3.51 \(\int (1+4 x+4 x^2+4 x^4)^4 \, dx\)

Optimal. Leaf size=97 \[ \frac{256 x^{17}}{17}+\frac{1024 x^{15}}{15}+\frac{512 x^{14}}{7}+\frac{1792 x^{13}}{13}+256 x^{12}+\frac{3328 x^{11}}{11}+384 x^{10}+\frac{4192 x^9}{9}+448 x^8+\frac{2752 x^7}{7}+\frac{992 x^6}{3}+\frac{1136 x^5}{5}+112 x^4+\frac{112 x^3}{3}+8 x^2+x \]

[Out]

x + 8*x^2 + (112*x^3)/3 + 112*x^4 + (1136*x^5)/5 + (992*x^6)/3 + (2752*x^7)/7 + 448*x^8 + (4192*x^9)/9 + 384*x
^10 + (3328*x^11)/11 + 256*x^12 + (1792*x^13)/13 + (512*x^14)/7 + (1024*x^15)/15 + (256*x^17)/17

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Rubi [A]  time = 0.027978, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2061} \[ \frac{256 x^{17}}{17}+\frac{1024 x^{15}}{15}+\frac{512 x^{14}}{7}+\frac{1792 x^{13}}{13}+256 x^{12}+\frac{3328 x^{11}}{11}+384 x^{10}+\frac{4192 x^9}{9}+448 x^8+\frac{2752 x^7}{7}+\frac{992 x^6}{3}+\frac{1136 x^5}{5}+112 x^4+\frac{112 x^3}{3}+8 x^2+x \]

Antiderivative was successfully verified.

[In]

Int[(1 + 4*x + 4*x^2 + 4*x^4)^4,x]

[Out]

x + 8*x^2 + (112*x^3)/3 + 112*x^4 + (1136*x^5)/5 + (992*x^6)/3 + (2752*x^7)/7 + 448*x^8 + (4192*x^9)/9 + 384*x
^10 + (3328*x^11)/11 + 256*x^12 + (1792*x^13)/13 + (512*x^14)/7 + (1024*x^15)/15 + (256*x^17)/17

Rule 2061

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

Rubi steps

\begin{align*} \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx &=\int \left (1+16 x+112 x^2+448 x^3+1136 x^4+1984 x^5+2752 x^6+3584 x^7+4192 x^8+3840 x^9+3328 x^{10}+3072 x^{11}+1792 x^{12}+1024 x^{13}+1024 x^{14}+256 x^{16}\right ) \, dx\\ &=x+8 x^2+\frac{112 x^3}{3}+112 x^4+\frac{1136 x^5}{5}+\frac{992 x^6}{3}+\frac{2752 x^7}{7}+448 x^8+\frac{4192 x^9}{9}+384 x^{10}+\frac{3328 x^{11}}{11}+256 x^{12}+\frac{1792 x^{13}}{13}+\frac{512 x^{14}}{7}+\frac{1024 x^{15}}{15}+\frac{256 x^{17}}{17}\\ \end{align*}

Mathematica [A]  time = 0.0014482, size = 97, normalized size = 1. \[ \frac{256 x^{17}}{17}+\frac{1024 x^{15}}{15}+\frac{512 x^{14}}{7}+\frac{1792 x^{13}}{13}+256 x^{12}+\frac{3328 x^{11}}{11}+384 x^{10}+\frac{4192 x^9}{9}+448 x^8+\frac{2752 x^7}{7}+\frac{992 x^6}{3}+\frac{1136 x^5}{5}+112 x^4+\frac{112 x^3}{3}+8 x^2+x \]

Antiderivative was successfully verified.

[In]

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

[Out]

x + 8*x^2 + (112*x^3)/3 + 112*x^4 + (1136*x^5)/5 + (992*x^6)/3 + (2752*x^7)/7 + 448*x^8 + (4192*x^9)/9 + 384*x
^10 + (3328*x^11)/11 + 256*x^12 + (1792*x^13)/13 + (512*x^14)/7 + (1024*x^15)/15 + (256*x^17)/17

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Maple [A]  time = 0.001, size = 78, normalized size = 0.8 \begin{align*} x+8\,{x}^{2}+{\frac{112\,{x}^{3}}{3}}+112\,{x}^{4}+{\frac{1136\,{x}^{5}}{5}}+{\frac{992\,{x}^{6}}{3}}+{\frac{2752\,{x}^{7}}{7}}+448\,{x}^{8}+{\frac{4192\,{x}^{9}}{9}}+384\,{x}^{10}+{\frac{3328\,{x}^{11}}{11}}+256\,{x}^{12}+{\frac{1792\,{x}^{13}}{13}}+{\frac{512\,{x}^{14}}{7}}+{\frac{1024\,{x}^{15}}{15}}+{\frac{256\,{x}^{17}}{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^4+4*x^2+4*x+1)^4,x)

