3.35.4 \(\int \frac {-25 e-4 x^4+4 x^4 \log (x)}{20 x^6} \, dx\)

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

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 14, 2304} \begin {gather*} \frac {e}{4 x^5}-\frac {\log (x)}{5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-25*E - 4*x^4 + 4*x^4*Log[x])/(20*x^6),x]

[Out]

E/(4*x^5) - Log[x]/(5*x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[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]]

Rule 2304

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} &=\frac {1}{20} \int \frac {-25 e-4 x^4+4 x^4 \log (x)}{x^6} \, dx\\ &=\frac {1}{20} \int \left (\frac {-25 e-4 x^4}{x^6}+\frac {4 \log (x)}{x^2}\right ) \, dx\\ &=\frac {1}{20} \int \frac {-25 e-4 x^4}{x^6} \, dx+\frac {1}{5} \int \frac {\log (x)}{x^2} \, dx\\ &=-\frac {1}{5 x}-\frac {\log (x)}{5 x}+\frac {1}{20} \int \left (-\frac {25 e}{x^6}-\frac {4}{x^2}\right ) \, dx\\ &=\frac {e}{4 x^5}-\frac {\log (x)}{5 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {e}{4 x^5}-\frac {\log (x)}{5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25*E - 4*x^4 + 4*x^4*Log[x])/(20*x^6),x]

[Out]

E/(4*x^5) - Log[x]/(5*x)

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fricas [A]  time = 0.48, size = 17, normalized size = 0.94 \begin {gather*} -\frac {4 \, x^{4} \log \relax (x) - 5 \, e}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(4*x^4*log(x)-25*exp(1)-4*x^4)/x^6,x, algorithm="fricas")

[Out]

-1/20*(4*x^4*log(x) - 5*e)/x^5

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giac [A]  time = 0.14, size = 17, normalized size = 0.94 \begin {gather*} -\frac {4 \, x^{4} \log \relax (x) - 5 \, e}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(4*x^4*log(x)-25*exp(1)-4*x^4)/x^6,x, algorithm="giac")

[Out]

-1/20*(4*x^4*log(x) - 5*e)/x^5

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maple [A]  time = 0.03, size = 16, normalized size = 0.89

method	result	size
default	\(-\frac {\ln \relax (x )}{5 x}+\frac {{\mathrm e}}{4 x^{5}}\)	\(16\)
risch	\(-\frac {\ln \relax (x )}{5 x}+\frac {{\mathrm e}}{4 x^{5}}\)	\(16\)
norman	\(\frac {-\frac {x^{4} \ln \relax (x )}{5}+\frac {{\mathrm e}}{4}}{x^{5}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/20*(4*x^4*ln(x)-25*exp(1)-4*x^4)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*ln(x)/x+1/4*exp(1)/x^5

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maxima [A]  time = 0.35, size = 15, normalized size = 0.83 \begin {gather*} -\frac {\log \relax (x)}{5 \, x} + \frac {e}{4 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(4*x^4*log(x)-25*exp(1)-4*x^4)/x^6,x, algorithm="maxima")

[Out]

-1/5*log(x)/x + 1/4*e/x^5

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mupad [B]  time = 2.03, size = 17, normalized size = 0.94 \begin {gather*} \frac {5\,\mathrm {e}-4\,x^4\,\ln \relax (x)}{20\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((5*exp(1))/4 - (x^4*log(x))/5 + x^4/5)/x^6,x)

[Out]

(5*exp(1) - 4*x^4*log(x))/(20*x^5)

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sympy [A]  time = 0.11, size = 14, normalized size = 0.78 \begin {gather*} - \frac {\log {\relax (x )}}{5 x} + \frac {e}{4 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(4*x**4*ln(x)-25*exp(1)-4*x**4)/x**6,x)

[Out]

-log(x)/(5*x) + E/(4*x**5)

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