3.1030 \(\int (a+b x) (a c-b c x) \, dx\)

Optimal. Leaf size=18 \[ a^2 c x-\frac{1}{3} b^2 c x^3 \]

[Out]

a^2*c*x - (b^2*c*x^3)/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0163604, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ a^2 c x-\frac{1}{3} b^2 c x^3 \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(a*c - b*c*x),x]

[Out]

a^2*c*x - (b^2*c*x^3)/3

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*Integral(c, x) - b**2*c*x**3/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.00187478, size = 18, normalized size = 1. \[ c \left (a^2 x-\frac{b^2 x^3}{3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

c*(a^2*x - (b^2*x^3)/3)

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 17, normalized size = 0.9 \[{a}^{2}cx-{\frac{{b}^{2}c{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c),x)

[Out]

a^2*c*x-1/3*b^2*c*x^3

_______________________________________________________________________________________

Maxima [A]  time = 1.34481, size = 22, normalized size = 1.22 \[ -\frac{1}{3} \, b^{2} c x^{3} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*b^2*c*x^3 + a^2*c*x

_______________________________________________________________________________________

Fricas [A]  time = 0.173883, size = 1, normalized size = 0.06 \[ -\frac{1}{3} x^{3} c b^{2} + x c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*x^3*c*b^2 + x*c*a^2

_______________________________________________________________________________________

Sympy [A]  time = 0.078144, size = 15, normalized size = 0.83 \[ a^{2} c x - \frac{b^{2} c x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c*x - b**2*c*x**3/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.202479, size = 22, normalized size = 1.22 \[ -\frac{1}{3} \, b^{2} c x^{3} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*b^2*c*x^3 + a^2*c*x