3.40.63 \(\int \frac {-4+4 x^5-\log (x)}{x^2} \, dx\) [3963]

Optimal. Leaf size=18 \[ \frac {5-2 x+x \left (6+x^4\right )+\log (x)}{x} \]

[Out]

(5-2*x+x*(x^4+6)+ln(x))/x

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Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {14, 2341} \begin {gather*} x^4+\frac {5}{x}+\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 + 4*x^5 - Log[x])/x^2,x]

[Out]

5/x + x^4 + Log[x]/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]]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4 \left (-1+x^5\right )}{x^2}-\frac {\log (x)}{x^2}\right ) \, dx\\ &=4 \int \frac {-1+x^5}{x^2} \, dx-\int \frac {\log (x)}{x^2} \, dx\\ &=\frac {1}{x}+\frac {\log (x)}{x}+4 \int \left (-\frac {1}{x^2}+x^3\right ) \, dx\\ &=\frac {5}{x}+x^4+\frac {\log (x)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 0.83 \begin {gather*} \frac {5}{x}+x^4+\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 4*x^5 - Log[x])/x^2,x]

[Out]

5/x + x^4 + Log[x]/x

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Maple [A]
time = 0.07, size = 16, normalized size = 0.89

method	result	size
norman	\(\frac {5+x^{5}+\ln \left (x \right )}{x}\)	\(12\)
default	\(x^{4}+\frac {\ln \left (x \right )}{x}+\frac {5}{x}\)	\(16\)
risch	\(\frac {\ln \left (x \right )}{x}+\frac {x^{5}+5}{x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(x)+4*x^5-4)/x^2,x,method=_RETURNVERBOSE)

[Out]

x^4+ln(x)/x+5/x

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Maxima [A]
time = 0.28, size = 15, normalized size = 0.83 \begin {gather*} x^{4} + \frac {\log \left (x\right )}{x} + \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)+4*x^5-4)/x^2,x, algorithm="maxima")

[Out]

x^4 + log(x)/x + 5/x

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Fricas [A]
time = 0.39, size = 11, normalized size = 0.61 \begin {gather*} \frac {x^{5} + \log \left (x\right ) + 5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)+4*x^5-4)/x^2,x, algorithm="fricas")

[Out]

(x^5 + log(x) + 5)/x

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Sympy [A]
time = 0.04, size = 10, normalized size = 0.56 \begin {gather*} x^{4} + \frac {\log {\left (x \right )}}{x} + \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(x)+4*x**5-4)/x**2,x)

[Out]

x**4 + log(x)/x + 5/x

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Giac [A]
time = 0.42, size = 15, normalized size = 0.83 \begin {gather*} x^{4} + \frac {\log \left (x\right )}{x} + \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)+4*x^5-4)/x^2,x, algorithm="giac")

[Out]

x^4 + log(x)/x + 5/x

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Mupad [B]
time = 2.65, size = 12, normalized size = 0.67 \begin {gather*} \frac {\ln \left (x\right )+5}{x}+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x) - 4*x^5 + 4)/x^2,x)

[Out]

(log(x) + 5)/x + x^4

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