^2+7 c^3\right ) \left (\frac{c}{d}+x\right )^5}{5 d^4}-\frac{8 c^5 \left (4 a d^2+c^3\right )^3 \left (\frac{c}{d}+x\right )^3}{3 d^6}+\frac{c^4 x \left (4 a d^2+c^3\right )^4}{d^8}-\frac{8}{15} c^2 d^6 \left (\frac{c}{d}+x\right )^{15}+\frac{1}{17} d^8 \left (\frac{c}{d}+x\right )^{17} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^4*(c^3 + 4*a*d^2)^4*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*(3
5*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

Rule 1106

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rule 1090

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(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]

Rubi steps

\begin{align*} \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx &=\operatorname{Subst}\left (\int \left (c \left (4 a+\frac{c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4\right )^4 \, dx,x,\frac{c}{d}+x\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{\left (c^4+4 a c d^2\right )^4}{d^8}-\frac{8 c^5 \left (c^3+4 a d^2\right )^3 x^2}{d^6}+\frac{24 c^6 \left (c^3+4 a d^2\right )^2 \left (\frac{7}{6}+\frac{2 a d^2}{3 c^3}\right ) x^4}{d^4}-\frac{32 c^7 \left (c^3+4 a d^2\right ) \left (\frac{7}{4}+\frac{3 a d^2}{c^3}\right ) x^6}{d^2}+16 c^8 \left (\frac{35}{8}+\frac{15 a d^2}{c^3}+\frac{6 a^2 d^4}{c^6}\right ) x^8-32 c^6 d^2 \left (\frac{7}{4}+\frac{3 a d^2}{c^3}\right ) x^{10}+24 c^4 d^4 \left (\frac{7}{6}+\frac{2 a d^2}{3 c^3}\right ) x^{12}-8 c^2 d^6 x^{14}+d^8 x^{16}\right ) \, dx,x,\frac{c}{d}+x\right )\\ &=\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 \left (\frac{c}{d}+x\right )^3}{3 d^6}+\frac{4 c^3 \left (c^3+4 a d^2\right )^2 \left (7 c^3+4 a d^2\right ) \left (\frac{c}{d}+x\right )^5}{5 d^4}-\frac{8 c^4 \left (c^3+4 a d^2\right ) \left (7 c^3+12 a d^2\right ) \left (\frac{c}{d}+x\right )^7}{7 d^2}+\frac{2}{9} c^2 \left (35 c^6+120 a c^3 d^2+48 a^2 d^4\right ) \left (\frac{c}{d}+x\right )^9-\frac{8}{11} c^3 d^2 \left (7 c^3+12 a d^2\right ) \left (\frac{c}{d}+x\right )^{11}+\frac{4}{13} c d^4 \left (7 c^3+4 a d^2\right ) \left (\frac{c}{d}+x\right )^{13}-\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}\\ \end{align*}

Mathematica [A]  time = 0.0355397, size = 285, normalized size = 1.06 \[ \frac{32}{9} c^2 x^9 \left (3 a^2 d^4+120 a c^3 d^2+8 c^6\right )+\frac{256}{5} a^2 c^3 x^5 \left (a d^2+6 c^3\right )+512 a^2 c^5 d x^6+256 a^3 c^4 d x^4+\frac{1024}{3} a^3 c^5 x^3+256 a^4 c^4 x+\frac{16}{13} c d^4 x^{13} \left (a d^2+70 c^3\right )+\frac{16}{3} c^2 d^3 x^{12} \left (3 a d^2+28 c^3\right )+\frac{64}{11} c^3 d^2 x^{11} \left (15 a d^2+28 c^3\right )+\frac{256}{5} c^4 d x^{10} \left (5 a d^2+2 c^3\right )+96 a c^3 d x^8 \left (a d^2+4 c^3\right )+\frac{256}{7} a c^4 x^7 \left (9 a d^2+4 c^3\right )+\frac{112}{15} c^2 d^6 x^{15}+32 c^3 d^5 x^{14}+c d^7 x^{16}+\frac{d^8 x^{17}}{17} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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)/1
5 + c*d^7*x^16 + (d^8*x^17)/17

