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

Optimal. Leaf size=30 \[ \frac{a^2 x^3}{3}+\frac{1}{2} a b x^4+\frac{b^2 x^5}{5} \]

[Out]

(a^2*x^3)/3 + (a*b*x^4)/2 + (b^2*x^5)/5

_______________________________________________________________________________________

Rubi [A]  time = 0.0286132, 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 \[ \frac{a^2 x^3}{3}+\frac{1}{2} a b x^4+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)^2,x]

[Out]

(a^2*x^3)/3 + (a*b*x^4)/2 + (b^2*x^5)/5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.01251, size = 24, normalized size = 0.8 \[ \frac{a^{2} x^{3}}{3} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**3/3 + a*b*x**4/2 + b**2*x**5/5

_______________________________________________________________________________________

Mathematica [A]  time = 0.002213, size = 30, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{1}{2} a b x^4+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x)^2,x]

[Out]

(a^2*x^3)/3 + (a*b*x^4)/2 + (b^2*x^5)/5

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 25, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{3}}{3}}+{\frac{ab{x}^{4}}{2}}+{\frac{{b}^{2}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^2,x)

[Out]

1/3*a^2*x^3+1/2*a*b*x^4+1/5*b^2*x^5

_______________________________________________________________________________________

Maxima [A]  time = 1.32527, size = 32, normalized size = 1.07 \[ \frac{1}{5} \, b^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^2*x^5 + 1/2*a*b*x^4 + 1/3*a^2*x^3

_______________________________________________________________________________________

Fricas [A]  time = 0.186504, size = 1, normalized size = 0.03 \[ \frac{1}{5} x^{5} b^{2} + \frac{1}{2} x^{4} b a + \frac{1}{3} x^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*x^5*b^2 + 1/2*x^4*b*a + 1/3*x^3*a^2

_______________________________________________________________________________________

Sympy [A]  time = 0.081426, size = 24, normalized size = 0.8 \[ \frac{a^{2} x^{3}}{3} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x+a)**2,x)

[Out]

a**2*x**3/3 + a*b*x**4/2 + b**2*x**5/5

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213879, size = 32, normalized size = 1.07 \[ \frac{1}{5} \, b^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^2*x^5 + 1/2*a*b*x^4 + 1/3*a^2*x^3