3.612 \(\int \frac{a+b x^4}{x^3} \, dx\)

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

[Out]

-a/(2*x^2) + (b*x^2)/2

________________________________________________________________________________________

Rubi [A]  time = 0.0046726, 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} \[ \frac{b x^2}{2}-\frac{a}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)/x^3,x]

[Out]

-a/(2*x^2) + (b*x^2)/2

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

Mathematica [A]  time = 0.000985, size = 17, normalized size = 1. \[ \frac{b x^2}{2}-\frac{a}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)/x^3,x]

[Out]

-a/(2*x^2) + (b*x^2)/2

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 14, normalized size = 0.8 \begin{align*} -{\frac{a}{2\,{x}^{2}}}+{\frac{b{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)/x^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.983496, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{2} \, b x^{2} - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^3,x, algorithm="maxima")

[Out]

1/2*b*x^2 - 1/2*a/x^2

________________________________________________________________________________________

Fricas [A]  time = 1.37554, size = 28, normalized size = 1.65 \begin{align*} \frac{b x^{4} - a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^3,x, algorithm="fricas")

[Out]

1/2*(b*x^4 - a)/x^2

________________________________________________________________________________________

Sympy [A]  time = 0.229365, size = 12, normalized size = 0.71 \begin{align*} - \frac{a}{2 x^{2}} + \frac{b x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)/x**3,x)

[Out]

-a/(2*x**2) + b*x**2/2

________________________________________________________________________________________

Giac [A]  time = 1.10798, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{2} \, b x^{2} - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^3,x, algorithm="giac")

[Out]

1/2*b*x^2 - 1/2*a/x^2