\(\int (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4)^3 \, dx\) [34]

Optimal result

Integrand size = 29, antiderivative size = 171 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^3 \, dx=64 a^3 c^3 x+64 a^2 c^4 x^3+48 a^2 c^3 d x^4+\frac {48}{5} a c^2 \left (4 c^3+a d^2\right ) x^5+64 a c^4 d x^6+\frac {32}{7} c^3 \left (2 c^3+9 a d^2\right ) x^7+12 c^2 d \left (2 c^3+a d^2\right ) x^8+\frac {4}{3} c d^2 \left (20 c^3+a d^2\right ) x^9+16 c^3 d^3 x^{10}+\frac {60}{11} c^2 d^4 x^{11}+c d^5 x^{12}+\frac {d^6 x^{13}}{13} \]

[Out]

64*a^3*c^3*x+64*a^2*c^4*x^3+48*a^2*c^3*d*x^4+48/5*a*c^2*(a*d^2+4*c^3)*x^5+64*a*c^4*d*x^6+32/7*c^3*(9*a*d^2+2*c
^3)*x^7+12*c^2*d*(a*d^2+2*c^3)*x^8+4/3*c*d^2*(a*d^2+20*c^3)*x^9+16*c^3*d^3*x^10+60/11*c^2*d^4*x^11+c*d^5*x^12+
1/13*d^6*x^13

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2086} \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^3 \, dx=64 a^3 c^3 x+64 a^2 c^4 x^3+48 a^2 c^3 d x^4+64 a c^4 d x^6+\frac {4}{3} c d^2 x^9 \left (a d^2+20 c^3\right )+\frac {32}{7} c^3 x^7 \left (9 a d^2+2 c^3\right )+12 c^2 d x^8 \left (a d^2+2 c^3\right )+\frac {48}{5} a c^2 x^5 \left (a d^2+4 c^3\right )+16 c^3 d^3 x^{10}+\frac {60}{11} c^2 d^4 x^{11}+c d^5 x^{12}+\frac {d^6 x^{13}}{13} \]

[In]

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

[Out]

64*a^3*c^3*x + 64*a^2*c^4*x^3 + 48*a^2*c^3*d*x^4 + (48*a*c^2*(4*c^3 + a*d^2)*x^5)/5 + 64*a*c^4*d*x^6 + (32*c^3
*(2*c^3 + 9*a*d^2)*x^7)/7 + 12*c^2*d*(2*c^3 + a*d^2)*x^8 + (4*c*d^2*(20*c^3 + a*d^2)*x^9)/3 + 16*c^3*d^3*x^10
+ (60*c^2*d^4*x^11)/11 + c*d^5*x^12 + (d^6*x^13)/13

Rule 2086

Int[(P_)^(p_), x_Symbol] :> Int[ExpandToSum[P^p, x], x] /; PolyQ[P, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (64 a^3 c^3+192 a^2 c^4 x^2+192 a^2 c^3 d x^3+48 a c^2 \left (4 c^3+a d^2\right ) x^4+384 a c^4 d x^5+32 c^3 \left (2 c^3+9 a d^2\right ) x^6+96 c^2 d \left (2 c^3+a d^2\right ) x^7+12 c d^2 \left (20 c^3+a d^2\right ) x^8+160 c^3 d^3 x^9+60 c^2 d^4 x^{10}+12 c d^5 x^{11}+d^6 x^{12}\right ) \, dx \\ & = 64 a^3 c^3 x+64 a^2 c^4 x^3+48 a^2 c^3 d x^4+\frac {48}{5} a c^2 \left (4 c^3+a d^2\right ) x^5+64 a c^4 d x^6+\frac {32}{7} c^3 \left (2 c^3+9 a d^2\right ) x^7+12 c^2 d \left (2 c^3+a d^2\right ) x^8+\frac {4}{3} c d^2 \left (20 c^3+a d^2\right ) x^9+16 c^3 d^3 x^{10}+\frac {60}{11} c^2 d^4 x^{11}+c d^5 x^{12}+\frac {d^6 x^{13}}{13} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^3 \, dx=64 a^3 c^3 x+64 a^2 c^4 x^3+48 a^2 c^3 d x^4+\frac {48}{5} a c^2 \left (4 c^3+a d^2\right ) x^5+64 a c^4 d x^6+\frac {32}{7} c^3 \left (2 c^3+9 a d^2\right ) x^7+12 c^2 d \left (2 c^3+a d^2\right ) x^8+\frac {4}{3} c d^2 \left (20 c^3+a d^2\right ) x^9+16 c^3 d^3 x^{10}+\frac {60}{11} c^2 d^4 x^{11}+c d^5 x^{12}+\frac {d^6 x^{13}}{13} \]

