3.614 \(\int \frac{a+b x^4}{x^5} \, dx\)

Optimal. Leaf size=13 \[ b \log (x)-\frac{a}{4 x^4} \]

[Out]

-a/(4*x^4) + b*Log[x]

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Rubi [A]  time = 0.0046414, 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} \[ b \log (x)-\frac{a}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)/x^5,x]

[Out]

-a/(4*x^4) + 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^4}{x^5} \, dx &=\int \left (\frac{a}{x^5}+\frac{b}{x}\right ) \, dx\\ &=-\frac{a}{4 x^4}+b \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0022041, size = 13, normalized size = 1. \[ b \log (x)-\frac{a}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)/x^5,x]

[Out]

-a/(4*x^4) + b*Log[x]

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Maple [A]  time = 0.006, size = 12, normalized size = 0.9 \begin{align*} -{\frac{a}{4\,{x}^{4}}}+b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)/x^5,x)

[Out]

-1/4*a/x^4+b*ln(x)

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Maxima [A]  time = 0.955545, size = 19, normalized size = 1.46 \begin{align*} \frac{1}{4} \, b \log \left (x^{4}\right ) - \frac{a}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^5,x, algorithm="maxima")

[Out]

1/4*b*log(x^4) - 1/4*a/x^4

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Fricas [A]  time = 1.47198, size = 41, normalized size = 3.15 \begin{align*} \frac{4 \, b x^{4} \log \left (x\right ) - a}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^5,x, algorithm="fricas")

[Out]

1/4*(4*b*x^4*log(x) - a)/x^4

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Sympy [A]  time = 0.264435, size = 10, normalized size = 0.77 \begin{align*} - \frac{a}{4 x^{4}} + b \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)/x**5,x)

[Out]

-a/(4*x**4) + b*log(x)

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Giac [A]  time = 1.09038, size = 27, normalized size = 2.08 \begin{align*} \frac{1}{4} \, b \log \left (x^{4}\right ) - \frac{b x^{4} + a}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^5,x, algorithm="giac")

[Out]

1/4*b*log(x^4) - 1/4*(b*x^4 + a)/x^4