3.1 \(\int x^2 (a+b x) (a c-b c x)^3 \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{3} a^4 c^3 x^3-\frac{1}{2} a^3 b c^3 x^4+\frac{1}{3} a b^3 c^3 x^6-\frac{1}{7} b^4 c^3 x^7 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0969724, antiderivative size = 55, 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 \[ \frac{1}{3} a^4 c^3 x^3-\frac{1}{2} a^3 b c^3 x^4+\frac{1}{3} a b^3 c^3 x^6-\frac{1}{7} b^4 c^3 x^7 \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.3337, size = 49, normalized size = 0.89 \[ \frac{a^{4} c^{3} x^{3}}{3} - \frac{a^{3} b c^{3} x^{4}}{2} + \frac{a b^{3} c^{3} x^{6}}{3} - \frac{b^{4} c^{3} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)*(-b*c*x+a*c)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00588609, size = 47, normalized size = 0.85 \[ c^3 \left (\frac{a^4 x^3}{3}-\frac{1}{2} a^3 b x^4+\frac{1}{3} a b^3 x^6-\frac{1}{7} b^4 x^7\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 48, normalized size = 0.9 \[{\frac{{a}^{4}{c}^{3}{x}^{3}}{3}}-{\frac{{a}^{3}b{c}^{3}{x}^{4}}{2}}+{\frac{a{b}^{3}{c}^{3}{x}^{6}}{3}}-{\frac{{b}^{4}{c}^{3}{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)*(-b*c*x+a*c)^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.35392, size = 63, normalized size = 1.15 \[ -\frac{1}{7} \, b^{4} c^{3} x^{7} + \frac{1}{3} \, a b^{3} c^{3} x^{6} - \frac{1}{2} \, a^{3} b c^{3} x^{4} + \frac{1}{3} \, a^{4} c^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.182293, size = 1, normalized size = 0.02 \[ -\frac{1}{7} x^{7} c^{3} b^{4} + \frac{1}{3} x^{6} c^{3} b^{3} a - \frac{1}{2} x^{4} c^{3} b a^{3} + \frac{1}{3} x^{3} c^{3} a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.063702, size = 49, normalized size = 0.89 \[ \frac{a^{4} c^{3} x^{3}}{3} - \frac{a^{3} b c^{3} x^{4}}{2} + \frac{a b^{3} c^{3} x^{6}}{3} - \frac{b^{4} c^{3} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x+a)*(-b*c*x+a*c)**3,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.324206, size = 63, normalized size = 1.15 \[ -\frac{1}{7} \, b^{4} c^{3} x^{7} + \frac{1}{3} \, a b^{3} c^{3} x^{6} - \frac{1}{2} \, a^{3} b c^{3} x^{4} + \frac{1}{3} \, a^{4} c^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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