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

Optimal. Leaf size=12 \[ b x-\frac{a}{2 x^2} \]

[Out]

-a/(2*x^2) + b*x

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Rubi [A]  time = 0.0040152, antiderivative size = 12, 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} \[ b x-\frac{a}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)/x^3,x]

[Out]

-a/(2*x^2) + b*x

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

Mathematica [A]  time = 0.000685, size = 12, normalized size = 1. \[ b x-\frac{a}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)/x^3,x]

[Out]

-a/(2*x^2) + b*x

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Maple [A]  time = 0.003, size = 11, normalized size = 0.9 \begin{align*} -{\frac{a}{2\,{x}^{2}}}+bx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.967883, size = 14, normalized size = 1.17 \begin{align*} b x - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x - 1/2*a/x^2

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Fricas [A]  time = 1.90293, size = 31, normalized size = 2.58 \begin{align*} \frac{2 \, b x^{3} - a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b*x^3 - a)/x^2

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Sympy [A]  time = 0.306722, size = 8, normalized size = 0.67 \begin{align*} - \frac{a}{2 x^{2}} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(2*x**2) + b*x

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Giac [A]  time = 1.14068, size = 14, normalized size = 1.17 \begin{align*} b x - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x - 1/2*a/x^2