3.118 \(\int \left (a+8 x-8 x^2+4 x^3-x^4\right )^2 \, dx\)

Optimal. Leaf size=72 \[ a^2 x+\frac{2}{5} (64-a) x^5-2 (16-a) x^4+\frac{16}{3} (4-a) x^3+8 a x^2+\frac{x^9}{9}-x^8+\frac{32 x^7}{7}-\frac{40 x^6}{3} \]

[Out]

a^2*x + 8*a*x^2 + (16*(4 - a)*x^3)/3 - 2*(16 - a)*x^4 + (2*(64 - a)*x^5)/5 - (40
*x^6)/3 + (32*x^7)/7 - x^8 + x^9/9

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Rubi [A]  time = 0.0600135, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ a^2 x+\frac{2}{5} (64-a) x^5-2 (16-a) x^4+\frac{16}{3} (4-a) x^3+8 a x^2+\frac{x^9}{9}-x^8+\frac{32 x^7}{7}-\frac{40 x^6}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2,x]

[Out]

a^2*x + 8*a*x^2 + (16*(4 - a)*x^3)/3 - 2*(16 - a)*x^4 + (2*(64 - a)*x^5)/5 - (40
*x^6)/3 + (32*x^7)/7 - x^8 + x^9/9

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**4+4*x**3-8*x**2+a+8*x)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.0140277, size = 66, normalized size = 0.92 \[ a^2 x-\frac{2}{5} (a-64) x^5+2 (a-16) x^4-\frac{16}{3} (a-4) x^3+8 a x^2+\frac{x^9}{9}-x^8+\frac{32 x^7}{7}-\frac{40 x^6}{3} \]

Antiderivative was successfully verified.

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

[Out]

a^2*x + 8*a*x^2 - (16*(-4 + a)*x^3)/3 + 2*(-16 + a)*x^4 - (2*(-64 + a)*x^5)/5 -
(40*x^6)/3 + (32*x^7)/7 - x^8 + x^9/9

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Maple [A]  time = 0.001, size = 63, normalized size = 0.9 \[{\frac{{x}^{9}}{9}}-{x}^{8}+{\frac{32\,{x}^{7}}{7}}-{\frac{40\,{x}^{6}}{3}}+{\frac{ \left ( -2\,a+128 \right ){x}^{5}}{5}}+{\frac{ \left ( 8\,a-128 \right ){x}^{4}}{4}}+{\frac{ \left ( -16\,a+64 \right ){x}^{3}}{3}}+8\,a{x}^{2}+{a}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+4*x^3-8*x^2+a+8*x)^2,x)

[Out]

1/9*x^9-x^8+32/7*x^7-40/3*x^6+1/5*(-2*a+128)*x^5+1/4*(8*a-128)*x^4+1/3*(-16*a+64
)*x^3+8*a*x^2+a^2*x

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Maxima [A]  time = 0.802978, size = 88, normalized size = 1.22 \[ \frac{1}{9} \, x^{9} - x^{8} + \frac{32}{7} \, x^{7} - \frac{40}{3} \, x^{6} + \frac{128}{5} \, x^{5} - 32 \, x^{4} + a^{2} x + \frac{64}{3} \, x^{3} - \frac{2}{15} \,{\left (3 \, x^{5} - 15 \, x^{4} + 40 \, x^{3} - 60 \, x^{2}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*x^9 - x^8 + 32/7*x^7 - 40/3*x^6 + 128/5*x^5 - 32*x^4 + a^2*x + 64/3*x^3 - 2/
15*(3*x^5 - 15*x^4 + 40*x^3 - 60*x^2)*a

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Fricas [A]  time = 0.259592, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} - x^{8} + \frac{32}{7} x^{7} - \frac{40}{3} x^{6} - \frac{2}{5} x^{5} a + \frac{128}{5} x^{5} + 2 x^{4} a - 32 x^{4} - \frac{16}{3} x^{3} a + \frac{64}{3} x^{3} + 8 x^{2} a + x a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*x^9 - x^8 + 32/7*x^7 - 40/3*x^6 - 2/5*x^5*a + 128/5*x^5 + 2*x^4*a - 32*x^4 -
16/3*x^3*a + 64/3*x^3 + 8*x^2*a + x*a^2

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Sympy [A]  time = 0.106636, size = 65, normalized size = 0.9 \[ a^{2} x + 8 a x^{2} + \frac{x^{9}}{9} - x^{8} + \frac{32 x^{7}}{7} - \frac{40 x^{6}}{3} + x^{5} \left (- \frac{2 a}{5} + \frac{128}{5}\right ) + x^{4} \left (2 a - 32\right ) + x^{3} \left (- \frac{16 a}{3} + \frac{64}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**4+4*x**3-8*x**2+a+8*x)**2,x)

[Out]

a**2*x + 8*a*x**2 + x**9/9 - x**8 + 32*x**7/7 - 40*x**6/3 + x**5*(-2*a/5 + 128/5
) + x**4*(2*a - 32) + x**3*(-16*a/3 + 64/3)

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GIAC/XCAS [A]  time = 0.262786, size = 88, normalized size = 1.22 \[ \frac{1}{9} \, x^{9} - x^{8} + \frac{32}{7} \, x^{7} - \frac{2}{5} \, a x^{5} - \frac{40}{3} \, x^{6} + 2 \, a x^{4} + \frac{128}{5} \, x^{5} - \frac{16}{3} \, a x^{3} - 32 \, x^{4} + a^{2} x + 8 \, a x^{2} + \frac{64}{3} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*x^9 - x^8 + 32/7*x^7 - 2/5*a*x^5 - 40/3*x^6 + 2*a*x^4 + 128/5*x^5 - 16/3*a*x
^3 - 32*x^4 + a^2*x + 8*a*x^2 + 64/3*x^3