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

Optimal. Leaf size=45 \[ \frac{16 x^9}{9}+\frac{32 x^7}{7}+\frac{16 x^6}{3}+\frac{24 x^5}{5}+8 x^4+8 x^3+4 x^2+x \]

[Out]

x + 4*x^2 + 8*x^3 + 8*x^4 + (24*x^5)/5 + (16*x^6)/3 + (32*x^7)/7 + (16*x^9)/9

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Rubi [A]  time = 0.0148994, antiderivative size = 45, 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} \[ \frac{16 x^9}{9}+\frac{32 x^7}{7}+\frac{16 x^6}{3}+\frac{24 x^5}{5}+8 x^4+8 x^3+4 x^2+x \]

Antiderivative was successfully verified.

[In]

Int[(1 + 4*x + 4*x^2 + 4*x^4)^2,x]

[Out]

x + 4*x^2 + 8*x^3 + 8*x^4 + (24*x^5)/5 + (16*x^6)/3 + (32*x^7)/7 + (16*x^9)/9

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 )^2 \, dx &=\int \left (1+8 x+24 x^2+32 x^3+24 x^4+32 x^5+32 x^6+16 x^8\right ) \, dx\\ &=x+4 x^2+8 x^3+8 x^4+\frac{24 x^5}{5}+\frac{16 x^6}{3}+\frac{32 x^7}{7}+\frac{16 x^9}{9}\\ \end{align*}

Mathematica [A]  time = 0.001273, size = 45, normalized size = 1. \[ \frac{16 x^9}{9}+\frac{32 x^7}{7}+\frac{16 x^6}{3}+\frac{24 x^5}{5}+8 x^4+8 x^3+4 x^2+x \]

Antiderivative was successfully verified.

[In]

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

[Out]

x + 4*x^2 + 8*x^3 + 8*x^4 + (24*x^5)/5 + (16*x^6)/3 + (32*x^7)/7 + (16*x^9)/9

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Maple [A]  time = 0.002, size = 38, normalized size = 0.8 \begin{align*} x+4\,{x}^{2}+8\,{x}^{3}+8\,{x}^{4}+{\frac{24\,{x}^{5}}{5}}+{\frac{16\,{x}^{6}}{3}}+{\frac{32\,{x}^{7}}{7}}+{\frac{16\,{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+4*x^2+8*x^3+8*x^4+24/5*x^5+16/3*x^6+32/7*x^7+16/9*x^9

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Maxima [A]  time = 1.78079, size = 50, normalized size = 1.11 \begin{align*} \frac{16}{9} \, x^{9} + \frac{32}{7} \, x^{7} + \frac{16}{3} \, x^{6} + \frac{24}{5} \, x^{5} + 8 \, x^{4} + 8 \, x^{3} + 4 \, 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)^2,x, algorithm="maxima")

[Out]

16/9*x^9 + 32/7*x^7 + 16/3*x^6 + 24/5*x^5 + 8*x^4 + 8*x^3 + 4*x^2 + x

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Fricas [A]  time = 1.0902, size = 96, normalized size = 2.13 \begin{align*} \frac{16}{9} x^{9} + \frac{32}{7} x^{7} + \frac{16}{3} x^{6} + \frac{24}{5} x^{5} + 8 x^{4} + 8 x^{3} + 4 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)^2,x, algorithm="fricas")

[Out]

16/9*x^9 + 32/7*x^7 + 16/3*x^6 + 24/5*x^5 + 8*x^4 + 8*x^3 + 4*x^2 + x

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Sympy [A]  time = 0.058814, size = 42, normalized size = 0.93 \begin{align*} \frac{16 x^{9}}{9} + \frac{32 x^{7}}{7} + \frac{16 x^{6}}{3} + \frac{24 x^{5}}{5} + 8 x^{4} + 8 x^{3} + 4 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)**2,x)

[Out]

16*x**9/9 + 32*x**7/7 + 16*x**6/3 + 24*x**5/5 + 8*x**4 + 8*x**3 + 4*x**2 + x

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Giac [A]  time = 1.12127, size = 50, normalized size = 1.11 \begin{align*} \frac{16}{9} \, x^{9} + \frac{32}{7} \, x^{7} + \frac{16}{3} \, x^{6} + \frac{24}{5} \, x^{5} + 8 \, x^{4} + 8 \, x^{3} + 4 \, 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)^2,x, algorithm="giac")

[Out]

16/9*x^9 + 32/7*x^7 + 16/3*x^6 + 24/5*x^5 + 8*x^4 + 8*x^3 + 4*x^2 + x