3.11 \(\int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right ) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0170615, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ 3 a b x+\frac{3 b^2 x^2}{2}+b c x^3+\frac{c^2 x^4}{4} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0000655965, size = 32, normalized size = 1. \[ 3 a b x+\frac{3 b^2 x^2}{2}+b c x^3+\frac{c^2 x^4}{4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 29, normalized size = 0.9 \[ 3\,abx+{\frac{3\,{b}^{2}{x}^{2}}{2}}+bc{x}^{3}+{\frac{{c}^{2}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.77087, size = 38, normalized size = 1.19 \[ \frac{1}{4} \, c^{2} x^{4} + b c x^{3} + \frac{3}{2} \, b^{2} x^{2} + 3 \, a b x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.2788, size = 1, normalized size = 0.03 \[ \frac{1}{4} x^{4} c^{2} + x^{3} c b + \frac{3}{2} x^{2} b^{2} + 3 x b a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.090594, size = 31, normalized size = 0.97 \[ 3 a b x + \frac{3 b^{2} x^{2}}{2} + b c x^{3} + \frac{c^{2} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.25942, size = 38, normalized size = 1.19 \[ \frac{1}{4} \, c^{2} x^{4} + b c x^{3} + \frac{3}{2} \, b^{2} x^{2} + 3 \, a b x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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