\(\int (1+4 x+4 x^2+4 x^4)^3 \, dx\) [52]

Optimal result

Integrand size = 17, antiderivative size = 69 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^3 \, dx=x+6 x^2+20 x^3+40 x^4+\frac {252 x^5}{5}+48 x^6+\frac {352 x^7}{7}+48 x^8+\frac {80 x^9}{3}+\frac {96 x^{10}}{5}+\frac {192 x^{11}}{11}+\frac {64 x^{13}}{13} \]

[Out]

x+6*x^2+20*x^3+40*x^4+252/5*x^5+48*x^6+352/7*x^7+48*x^8+80/3*x^9+96/5*x^10+192/11*x^11+64/13*x^13

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2086} \[ \int \left (1+4 x+4 x^2+4 x^4\right )^3 \, dx=\frac {64 x^{13}}{13}+\frac {192 x^{11}}{11}+\frac {96 x^{10}}{5}+\frac {80 x^9}{3}+48 x^8+\frac {352 x^7}{7}+48 x^6+\frac {252 x^5}{5}+40 x^4+20 x^3+6 x^2+x \]

[In]

Int[(1 + 4*x + 4*x^2 + 4*x^4)^3,x]

[Out]

x + 6*x^2 + 20*x^3 + 40*x^4 + (252*x^5)/5 + 48*x^6 + (352*x^7)/7 + 48*x^8 + (80*x^9)/3 + (96*x^10)/5 + (192*x^
11)/11 + (64*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 (1+12 x+60 x^2+160 x^3+252 x^4+288 x^5+352 x^6+384 x^7+240 x^8+192 x^9+192 x^{10}+64 x^{12}\right ) \, dx \\ & = x+6 x^2+20 x^3+40 x^4+\frac {252 x^5}{5}+48 x^6+\frac {352 x^7}{7}+48 x^8+\frac {80 x^9}{3}+\frac {96 x^{10}}{5}+\frac {192 x^{11}}{11}+\frac {64 x^{13}}{13} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^3 \, dx=x+6 x^2+20 x^3+40 x^4+\frac {252 x^5}{5}+48 x^6+\frac {352 x^7}{7}+48 x^8+\frac {80 x^9}{3}+\frac {96 x^{10}}{5}+\frac {192 x^{11}}{11}+\frac {64 x^{13}}{13} \]

[In]

Integrate[(1 + 4*x + 4*x^2 + 4*x^4)^3,x]

[Out]

x + 6*x^2 + 20*x^3 + 40*x^4 + (252*x^5)/5 + 48*x^6 + (352*x^7)/7 + 48*x^8 + (80*x^9)/3 + (96*x^10)/5 + (192*x^
11)/11 + (64*x^13)/13

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84

method	result	size
gosper	\(x +6 x^{2}+20 x^{3}+40 x^{4}+\frac {252}{5} x^{5}+48 x^{6}+\frac {352}{7} x^{7}+48 x^{8}+\frac {80}{3} x^{9}+\frac {96}{5} x^{10}+\frac {192}{11} x^{11}+\frac {64}{13} x^{13}\)	\(58\)
default	\(x +6 x^{2}+20 x^{3}+40 x^{4}+\frac {252}{5} x^{5}+48 x^{6}+\frac {352}{7} x^{7}+48 x^{8}+\frac {80}{3} x^{9}+\frac {96}{5} x^{10}+\frac {192}{11} x^{11}+\frac {64}{13} x^{13}\)	\(58\)
norman	\(x +6 x^{2}+20 x^{3}+40 x^{4}+\frac {252}{5} x^{5}+48 x^{6}+\frac {352}{7} x^{7}+48 x^{8}+\frac {80}{3} x^{9}+\frac {96}{5} x^{10}+\frac {192}{11} x^{11}+\frac {64}{13} x^{13}\)	\(58\)
risch	\(x +6 x^{2}+20 x^{3}+40 x^{4}+\frac {252}{5} x^{5}+48 x^{6}+\frac {352}{7} x^{7}+48 x^{8}+\frac {80}{3} x^{9}+\frac {96}{5} x^{10}+\frac {192}{11} x^{11}+\frac {64}{13} x^{13}\)	\(58\)
parallelrisch	\(x +6 x^{2}+20 x^{3}+40 x^{4}+\frac {252}{5} x^{5}+48 x^{6}+\frac {352}{7} x^{7}+48 x^{8}+\frac {80}{3} x^{9}+\frac {96}{5} x^{10}+\frac {192}{11} x^{11}+\frac {64}{13} x^{13}\)	\(58\)