[In]

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

[Out]

64*a^3*c^3*x + 64*a^2*c^4*x^3 + 48*a^2*c^3*d*x^4 + (48*a*c^2*(4*c^3 + a*d^2)*x^5)/5 + 64*a*c^4*d*x^6 + (32*c^3
*(2*c^3 + 9*a*d^2)*x^7)/7 + 12*c^2*d*(2*c^3 + a*d^2)*x^8 + (4*c*d^2*(20*c^3 + a*d^2)*x^9)/3 + 16*c^3*d^3*x^10
+ (60*c^2*d^4*x^11)/11 + c*d^5*x^12 + (d^6*x^13)/13

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.95

method	result	size
norman	\(\frac {d^{6} x^{13}}{13}+c \,d^{5} x^{12}+\frac {60 c^{2} d^{4} x^{11}}{11}+16 c^{3} d^{3} x^{10}+\left (\frac {4}{3} a c \,d^{4}+\frac {80}{3} d^{2} c^{4}\right ) x^{9}+\left (12 a \,c^{2} d^{3}+24 c^{5} d \right ) x^{8}+\left (\frac {288}{7} a \,c^{3} d^{2}+\frac {64}{7} c^{6}\right ) x^{7}+64 a \,c^{4} d \,x^{6}+\left (\frac {48}{5} a^{2} c^{2} d^{2}+\frac {192}{5} a \,c^{5}\right ) x^{5}+48 a^{2} c^{3} d \,x^{4}+64 a^{2} c^{4} x^{3}+64 a^{3} c^{3} x\)	\(163\)
gosper	\(\frac {1}{13} d^{6} x^{13}+c \,d^{5} x^{12}+\frac {60}{11} c^{2} d^{4} x^{11}+16 c^{3} d^{3} x^{10}+\frac {4}{3} x^{9} a c \,d^{4}+\frac {80}{3} x^{9} d^{2} c^{4}+12 a \,c^{2} d^{3} x^{8}+24 c^{5} d \,x^{8}+\frac {288}{7} x^{7} a \,c^{3} d^{2}+\frac {64}{7} x^{7} c^{6}+64 a \,c^{4} d \,x^{6}+\frac {48}{5} x^{5} a^{2} c^{2} d^{2}+\frac {192}{5} x^{5} a \,c^{5}+48 a^{2} c^{3} d \,x^{4}+64 a^{2} c^{4} x^{3}+64 a^{3} c^{3} x\)	\(167\)
risch	\(\frac {1}{13} d^{6} x^{13}+c \,d^{5} x^{12}+\frac {60}{11} c^{2} d^{4} x^{11}+16 c^{3} d^{3} x^{10}+\frac {4}{3} x^{9} a c \,d^{4}+\frac {80}{3} x^{9} d^{2} c^{4}+12 a \,c^{2} d^{3} x^{8}+24 c^{5} d \,x^{8}+\frac {288}{7} x^{7} a \,c^{3} d^{2}+\frac {64}{7} x^{7} c^{6}+64 a \,c^{4} d \,x^{6}+\frac {48}{5} x^{5} a^{2} c^{2} d^{2}+\frac {192}{5} x^{5} a \,c^{5}+48 a^{2} c^{3} d \,x^{4}+64 a^{2} c^{4} x^{3}+64 a^{3} c^{3} x\)	\(167\)
parallelrisch	\(\frac {1}{13} d^{6} x^{13}+c \,d^{5} x^{12}+\frac {60}{11} c^{2} d^{4} x^{11}+16 c^{3} d^{3} x^{10}+\frac {4}{3} x^{9} a c \,d^{4}+\frac {80}{3} x^{9} d^{2} c^{4}+12 a \,c^{2} d^{3} x^{8}+24 c^{5} d \,x^{8}+\frac {288}{7} x^{7} a \,c^{3} d^{2}+\frac {64}{7} x^{7} c^{6}+64 a \,c^{4} d \,x^{6}+\frac {48}{5} x^{5} a^{2} c^{2} d^{2}+\frac {192}{5} x^{5} a \,c^{5}+48 a^{2} c^{3} d \,x^{4}+64 a^{2} c^{4} x^{3}+64 a^{3} c^{3} x\)	\(167\)
default	\(\frac {d^{6} x^{13}}{13}+c \,d^{5} x^{12}+\frac {60 c^{2} d^{4} x^{11}}{11}+16 c^{3} d^{3} x^{10}+\frac {\left (4 a c \,d^{4}+224 d^{2} c^{4}+d^{2} \left (8 d^{2} a c +16 c^{4}\right )\right ) x^{9}}{9}+\frac {\left (64 a \,c^{2} d^{3}+128 c^{5} d +4 c d \left (8 d^{2} a c +16 c^{4}\right )\right ) x^{8}}{8}+\frac {\left (256 a \,c^{3} d^{2}+4 c^{2} \left (8 d^{2} a c +16 c^{4}\right )\right ) x^{7}}{7}+64 a \,c^{4} d \,x^{6}+\frac {\left (4 a c \left (8 d^{2} a c +16 c^{4}\right )+128 a \,c^{5}+16 a^{2} c^{2} d^{2}\right ) x^{5}}{5}+48 a^{2} c^{3} d \,x^{4}+64 a^{2} c^{4} x^{3}+64 a^{3} c^{3} x\)	\(231\)

