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

Optimal. Leaf size=16 \[ \frac{\left (a+b x^2\right )^4}{8 b} \]

[Out]

(a + b*x^2)^4/(8*b)

_______________________________________________________________________________________

Rubi [A]  time = 0.012089, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a+b x^2\right )^4}{8 b} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^2)^3,x]

[Out]

(a + b*x^2)^4/(8*b)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.24312, size = 10, normalized size = 0.62 \[ \frac{\left (a + b x^{2}\right )^{4}}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**2)**4/(8*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00356205, size = 16, normalized size = 1. \[ \frac{\left (a+b x^2\right )^4}{8 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x^2)^3,x]

[Out]

(a + b*x^2)^4/(8*b)

_______________________________________________________________________________________

Maple [B]  time = 0.001, size = 36, normalized size = 2.3 \[{\frac{{b}^{3}{x}^{8}}{8}}+{\frac{a{b}^{2}{x}^{6}}{2}}+{\frac{3\,{a}^{2}b{x}^{4}}{4}}+{\frac{{a}^{3}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2+a)^3,x)

[Out]

1/8*b^3*x^8+1/2*a*b^2*x^6+3/4*a^2*b*x^4+1/2*a^3*x^2

_______________________________________________________________________________________

Maxima [A]  time = 1.3496, size = 19, normalized size = 1.19 \[ \frac{{\left (b x^{2} + a\right )}^{4}}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b*x^2 + a)^4/b

_______________________________________________________________________________________

Fricas [A]  time = 0.179606, size = 1, normalized size = 0.06 \[ \frac{1}{8} x^{8} b^{3} + \frac{1}{2} x^{6} b^{2} a + \frac{3}{4} x^{4} b a^{2} + \frac{1}{2} x^{2} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^8*b^3 + 1/2*x^6*b^2*a + 3/4*x^4*b*a^2 + 1/2*x^2*a^3

_______________________________________________________________________________________

Sympy [A]  time = 0.098184, size = 37, normalized size = 2.31 \[ \frac{a^{3} x^{2}}{2} + \frac{3 a^{2} b x^{4}}{4} + \frac{a b^{2} x^{6}}{2} + \frac{b^{3} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x**2+a)**3,x)

[Out]

a**3*x**2/2 + 3*a**2*b*x**4/4 + a*b**2*x**6/2 + b**3*x**8/8

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207988, size = 19, normalized size = 1.19 \[ \frac{{\left (b x^{2} + a\right )}^{4}}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b*x^2 + a)^4/b