3.2 \(\int x (a+b x) (a c-b c x)^3 \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{2} a^4 c^3 x^2-\frac{2}{3} a^3 b c^3 x^3+\frac{2}{5} a b^3 c^3 x^5-\frac{1}{6} b^4 c^3 x^6 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0799337, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{1}{2} a^4 c^3 x^2-\frac{2}{3} a^3 b c^3 x^3+\frac{2}{5} a b^3 c^3 x^5-\frac{1}{6} b^4 c^3 x^6 \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{4} c^{3} \int x\, dx - \frac{2 a^{3} b c^{3} x^{3}}{3} + \frac{2 a b^{3} c^{3} x^{5}}{5} - \frac{b^{4} c^{3} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)*(-b*c*x+a*c)**3,x)

[Out]

a**4*c**3*Integral(x, x) - 2*a**3*b*c**3*x**3/3 + 2*a*b**3*c**3*x**5/5 - b**4*c*
*3*x**6/6

_______________________________________________________________________________________

Mathematica [A]  time = 0.00362285, size = 47, normalized size = 0.85 \[ c^3 \left (\frac{a^4 x^2}{2}-\frac{2}{3} a^3 b x^3+\frac{2}{5} a b^3 x^5-\frac{1}{6} b^4 x^6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 48, normalized size = 0.9 \[{\frac{{a}^{4}{c}^{3}{x}^{2}}{2}}-{\frac{2\,{a}^{3}b{c}^{3}{x}^{3}}{3}}+{\frac{2\,a{b}^{3}{c}^{3}{x}^{5}}{5}}-{\frac{{b}^{4}{c}^{3}{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)*(-b*c*x+a*c)^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.3479, size = 63, normalized size = 1.15 \[ -\frac{1}{6} \, b^{4} c^{3} x^{6} + \frac{2}{5} \, a b^{3} c^{3} x^{5} - \frac{2}{3} \, a^{3} b c^{3} x^{3} + \frac{1}{2} \, a^{4} c^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)*x,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.179631, size = 1, normalized size = 0.02 \[ -\frac{1}{6} x^{6} c^{3} b^{4} + \frac{2}{5} x^{5} c^{3} b^{3} a - \frac{2}{3} x^{3} c^{3} b a^{3} + \frac{1}{2} x^{2} c^{3} a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)*x,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.070649, size = 53, normalized size = 0.96 \[ \frac{a^{4} c^{3} x^{2}}{2} - \frac{2 a^{3} b c^{3} x^{3}}{3} + \frac{2 a b^{3} c^{3} x^{5}}{5} - \frac{b^{4} c^{3} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)*(-b*c*x+a*c)**3,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.244508, size = 63, normalized size = 1.15 \[ -\frac{1}{6} \, b^{4} c^{3} x^{6} + \frac{2}{5} \, a b^{3} c^{3} x^{5} - \frac{2}{3} \, a^{3} b c^{3} x^{3} + \frac{1}{2} \, a^{4} c^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)*x,x, algorithm="giac")

[Out]

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