3.624 \(\int \left (a+b x^4\right )^2 \, dx\)

Optimal. Leaf size=25 \[ a^2 x+\frac{2}{5} a b x^5+\frac{b^2 x^9}{9} \]

[Out]

a^2*x + (2*a*b*x^5)/5 + (b^2*x^9)/9

_______________________________________________________________________________________

Rubi [A]  time = 0.0196626, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ a^2 x+\frac{2}{5} a b x^5+\frac{b^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^2,x]

[Out]

a^2*x + (2*a*b*x^5)/5 + (b^2*x^9)/9

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{9}}{9} + \int a^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**2,x)

[Out]

2*a*b*x**5/5 + b**2*x**9/9 + Integral(a**2, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00145464, size = 25, normalized size = 1. \[ a^2 x+\frac{2}{5} a b x^5+\frac{b^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^2,x]

[Out]

a^2*x + (2*a*b*x^5)/5 + (b^2*x^9)/9

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 22, normalized size = 0.9 \[ x{a}^{2}+{\frac{2\,{x}^{5}ab}{5}}+{\frac{{b}^{2}{x}^{9}}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^2,x)

[Out]

x*a^2+2/5*x^5*a*b+1/9*b^2*x^9

_______________________________________________________________________________________

Maxima [A]  time = 1.4217, size = 28, normalized size = 1.12 \[ \frac{1}{9} \, b^{2} x^{9} + \frac{2}{5} \, a b x^{5} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2,x, algorithm="maxima")

[Out]

1/9*b^2*x^9 + 2/5*a*b*x^5 + a^2*x

_______________________________________________________________________________________

Fricas [A]  time = 0.205749, size = 1, normalized size = 0.04 \[ \frac{1}{9} x^{9} b^{2} + \frac{2}{5} x^{5} b a + x a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2,x, algorithm="fricas")

[Out]

1/9*x^9*b^2 + 2/5*x^5*b*a + x*a^2

_______________________________________________________________________________________

Sympy [A]  time = 0.082973, size = 22, normalized size = 0.88 \[ a^{2} x + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**2,x)

[Out]

a**2*x + 2*a*b*x**5/5 + b**2*x**9/9

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218151, size = 28, normalized size = 1.12 \[ \frac{1}{9} \, b^{2} x^{9} + \frac{2}{5} \, a b x^{5} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2,x, algorithm="giac")

[Out]

1/9*b^2*x^9 + 2/5*a*b*x^5 + a^2*x