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

Optimal. Leaf size=30 \[ \frac{a^2 x^4}{4}+\frac{2}{5} a b x^5+\frac{b^2 x^6}{6} \]

[Out]

(a^2*x^4)/4 + (2*a*b*x^5)/5 + (b^2*x^6)/6

_______________________________________________________________________________________

Rubi [A]  time = 0.0335317, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a^2 x^4}{4}+\frac{2}{5} a b x^5+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x)^2,x]

[Out]

(a^2*x^4)/4 + (2*a*b*x^5)/5 + (b^2*x^6)/6

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.31858, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{4}}{4} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**4/4 + 2*a*b*x**5/5 + b**2*x**6/6

_______________________________________________________________________________________

Mathematica [A]  time = 0.00238131, size = 30, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{2}{5} a b x^5+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x)^2,x]

[Out]

(a^2*x^4)/4 + (2*a*b*x^5)/5 + (b^2*x^6)/6

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{2\,ab{x}^{5}}{5}}+{\frac{{b}^{2}{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^2,x)

[Out]

1/4*a^2*x^4+2/5*a*b*x^5+1/6*b^2*x^6

_______________________________________________________________________________________

Maxima [A]  time = 1.33234, size = 32, normalized size = 1.07 \[ \frac{1}{6} \, b^{2} x^{6} + \frac{2}{5} \, a b x^{5} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^2*x^6 + 2/5*a*b*x^5 + 1/4*a^2*x^4

_______________________________________________________________________________________

Fricas [A]  time = 0.173221, size = 1, normalized size = 0.03 \[ \frac{1}{6} x^{6} b^{2} + \frac{2}{5} x^{5} b a + \frac{1}{4} x^{4} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^6*b^2 + 2/5*x^5*b*a + 1/4*x^4*a^2

_______________________________________________________________________________________

Sympy [A]  time = 0.08705, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{4}}{4} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x+a)**2,x)

[Out]

a**2*x**4/4 + 2*a*b*x**5/5 + b**2*x**6/6

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.209657, size = 32, normalized size = 1.07 \[ \frac{1}{6} \, b^{2} x^{6} + \frac{2}{5} \, a b x^{5} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^2*x^6 + 2/5*a*b*x^5 + 1/4*a^2*x^4