[In]

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

[Out]

1/13*d^6*x^13+c*d^5*x^12+60/11*c^2*d^4*x^11+16*c^3*d^3*x^10+(4/3*a*c*d^4+80/3*d^2*c^4)*x^9+(12*a*c^2*d^3+24*c^
5*d)*x^8+(288/7*a*c^3*d^2+64/7*c^6)*x^7+64*a*c^4*d*x^6+(48/5*a^2*c^2*d^2+192/5*a*c^5)*x^5+48*a^2*c^3*d*x^4+64*
a^2*c^4*x^3+64*a^3*c^3*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.95 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^3 \, dx=\frac {1}{13} \, d^{6} x^{13} + c d^{5} x^{12} + \frac {60}{11} \, c^{2} d^{4} x^{11} + 16 \, c^{3} d^{3} x^{10} + 64 \, a c^{4} d x^{6} + 48 \, a^{2} c^{3} d x^{4} + \frac {4}{3} \, {\left (20 \, c^{4} d^{2} + a c d^{4}\right )} x^{9} + 64 \, a^{2} c^{4} x^{3} + 12 \, {\left (2 \, c^{5} d + a c^{2} d^{3}\right )} x^{8} + \frac {32}{7} \, {\left (2 \, c^{6} + 9 \, a c^{3} d^{2}\right )} x^{7} + 64 \, a^{3} c^{3} x + \frac {48}{5} \, {\left (4 \, a c^{5} + a^{2} c^{2} d^{2}\right )} x^{5} \]

[In]

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

[Out]

1/13*d^6*x^13 + c*d^5*x^12 + 60/11*c^2*d^4*x^11 + 16*c^3*d^3*x^10 + 64*a*c^4*d*x^6 + 48*a^2*c^3*d*x^4 + 4/3*(2
0*c^4*d^2 + a*c*d^4)*x^9 + 64*a^2*c^4*x^3 + 12*(2*c^5*d + a*c^2*d^3)*x^8 + 32/7*(2*c^6 + 9*a*c^3*d^2)*x^7 + 64
*a^3*c^3*x + 48/5*(4*a*c^5 + a^2*c^2*d^2)*x^5

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.05 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^3 \, dx=64 a^{3} c^{3} x + 64 a^{2} c^{4} x^{3} + 48 a^{2} c^{3} d x^{4} + 64 a c^{4} d x^{6} + 16 c^{3} d^{3} x^{10} + \frac {60 c^{2} d^{4} x^{11}}{11} + c d^{5} x^{12} + \frac {d^{6} x^{13}}{13} + x^{9} \cdot \left (\frac {4 a c d^{4}}{3} + \frac {80 c^{4} d^{2}}{3}\right ) + x^{8} \cdot \left (12 a c^{2} d^{3} + 24 c^{5} d\right ) + x^{7} \cdot \left (\frac {288 a c^{3} d^{2}}{7} + \frac {64 c^{6}}{7}\right ) + x^{5} \cdot \left (\frac {48 a^{2} c^{2} d^{2}}{5} + \frac {192 a c^{5}}{5}\right ) \]

[In]

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

[Out]

64*a**3*c**3*x + 64*a**2*c**4*x**3 + 48*a**2*c**3*d*x**4 + 64*a*c**4*d*x**6 + 16*c**3*d**3*x**10 + 60*c**2*d**
4*x**11/11 + c*d**5*x**12 + d**6*x**13/13 + x**9*(4*a*c*d**4/3 + 80*c**4*d**2/3) + x**8*(12*a*c**2*d**3 + 24*c
**5*d) + x**7*(288*a*c**3*d**2/7 + 64*c**6/7) + x**5*(48*a**2*c**2*d**2/5 + 192*a*c**5/5)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.20 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^3 \, dx=\frac {1}{13} \, d^{6} x^{13} + c d^{5} x^{12} + \frac {48}{11} \, c^{2} d^{4} x^{11} + \frac {32}{5} \, c^{3} d^{3} x^{10} + \frac {64}{7} \, c^{6} x^{7} + 64 \, a^{3} c^{3} x + \frac {16}{5} \, {\left (3 \, d^{2} x^{5} + 15 \, c d x^{4} + 20 \, c^{2} x^{3}\right )} a^{2} c^{2} + \frac {8}{3} \, {\left (2 \, d^{2} x^{9} + 9 \, c d x^{8}\right )} c^{4} + \frac {4}{105} \, {\left (35 \, d^{4} x^{9} + 315 \, c d^{3} x^{8} + 720 \, c^{2} d^{2} x^{7} + 1008 \, c^{4} x^{5} + 120 \, {\left (3 \, d^{2} x^{7} + 14 \, c d x^{6}\right )} c^{2}\right )} a c + \frac {4}{165} \, {\left (45 \, d^{4} x^{11} + 396 \, c d^{3} x^{10} + 880 \, c^{2} d^{2} x^{9}\right )} c^{2} \]

