3.14 \(\int x^2 (a+b x) (a c-b c x)^4 \, dx\)

Optimal. Leaf size=80 \[ -\frac{2 a^3 c^4 (a-b x)^5}{5 b^3}+\frac{5 a^2 c^4 (a-b x)^6}{6 b^3}+\frac{c^4 (a-b x)^8}{8 b^3}-\frac{4 a c^4 (a-b x)^7}{7 b^3} \]

[Out]

(-2*a^3*c^4*(a - b*x)^5)/(5*b^3) + (5*a^2*c^4*(a - b*x)^6)/(6*b^3) - (4*a*c^4*(a
- b*x)^7)/(7*b^3) + (c^4*(a - b*x)^8)/(8*b^3)

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Rubi [A]  time = 0.130913, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{2 a^3 c^4 (a-b x)^5}{5 b^3}+\frac{5 a^2 c^4 (a-b x)^6}{6 b^3}+\frac{c^4 (a-b x)^8}{8 b^3}-\frac{4 a c^4 (a-b x)^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(-2*a^3*c^4*(a - b*x)^5)/(5*b^3) + (5*a^2*c^4*(a - b*x)^6)/(6*b^3) - (4*a*c^4*(a
- b*x)^7)/(7*b^3) + (c^4*(a - b*x)^8)/(8*b^3)

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Rubi in Sympy [A]  time = 33.5262, size = 85, normalized size = 1.06 \[ \frac{a^{5} c^{4} x^{3}}{3} - \frac{3 a^{4} b c^{4} x^{4}}{4} + \frac{2 a^{3} b^{2} c^{4} x^{5}}{5} + \frac{a^{2} b^{3} c^{4} x^{6}}{3} - \frac{3 a b^{4} c^{4} x^{7}}{7} + \frac{b^{5} c^{4} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)*(-b*c*x+a*c)**4,x)

[Out]

a**5*c**4*x**3/3 - 3*a**4*b*c**4*x**4/4 + 2*a**3*b**2*c**4*x**5/5 + a**2*b**3*c*
*4*x**6/3 - 3*a*b**4*c**4*x**7/7 + b**5*c**4*x**8/8

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Mathematica [A]  time = 0.00470055, size = 87, normalized size = 1.09 \[ \frac{1}{3} a^5 c^4 x^3-\frac{3}{4} a^4 b c^4 x^4+\frac{2}{5} a^3 b^2 c^4 x^5+\frac{1}{3} a^2 b^3 c^4 x^6-\frac{3}{7} a b^4 c^4 x^7+\frac{1}{8} b^5 c^4 x^8 \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(a^5*c^4*x^3)/3 - (3*a^4*b*c^4*x^4)/4 + (2*a^3*b^2*c^4*x^5)/5 + (a^2*b^3*c^4*x^6
)/3 - (3*a*b^4*c^4*x^7)/7 + (b^5*c^4*x^8)/8

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Maple [A]  time = 0.001, size = 76, normalized size = 1. \[{\frac{{b}^{5}{c}^{4}{x}^{8}}{8}}-{\frac{3\,a{b}^{4}{c}^{4}{x}^{7}}{7}}+{\frac{{a}^{2}{c}^{4}{b}^{3}{x}^{6}}{3}}+{\frac{2\,{a}^{3}{c}^{4}{b}^{2}{x}^{5}}{5}}-{\frac{3\,{a}^{4}{c}^{4}b{x}^{4}}{4}}+{\frac{{a}^{5}{c}^{4}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)*(-b*c*x+a*c)^4,x)

[Out]

1/8*b^5*c^4*x^8-3/7*a*b^4*c^4*x^7+1/3*a^2*c^4*b^3*x^6+2/5*a^3*c^4*b^2*x^5-3/4*a^
4*c^4*b*x^4+1/3*a^5*c^4*x^3

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Maxima [A]  time = 1.34347, size = 101, normalized size = 1.26 \[ \frac{1}{8} \, b^{5} c^{4} x^{8} - \frac{3}{7} \, a b^{4} c^{4} x^{7} + \frac{1}{3} \, a^{2} b^{3} c^{4} x^{6} + \frac{2}{5} \, a^{3} b^{2} c^{4} x^{5} - \frac{3}{4} \, a^{4} b c^{4} x^{4} + \frac{1}{3} \, a^{5} c^{4} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^5*c^4*x^8 - 3/7*a*b^4*c^4*x^7 + 1/3*a^2*b^3*c^4*x^6 + 2/5*a^3*b^2*c^4*x^5
- 3/4*a^4*b*c^4*x^4 + 1/3*a^5*c^4*x^3

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Fricas [A]  time = 0.181705, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} c^{4} b^{5} - \frac{3}{7} x^{7} c^{4} b^{4} a + \frac{1}{3} x^{6} c^{4} b^{3} a^{2} + \frac{2}{5} x^{5} c^{4} b^{2} a^{3} - \frac{3}{4} x^{4} c^{4} b a^{4} + \frac{1}{3} x^{3} c^{4} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^8*c^4*b^5 - 3/7*x^7*c^4*b^4*a + 1/3*x^6*c^4*b^3*a^2 + 2/5*x^5*c^4*b^2*a^3
- 3/4*x^4*c^4*b*a^4 + 1/3*x^3*c^4*a^5

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Sympy [A]  time = 0.074648, size = 85, normalized size = 1.06 \[ \frac{a^{5} c^{4} x^{3}}{3} - \frac{3 a^{4} b c^{4} x^{4}}{4} + \frac{2 a^{3} b^{2} c^{4} x^{5}}{5} + \frac{a^{2} b^{3} c^{4} x^{6}}{3} - \frac{3 a b^{4} c^{4} x^{7}}{7} + \frac{b^{5} c^{4} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x+a)*(-b*c*x+a*c)**4,x)

[Out]

a**5*c**4*x**3/3 - 3*a**4*b*c**4*x**4/4 + 2*a**3*b**2*c**4*x**5/5 + a**2*b**3*c*
*4*x**6/3 - 3*a*b**4*c**4*x**7/7 + b**5*c**4*x**8/8

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GIAC/XCAS [A]  time = 0.219836, size = 101, normalized size = 1.26 \[ \frac{1}{8} \, b^{5} c^{4} x^{8} - \frac{3}{7} \, a b^{4} c^{4} x^{7} + \frac{1}{3} \, a^{2} b^{3} c^{4} x^{6} + \frac{2}{5} \, a^{3} b^{2} c^{4} x^{5} - \frac{3}{4} \, a^{4} b c^{4} x^{4} + \frac{1}{3} \, a^{5} c^{4} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^5*c^4*x^8 - 3/7*a*b^4*c^4*x^7 + 1/3*a^2*b^3*c^4*x^6 + 2/5*a^3*b^2*c^4*x^5
- 3/4*a^4*b*c^4*x^4 + 1/3*a^5*c^4*x^3