3.1807 \(\int (a+\frac{b}{x^2}) x^3 \, dx\)

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

[Out]

(b*x^2)/2 + (a*x^4)/4

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Rubi [A]  time = 0.0045587, 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, Rules used = {14} \[ \frac{a x^4}{4}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)*x^3,x]

[Out]

(b*x^2)/2 + (a*x^4)/4

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x^2}\right ) x^3 \, dx &=\int \left (b x+a x^3\right ) \, dx\\ &=\frac{b x^2}{2}+\frac{a x^4}{4}\\ \end{align*}

Mathematica [A]  time = 0.0009498, 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

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Maple [A]  time = 0., size = 14, normalized size = 0.8 \begin{align*}{\frac{b{x}^{2}}{2}}+{\frac{a{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)*x^3,x)

[Out]

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

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Maxima [A]  time = 0.950609, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{4} \, a x^{4} + \frac{1}{2} \, b x^{2} \end{align*}

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

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Fricas [A]  time = 1.37498, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{4} \, a x^{4} + \frac{1}{2} \, b x^{2} \end{align*}

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

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Sympy [A]  time = 0.054884, size = 12, normalized size = 0.71 \begin{align*} \frac{a x^{4}}{4} + \frac{b x^{2}}{2} \end{align*}

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

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Giac [A]  time = 1.16419, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{4} \, a x^{4} + \frac{1}{2} \, b x^{2} \end{align*}

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