[Out]

x+8*x^2+112/3*x^3+112*x^4+1136/5*x^5+992/3*x^6+2752/7*x^7+448*x^8+4192/9*x^9+384*x^10+3328/11*x^11+256*x^12+17
92/13*x^13+512/7*x^14+1024/15*x^15+256/17*x^17

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Maxima [A]  time = 1.15065, size = 104, normalized size = 1.07 \begin{align*} \frac{256}{17} \, x^{17} + \frac{1024}{15} \, x^{15} + \frac{512}{7} \, x^{14} + \frac{1792}{13} \, x^{13} + 256 \, x^{12} + \frac{3328}{11} \, x^{11} + 384 \, x^{10} + \frac{4192}{9} \, x^{9} + 448 \, x^{8} + \frac{2752}{7} \, x^{7} + \frac{992}{3} \, x^{6} + \frac{1136}{5} \, x^{5} + 112 \, x^{4} + \frac{112}{3} \, x^{3} + 8 \, x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

256/17*x^17 + 1024/15*x^15 + 512/7*x^14 + 1792/13*x^13 + 256*x^12 + 3328/11*x^11 + 384*x^10 + 4192/9*x^9 + 448
*x^8 + 2752/7*x^7 + 992/3*x^6 + 1136/5*x^5 + 112*x^4 + 112/3*x^3 + 8*x^2 + x

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Fricas [A]  time = 1.11911, size = 254, normalized size = 2.62 \begin{align*} \frac{256}{17} x^{17} + \frac{1024}{15} x^{15} + \frac{512}{7} x^{14} + \frac{1792}{13} x^{13} + 256 x^{12} + \frac{3328}{11} x^{11} + 384 x^{10} + \frac{4192}{9} x^{9} + 448 x^{8} + \frac{2752}{7} x^{7} + \frac{992}{3} x^{6} + \frac{1136}{5} x^{5} + 112 x^{4} + \frac{112}{3} x^{3} + 8 x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

256/17*x^17 + 1024/15*x^15 + 512/7*x^14 + 1792/13*x^13 + 256*x^12 + 3328/11*x^11 + 384*x^10 + 4192/9*x^9 + 448
*x^8 + 2752/7*x^7 + 992/3*x^6 + 1136/5*x^5 + 112*x^4 + 112/3*x^3 + 8*x^2 + x

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Sympy [A]  time = 0.067993, size = 94, normalized size = 0.97 \begin{align*} \frac{256 x^{17}}{17} + \frac{1024 x^{15}}{15} + \frac{512 x^{14}}{7} + \frac{1792 x^{13}}{13} + 256 x^{12} + \frac{3328 x^{11}}{11} + 384 x^{10} + \frac{4192 x^{9}}{9} + 448 x^{8} + \frac{2752 x^{7}}{7} + \frac{992 x^{6}}{3} + \frac{1136 x^{5}}{5} + 112 x^{4} + \frac{112 x^{3}}{3} + 8 x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

256*x**17/17 + 1024*x**15/15 + 512*x**14/7 + 1792*x**13/13 + 256*x**12 + 3328*x**11/11 + 384*x**10 + 4192*x**9
/9 + 448*x**8 + 2752*x**7/7 + 992*x**6/3 + 1136*x**5/5 + 112*x**4 + 112*x**3/3 + 8*x**2 + x

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Giac [A]  time = 1.11347, size = 104, normalized size = 1.07 \begin{align*} \frac{256}{17} \, x^{17} + \frac{1024}{15} \, x^{15} + \frac{512}{7} \, x^{14} + \frac{1792}{13} \, x^{13} + 256 \, x^{12} + \frac{3328}{11} \, x^{11} + 384 \, x^{10} + \frac{4192}{9} \, x^{9} + 448 \, x^{8} + \frac{2752}{7} \, x^{7} + \frac{992}{3} \, x^{6} + \frac{1136}{5} \, x^{5} + 112 \, x^{4} + \frac{112}{3} \, x^{3} + 8 \, x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

256/17*x^17 + 1024/15*x^15 + 512/7*x^14 + 1792/13*x^13 + 256*x^12 + 3328/11*x^11 + 384*x^10 + 4192/9*x^9 + 448
*x^8 + 2752/7*x^7 + 992/3*x^6 + 1136/5*x^5 + 112*x^4 + 112/3*x^3 + 8*x^2 + x