3.47 \(\int (8+8 x-x^3+8 x^4)^2 \, dx\)

Optimal. Leaf size=54 \[ \frac{64 x^9}{9}-2 x^8+\frac{x^7}{7}+\frac{64 x^6}{3}+\frac{112 x^5}{5}-4 x^4+\frac{64 x^3}{3}+64 x^2+64 x \]

[Out]

64*x + 64*x^2 + (64*x^3)/3 - 4*x^4 + (112*x^5)/5 + (64*x^6)/3 + x^7/7 - 2*x^8 + (64*x^9)/9

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Rubi [A]  time = 0.0165229, antiderivative size = 54, 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} \[ \frac{64 x^9}{9}-2 x^8+\frac{x^7}{7}+\frac{64 x^6}{3}+\frac{112 x^5}{5}-4 x^4+\frac{64 x^3}{3}+64 x^2+64 x \]

Antiderivative was successfully verified.

[In]

Int[(8 + 8*x - x^3 + 8*x^4)^2,x]

[Out]

64*x + 64*x^2 + (64*x^3)/3 - 4*x^4 + (112*x^5)/5 + (64*x^6)/3 + x^7/7 - 2*x^8 + (64*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 (8+8 x-x^3+8 x^4\right )^2 \, dx &=\int \left (64+128 x+64 x^2-16 x^3+112 x^4+128 x^5+x^6-16 x^7+64 x^8\right ) \, dx\\ &=64 x+64 x^2+\frac{64 x^3}{3}-4 x^4+\frac{112 x^5}{5}+\frac{64 x^6}{3}+\frac{x^7}{7}-2 x^8+\frac{64 x^9}{9}\\ \end{align*}

Mathematica [A]  time = 0.0012604, size = 54, normalized size = 1. \[ \frac{64 x^9}{9}-2 x^8+\frac{x^7}{7}+\frac{64 x^6}{3}+\frac{112 x^5}{5}-4 x^4+\frac{64 x^3}{3}+64 x^2+64 x \]

Antiderivative was successfully verified.

[In]

Integrate[(8 + 8*x - x^3 + 8*x^4)^2,x]

[Out]

64*x + 64*x^2 + (64*x^3)/3 - 4*x^4 + (112*x^5)/5 + (64*x^6)/3 + x^7/7 - 2*x^8 + (64*x^9)/9

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Maple [A]  time = 0.001, size = 45, normalized size = 0.8 \begin{align*} 64\,x+64\,{x}^{2}+{\frac{64\,{x}^{3}}{3}}-4\,{x}^{4}+{\frac{112\,{x}^{5}}{5}}+{\frac{64\,{x}^{6}}{3}}+{\frac{{x}^{7}}{7}}-2\,{x}^{8}+{\frac{64\,{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^4-x^3+8*x+8)^2,x)

[Out]

64*x+64*x^2+64/3*x^3-4*x^4+112/5*x^5+64/3*x^6+1/7*x^7-2*x^8+64/9*x^9

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Maxima [A]  time = 1.13361, size = 59, normalized size = 1.09 \begin{align*} \frac{64}{9} \, x^{9} - 2 \, x^{8} + \frac{1}{7} \, x^{7} + \frac{64}{3} \, x^{6} + \frac{112}{5} \, x^{5} - 4 \, x^{4} + \frac{64}{3} \, x^{3} + 64 \, x^{2} + 64 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4-x^3+8*x+8)^2,x, algorithm="maxima")

[Out]

64/9*x^9 - 2*x^8 + 1/7*x^7 + 64/3*x^6 + 112/5*x^5 - 4*x^4 + 64/3*x^3 + 64*x^2 + 64*x

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Fricas [A]  time = 1.06795, size = 116, normalized size = 2.15 \begin{align*} \frac{64}{9} x^{9} - 2 x^{8} + \frac{1}{7} x^{7} + \frac{64}{3} x^{6} + \frac{112}{5} x^{5} - 4 x^{4} + \frac{64}{3} x^{3} + 64 x^{2} + 64 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4-x^3+8*x+8)^2,x, algorithm="fricas")

[Out]

64/9*x^9 - 2*x^8 + 1/7*x^7 + 64/3*x^6 + 112/5*x^5 - 4*x^4 + 64/3*x^3 + 64*x^2 + 64*x

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Sympy [A]  time = 0.058537, size = 49, normalized size = 0.91 \begin{align*} \frac{64 x^{9}}{9} - 2 x^{8} + \frac{x^{7}}{7} + \frac{64 x^{6}}{3} + \frac{112 x^{5}}{5} - 4 x^{4} + \frac{64 x^{3}}{3} + 64 x^{2} + 64 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**4-x**3+8*x+8)**2,x)

[Out]

64*x**9/9 - 2*x**8 + x**7/7 + 64*x**6/3 + 112*x**5/5 - 4*x**4 + 64*x**3/3 + 64*x**2 + 64*x

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Giac [A]  time = 1.10815, size = 59, normalized size = 1.09 \begin{align*} \frac{64}{9} \, x^{9} - 2 \, x^{8} + \frac{1}{7} \, x^{7} + \frac{64}{3} \, x^{6} + \frac{112}{5} \, x^{5} - 4 \, x^{4} + \frac{64}{3} \, x^{3} + 64 \, x^{2} + 64 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4-x^3+8*x+8)^2,x, algorithm="giac")

[Out]

64/9*x^9 - 2*x^8 + 1/7*x^7 + 64/3*x^6 + 112/5*x^5 - 4*x^4 + 64/3*x^3 + 64*x^2 + 64*x