[In]

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

[Out]

1/13*d^6*x^13 + c*d^5*x^12 + 48/11*c^2*d^4*x^11 + 32/5*c^3*d^3*x^10 + 64/7*c^6*x^7 + 64*a^3*c^3*x + 16/5*(3*d^
2*x^5 + 15*c*d*x^4 + 20*c^2*x^3)*a^2*c^2 + 8/3*(2*d^2*x^9 + 9*c*d*x^8)*c^4 + 4/105*(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*c + 4/165*(45*d^4*x^11 + 396*c*d^3*x^1
0 + 880*c^2*d^2*x^9)*c^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.97 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^3 \, dx=\frac {1}{13} \, d^{6} x^{13} + c d^{5} x^{12} + \frac {60}{11} \, c^{2} d^{4} x^{11} + 16 \, c^{3} d^{3} x^{10} + \frac {80}{3} \, c^{4} d^{2} x^{9} + \frac {4}{3} \, a c d^{4} x^{9} + 24 \, c^{5} d x^{8} + 12 \, a c^{2} d^{3} x^{8} + \frac {64}{7} \, c^{6} x^{7} + \frac {288}{7} \, a c^{3} d^{2} x^{7} + 64 \, a c^{4} d x^{6} + \frac {192}{5} \, a c^{5} x^{5} + \frac {48}{5} \, a^{2} c^{2} d^{2} x^{5} + 48 \, a^{2} c^{3} d x^{4} + 64 \, a^{2} c^{4} x^{3} + 64 \, a^{3} c^{3} x \]

[In]

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

[Out]

1/13*d^6*x^13 + c*d^5*x^12 + 60/11*c^2*d^4*x^11 + 16*c^3*d^3*x^10 + 80/3*c^4*d^2*x^9 + 4/3*a*c*d^4*x^9 + 24*c^
5*d*x^8 + 12*a*c^2*d^3*x^8 + 64/7*c^6*x^7 + 288/7*a*c^3*d^2*x^7 + 64*a*c^4*d*x^6 + 192/5*a*c^5*x^5 + 48/5*a^2*
c^2*d^2*x^5 + 48*a^2*c^3*d*x^4 + 64*a^2*c^4*x^3 + 64*a^3*c^3*x

Mupad [B] (verification not implemented)

Time = 10.58 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.94 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^3 \, dx=x^8\,\left (24\,c^5\,d+12\,a\,c^2\,d^3\right )+x^9\,\left (\frac {80\,c^4\,d^2}{3}+\frac {4\,a\,c\,d^4}{3}\right )+\frac {d^6\,x^{13}}{13}+x^7\,\left (\frac {64\,c^6}{7}+\frac {288\,a\,c^3\,d^2}{7}\right )+64\,a^3\,c^3\,x+c\,d^5\,x^{12}+64\,a^2\,c^4\,x^3+16\,c^3\,d^3\,x^{10}+\frac {60\,c^2\,d^4\,x^{11}}{11}+48\,a^2\,c^3\,d\,x^4+\frac {48\,a\,c^2\,x^5\,\left (4\,c^3+a\,d^2\right )}{5}+64\,a\,c^4\,d\,x^6 \]

[In]

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

[Out]

x^8*(24*c^5*d + 12*a*c^2*d^3) + x^9*((80*c^4*d^2)/3 + (4*a*c*d^4)/3) + (d^6*x^13)/13 + x^7*((64*c^6)/7 + (288*
a*c^3*d^2)/7) + 64*a^3*c^3*x + c*d^5*x^12 + 64*a^2*c^4*x^3 + 16*c^3*d^3*x^10 + (60*c^2*d^4*x^11)/11 + 48*a^2*c
^3*d*x^4 + (48*a*c^2*x^5*(a*d^2 + 4*c^3))/5 + 64*a*c^4*d*x^6