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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0181635, 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. \[ a^3 x+\frac{3}{2} a^2 b x^2+a b^2 x^3+\frac{b^3 x^4}{4} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0000883153, size = 35, normalized size = 1. \[ a^3 x+\frac{3}{2} a^2 b x^2+a b^2 x^3+\frac{b^3 x^4}{4} \]

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

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

Maxima [A]  time = 0.757947, size = 42, normalized size = 1.2 \[ \frac{1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.23837, size = 1, normalized size = 0.03 \[ \frac{1}{4} x^{4} b^{3} + x^{3} b^{2} a + \frac{3}{2} x^{2} b a^{2} + x a^{3} \]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.084519, size = 32, normalized size = 0.91 \[ a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4} \]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.259653, size = 42, normalized size = 1.2 \[ \frac{1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \]

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