[In]

int((4*x^4+4*x^2+4*x+1)^3,x,method=_RETURNVERBOSE)

[Out]

x+6*x^2+20*x^3+40*x^4+252/5*x^5+48*x^6+352/7*x^7+48*x^8+80/3*x^9+96/5*x^10+192/11*x^11+64/13*x^13

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^3 \, dx=\frac {64}{13} \, x^{13} + \frac {192}{11} \, x^{11} + \frac {96}{5} \, x^{10} + \frac {80}{3} \, x^{9} + 48 \, x^{8} + \frac {352}{7} \, x^{7} + 48 \, x^{6} + \frac {252}{5} \, x^{5} + 40 \, x^{4} + 20 \, x^{3} + 6 \, x^{2} + x \]

[In]

integrate((4*x^4+4*x^2+4*x+1)^3,x, algorithm="fricas")

[Out]

64/13*x^13 + 192/11*x^11 + 96/5*x^10 + 80/3*x^9 + 48*x^8 + 352/7*x^7 + 48*x^6 + 252/5*x^5 + 40*x^4 + 20*x^3 +
6*x^2 + x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^3 \, dx=\frac {64 x^{13}}{13} + \frac {192 x^{11}}{11} + \frac {96 x^{10}}{5} + \frac {80 x^{9}}{3} + 48 x^{8} + \frac {352 x^{7}}{7} + 48 x^{6} + \frac {252 x^{5}}{5} + 40 x^{4} + 20 x^{3} + 6 x^{2} + x \]

[In]

integrate((4*x**4+4*x**2+4*x+1)**3,x)

[Out]

64*x**13/13 + 192*x**11/11 + 96*x**10/5 + 80*x**9/3 + 48*x**8 + 352*x**7/7 + 48*x**6 + 252*x**5/5 + 40*x**4 +
20*x**3 + 6*x**2 + x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^3 \, dx=\frac {64}{13} \, x^{13} + \frac {192}{11} \, x^{11} + \frac {96}{5} \, x^{10} + \frac {80}{3} \, x^{9} + 48 \, x^{8} + \frac {352}{7} \, x^{7} + 48 \, x^{6} + \frac {252}{5} \, x^{5} + 40 \, x^{4} + 20 \, x^{3} + 6 \, x^{2} + x \]

[In]

integrate((4*x^4+4*x^2+4*x+1)^3,x, algorithm="maxima")

[Out]

64/13*x^13 + 192/11*x^11 + 96/5*x^10 + 80/3*x^9 + 48*x^8 + 352/7*x^7 + 48*x^6 + 252/5*x^5 + 40*x^4 + 20*x^3 +
6*x^2 + x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^3 \, dx=\frac {64}{13} \, x^{13} + \frac {192}{11} \, x^{11} + \frac {96}{5} \, x^{10} + \frac {80}{3} \, x^{9} + 48 \, x^{8} + \frac {352}{7} \, x^{7} + 48 \, x^{6} + \frac {252}{5} \, x^{5} + 40 \, x^{4} + 20 \, x^{3} + 6 \, x^{2} + x \]

[In]

integrate((4*x^4+4*x^2+4*x+1)^3,x, algorithm="giac")

[Out]

64/13*x^13 + 192/11*x^11 + 96/5*x^10 + 80/3*x^9 + 48*x^8 + 352/7*x^7 + 48*x^6 + 252/5*x^5 + 40*x^4 + 20*x^3 +
6*x^2 + x

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^3 \, dx=\frac {64\,x^{13}}{13}+\frac {192\,x^{11}}{11}+\frac {96\,x^{10}}{5}+\frac {80\,x^9}{3}+48\,x^8+\frac {352\,x^7}{7}+48\,x^6+\frac {252\,x^5}{5}+40\,x^4+20\,x^3+6\,x^2+x \]

[In]

int((4*x + 4*x^2 + 4*x^4 + 1)^3,x)

[Out]

x + 6*x^2 + 20*x^3 + 40*x^4 + (252*x^5)/5 + 48*x^6 + (352*x^7)/7 + 48*x^8 + (80*x^9)/3 + (96*x^10)/5 + (192*x^
11)/11 + (64*x^13)/13