3.1805 \(\int \left (a+\frac{b}{x^2}\right ) x^3 \, dx\)

Optimal. Leaf size=17 \[ \frac{a x^4}{4}+\frac{b x^2}{2} \]

[Out]

(b*x^2)/2 + (a*x^4)/4

_______________________________________________________________________________________

Rubi [A]  time = 0.0161627, antiderivative size = 17, 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 \[ \frac{a x^4}{4}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)*x^3,x]

[Out]

(b*x^2)/2 + (a*x^4)/4

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a x^{4}}{4} + b \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**4/4 + b*Integral(x, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00180982, size = 17, normalized size = 1. \[ \frac{a x^4}{4}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)*x^3,x]

[Out]

(b*x^2)/2 + (a*x^4)/4

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 14, normalized size = 0.8 \[{\frac{b{x}^{2}}{2}}+{\frac{a{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*x^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.43386, size = 18, normalized size = 1.06 \[ \frac{1}{4} \, a x^{4} + \frac{1}{2} \, b x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a*x^4 + 1/2*b*x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.222844, size = 18, normalized size = 1.06 \[ \frac{1}{4} \, a x^{4} + \frac{1}{2} \, b x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a*x^4 + 1/2*b*x^2

_______________________________________________________________________________________

Sympy [A]  time = 0.069796, size = 12, normalized size = 0.71 \[ \frac{a x^{4}}{4} + \frac{b x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*x**3,x)

[Out]

a*x**4/4 + b*x**2/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.226542, size = 18, normalized size = 1.06 \[ \frac{1}{4} \, a x^{4} + \frac{1}{2} \, b x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a*x^4 + 1/2*b*x^2