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

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 = 0.0058015, 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., Rules used = {} \[ 3 a b x+\frac{3 b^2 x^2}{2}+b c x^3+\frac{c^2 x^4}{4} \]

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

Rubi steps

\begin{align*} \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right ) \, dx &=3 a b x+\frac{3 b^2 x^2}{2}+b c x^3+\frac{c^2 x^4}{4}\\ \end{align*}

Mathematica [A]  time = 0.0000507, 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., size = 29, normalized size = 0.9 \begin{align*} 3\,abx+{\frac{3\,{b}^{2}{x}^{2}}{2}}+bc{x}^{3}+{\frac{{c}^{2}{x}^{4}}{4}} \end{align*}

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 = 1.14607, size = 38, normalized size = 1.19 \begin{align*} \frac{1}{4} \, c^{2} x^{4} + b c x^{3} + \frac{3}{2} \, b^{2} x^{2} + 3 \, a b x \end{align*}

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 = 1.15206, size = 63, normalized size = 1.97 \begin{align*} \frac{1}{4} x^{4} c^{2} + x^{3} c b + \frac{3}{2} x^{2} b^{2} + 3 x b a \end{align*}

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 = 0.060498, size = 31, normalized size = 0.97 \begin{align*} 3 a b x + \frac{3 b^{2} x^{2}}{2} + b c x^{3} + \frac{c^{2} x^{4}}{4} \end{align*}

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 [A]  time = 1.06694, size = 38, normalized size = 1.19 \begin{align*} \frac{1}{4} \, c^{2} x^{4} + b c x^{3} + \frac{3}{2} \, b^{2} x^{2} + 3 \, a b x \end{align*}

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