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

Optimal. Leaf size=13 \[ b \log (x)-\frac{a}{3 x^3} \]

[Out]

-a/(3*x^3) + b*Log[x]

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Rubi [A]  time = 0.0042974, antiderivative size = 13, 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} \[ b \log (x)-\frac{a}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)/x^4,x]

[Out]

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

Mathematica [A]  time = 0.0021588, size = 13, normalized size = 1. \[ b \log (x)-\frac{a}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)/x^4,x]

[Out]

-a/(3*x^3) + b*Log[x]

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Maple [A]  time = 0.006, size = 12, normalized size = 0.9 \begin{align*} -{\frac{a}{3\,{x}^{3}}}+b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a/x^3+b*ln(x)

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Maxima [A]  time = 0.979164, size = 19, normalized size = 1.46 \begin{align*} \frac{1}{3} \, b \log \left (x^{3}\right ) - \frac{a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*log(x^3) - 1/3*a/x^3

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Fricas [A]  time = 1.85722, size = 41, normalized size = 3.15 \begin{align*} \frac{3 \, b x^{3} \log \left (x\right ) - a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*b*x^3*log(x) - a)/x^3

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Sympy [A]  time = 0.391207, size = 10, normalized size = 0.77 \begin{align*} - \frac{a}{3 x^{3}} + b \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(3*x**3) + b*log(x)

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Giac [A]  time = 1.11113, size = 24, normalized size = 1.85 \begin{align*} b \log \left ({\left | x \right |}\right ) - \frac{b x^{3} + a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(abs(x)) - 1/3*(b*x^3 + a)/x^3