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

Optimal. Leaf size=30 \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0539066, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \int ^{x^{2}} x\, dx}{2} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*Integral(x, (x, x**2))/2 + a*b*x**6/3 + b**2*x**8/8

_______________________________________________________________________________________

Mathematica [A]  time = 0.00134393, size = 30, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0., size = 25, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{ab{x}^{6}}{3}}+{\frac{{b}^{2}{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.32923, size = 32, normalized size = 1.07 \[ \frac{1}{8} \, b^{2} x^{8} + \frac{1}{3} \, a b x^{6} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.18926, size = 1, normalized size = 0.03 \[ \frac{1}{8} x^{8} b^{2} + \frac{1}{3} x^{6} b a + \frac{1}{4} x^{4} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.084781, size = 24, normalized size = 0.8 \[ \frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)**2,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.208328, size = 32, normalized size = 1.07 \[ \frac{1}{8} \, b^{2} x^{8} + \frac{1}{3} \, a b x^{6} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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