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

Optimal. Leaf size=35 \[ \frac{3}{2} a^2 b x^2+a^3 x+a b^2 x^3+\frac{b^3 x^4}{4} \]

[Out]

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

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Rubi [A]  time = 0.0068023, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \frac{3}{2} a^2 b x^2+a^3 x+a b^2 x^3+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0000398, size = 35, normalized size = 1. \[ \frac{3}{2} a^2 b x^2+a^3 x+a b^2 x^3+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 32, normalized size = 0.9 \begin{align*}{a}^{3}x+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+a{b}^{2}{x}^{3}+{\frac{{b}^{3}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(b^3*x^3+3*a*b^2*x^2+3*a^2*b*x+a^3,x)

[Out]

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

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Maxima [A]  time = 1.09014, size = 42, normalized size = 1.2 \begin{align*} \frac{1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.08271, size = 66, normalized size = 1.89 \begin{align*} \frac{1}{4} x^{4} b^{3} + x^{3} b^{2} a + \frac{3}{2} x^{2} b a^{2} + x a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.061812, size = 32, normalized size = 0.91 \begin{align*} a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b**3*x**3+3*a*b**2*x**2+3*a**2*b*x+a**3,x)

[Out]

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

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Giac [A]  time = 1.08373, size = 42, normalized size = 1.2 \begin{align*} \frac{1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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