3.1040 \(\int (a+b x)^2 (a c-b c x) \, dx\)

Optimal. Leaf size=32 \[ \frac{2 a c (a+b x)^3}{3 b}-\frac{c (a+b x)^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0412295, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 a c (a+b x)^3}{3 b}-\frac{c (a+b x)^4}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.49228, size = 26, normalized size = 0.81 \[ \frac{2 a c \left (a + b x\right )^{3}}{3 b} - \frac{c \left (a + b x\right )^{4}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00253299, size = 40, normalized size = 1.25 \[ c \left (a^3 x+\frac{1}{2} a^2 b x^2-\frac{1}{3} a b^2 x^3-\frac{1}{4} b^3 x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 37, normalized size = 1.2 \[ -{\frac{{b}^{3}c{x}^{4}}{4}}-{\frac{a{b}^{2}c{x}^{3}}{3}}+{\frac{{a}^{2}bc{x}^{2}}{2}}+{a}^{3}cx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(-b*c*x+a*c),x)

[Out]

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

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Maxima [A]  time = 1.34597, size = 49, normalized size = 1.53 \[ -\frac{1}{4} \, b^{3} c x^{4} - \frac{1}{3} \, a b^{2} c x^{3} + \frac{1}{2} \, a^{2} b c x^{2} + a^{3} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.19699, size = 1, normalized size = 0.03 \[ -\frac{1}{4} x^{4} c b^{3} - \frac{1}{3} x^{3} c b^{2} a + \frac{1}{2} x^{2} c b a^{2} + x c a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.109278, size = 39, normalized size = 1.22 \[ a^{3} c x + \frac{a^{2} b c x^{2}}{2} - \frac{a b^{2} c x^{3}}{3} - \frac{b^{3} c x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.202634, size = 49, normalized size = 1.53 \[ -\frac{1}{4} \, b^{3} c x^{4} - \frac{1}{3} \, a b^{2} c x^{3} + \frac{1}{2} \, a^{2} b c x^{2} + a^{3} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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