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Maple [A]  time = 0.003, size = 392, normalized size = 1.5 \begin{align*}{\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\,ac{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\,ac{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\,ac{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\,ac{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\,ac{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\,ac{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\,ac{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\,ac{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\,{a}^{4}{c}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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+1/13*(2*(8*a*c*d^2+16*c^4)*d^4+1088*c^4*d^4)*x^13
+1/12*(64*a*c^2*d^5+16*(8*a*c*d^2+16*c^4)*c*d^3+1536*c^5*d^3)*x^12+1/11*(576*a*c^3*d^4+48*(8*a*c*d^2+16*c^4)*c
^2*d^2+1024*c^6*d^2)*x^11+1/10*(2048*a*c^4*d^3+64*(8*a*c*d^2+16*c^4)*c^3*d)*x^10+1/9*(32*a^2*c^2*d^4+3584*a*c^
5*d^2+(8*a*c*d^2+16*c^4)^2)*x^9+1/8*(256*a^2*c^3*d^3+2048*a*c^6*d+64*a*c^2*d*(8*a*c*d^2+16*c^4))*x^8+1/7*(1792
*a^2*c^4*d^2+64*a*c^3*(8*a*c*d^2+16*c^4))*x^7+512*a^2*c^5*d*x^6+1/5*(32*a^2*c^2*(8*a*c*d^2+16*c^4)+1024*a^2*c^
6)*x^5+256*a^3*c^4*d*x^4+1024/3*a^3*c^5*x^3+256*a^4*c^4*x

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Maxima [A]  time = 1.18094, size = 502, normalized size = 1.86 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + 25
6*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 + 3
2/105*(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 +
65520*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

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Fricas [A]  time = 1.03737, size = 684, normalized size = 2.53 \begin{align*} \frac{1}{17} x^{17} d^{8} + x^{16} d^{7} c + \frac{112}{15} x^{15} d^{6} c^{2} + 32 x^{14} d^{5} c^{3} + \frac{1120}{13} x^{13} d^{4} c^{4} + \frac{16}{13} x^{13} d^{6} c a + \frac{448}{3} x^{12} d^{3} c^{5} + 16 x^{12} d^{5} c^{2} a + \frac{1792}{11} x^{11} d^{2} c^{6} + \frac{960}{11} x^{11} d^{4} c^{3} a + \frac{512}{5} x^{10} d c^{7} + 256 x^{10} d^{3} c^{4} a + \frac{256}{9} x^{9} c^{8} + \frac{1280}{3} x^{9} d^{2} c^{5} a + \frac{32}{3} x^{9} d^{4} c^{2} a^{2} + 384 x^{8} d c^{6} a + 96 x^{8} d^{3} c^{3} a^{2} + \frac{1024}{7} x^{7} c^{7} a + \frac{2304}{7} x^{7} d^{2} c^{4} a^{2} + 512 x^{6} d c^{5} a^{2} + \frac{1536}{5} x^{5} c^{6} a^{2} + \frac{256}{5} x^{5} d^{2} c^{3} a^{3} + 256 x^{4} d c^{4} a^{3} + \frac{1024}{3} x^{3} c^{5} a^{3} + 256 x c^{4} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*x^17*d^8 + x^16*d^7*c + 112/15*x^15*d^6*c^2 + 32*x^14*d^5*c^3 + 1120/13*x^13*d^4*c^4 + 16/13*x^13*d^6*c*a
+ 448/3*x^12*d^3*c^5 + 16*x^12*d^5*c^2*a + 1792/11*x^11*d^2*c^6 + 960/11*x^11*d^4*c^3*a + 512/5*x^10*d*c^7 +
256*x^10*d^3*c^4*a + 256/9*x^9*c^8 + 1280/3*x^9*d^2*c^5*a + 32/3*x^9*d^4*c^2*a^2 + 384*x^8*d*c^6*a + 96*x^8*d^
3*c^3*a^2 + 1024/7*x^7*c^7*a + 2304/7*x^7*d^2*c^4*a^2 + 512*x^6*d*c^5*a^2 + 1536/5*x^5*c^6*a^2 + 256/5*x^5*d^2
*c^3*a^3 + 256*x^4*d*c^4*a^3 + 1024/3*x^3*c^5*a^3 + 256*x*c^4*a^4

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Sympy [A]  time = 0.118807, size = 299, normalized size = 1.11 \begin{align*} 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} \left (\frac{16 a c d^{6}}{13} + \frac{1120 c^{4} d^{4}}{13}\right ) + x^{12} \left (16 a c^{2} d^{5} + \frac{448 c^{5} d^{3}}{3}\right ) + x^{11} \left (\frac{960 a c^{3} d^{4}}{11} + \frac{1792 c^{6} d^{2}}{11}\right ) + x^{10} \left (256 a c^{4} d^{3} + \frac{512 c^{7} d}{5}\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^{8} \left (96 a^{2} c^{3} d^{3} + 384 a c^{6} d\right ) + x^{7} \left (\frac{2304 a^{2} c^{4} d^{2}}{7} + \frac{1024 a c^{7}}{7}\right ) + x^{5} \left (\frac{256 a^{3} c^{3} d^{2}}{5} + \frac{1536 a^{2} c^{6}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + 1
12*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)

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Giac [A]  time = 1.14298, size = 374, normalized size = 1.39 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 +
256*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