3.100.28 \(\int \frac {13122 x^4+e (-160 x^2-6561 x^3+12800 x^4)+2 e \log (x)-2 e \log ^2(x)}{6561 e x^3} \, dx\)

Optimal. Leaf size=28 \[ x \left (-1+\frac {x}{e}\right )+\left (-x+\frac {1}{81} \left (x+\frac {\log (x)}{x}\right )\right )^2 \]

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Rubi [A]  time = 0.08, antiderivative size = 36, normalized size of antiderivative = 1.29, number of steps used = 8, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 14, 2304, 2305} \begin {gather*} \frac {(6561+6400 e) x^2}{6561 e}+\frac {\log ^2(x)}{6561 x^2}-x-\frac {160 \log (x)}{6561} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(13122*x^4 + E*(-160*x^2 - 6561*x^3 + 12800*x^4) + 2*E*Log[x] - 2*E*Log[x]^2)/(6561*E*x^3),x]

[Out]

-x + ((6561 + 6400*E)*x^2)/(6561*E) - (160*Log[x])/6561 + Log[x]^2/(6561*x^2)

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
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {13122 x^4+e \left (-160 x^2-6561 x^3+12800 x^4\right )+2 e \log (x)-2 e \log ^2(x)}{x^3} \, dx}{6561 e}\\ &=\frac {\int \left (\frac {-160 e-6561 e x+2 (6561+6400 e) x^2}{x}+\frac {2 e \log (x)}{x^3}-\frac {2 e \log ^2(x)}{x^3}\right ) \, dx}{6561 e}\\ &=\frac {2 \int \frac {\log (x)}{x^3} \, dx}{6561}-\frac {2 \int \frac {\log ^2(x)}{x^3} \, dx}{6561}+\frac {\int \frac {-160 e-6561 e x+2 (6561+6400 e) x^2}{x} \, dx}{6561 e}\\ &=-\frac {1}{13122 x^2}-\frac {\log (x)}{6561 x^2}+\frac {\log ^2(x)}{6561 x^2}-\frac {2 \int \frac {\log (x)}{x^3} \, dx}{6561}+\frac {\int \left (-6561 e-\frac {160 e}{x}+2 (6561+6400 e) x\right ) \, dx}{6561 e}\\ &=-x+\frac {(6561+6400 e) x^2}{6561 e}-\frac {160 \log (x)}{6561}+\frac {\log ^2(x)}{6561 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 35, normalized size = 1.25 \begin {gather*} \frac {-6561 e x+(6561+6400 e) x^2-160 e \log (x)+\frac {e \log ^2(x)}{x^2}}{6561 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(13122*x^4 + E*(-160*x^2 - 6561*x^3 + 12800*x^4) + 2*E*Log[x] - 2*E*Log[x]^2)/(6561*E*x^3),x]

[Out]

(-6561*E*x + (6561 + 6400*E)*x^2 - 160*E*Log[x] + (E*Log[x]^2)/x^2)/(6561*E)

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fricas [A]  time = 0.69, size = 43, normalized size = 1.54 \begin {gather*} \frac {{\left (6561 \, x^{4} - 160 \, x^{2} e \log \relax (x) + e \log \relax (x)^{2} + {\left (6400 \, x^{4} - 6561 \, x^{3}\right )} e\right )} e^{\left (-1\right )}}{6561 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6561*(-2*exp(1)*log(x)^2+2*exp(1)*log(x)+(12800*x^4-6561*x^3-160*x^2)*exp(1)+13122*x^4)/x^3/exp(1)
,x, algorithm="fricas")

[Out]

1/6561*(6561*x^4 - 160*x^2*e*log(x) + e*log(x)^2 + (6400*x^4 - 6561*x^3)*e)*e^(-1)/x^2

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giac [A]  time = 1.77, size = 43, normalized size = 1.54 \begin {gather*} \frac {{\left (6400 \, x^{4} e + 6561 \, x^{4} - 6561 \, x^{3} e - 160 \, x^{2} e \log \relax (x) + e \log \relax (x)^{2}\right )} e^{\left (-1\right )}}{6561 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6561*(-2*exp(1)*log(x)^2+2*exp(1)*log(x)+(12800*x^4-6561*x^3-160*x^2)*exp(1)+13122*x^4)/x^3/exp(1)
,x, algorithm="giac")

[Out]

1/6561*(6400*x^4*e + 6561*x^4 - 6561*x^3*e - 160*x^2*e*log(x) + e*log(x)^2)*e^(-1)/x^2

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maple [A]  time = 0.06, size = 29, normalized size = 1.04

method	result	size
risch	\(\frac {\ln \relax (x )^{2}}{6561 x^{2}}+\frac {6400 x^{2}}{6561}-x +x^{2} {\mathrm e}^{-1}-\frac {160 \ln \relax (x )}{6561}\)	\(29\)
norman	\(\frac {-\frac {160 x^{2} \ln \relax (x )}{6561}-x^{3}+\frac {\ln \relax (x )^{2}}{6561}+\frac {\left (6400 \,{\mathrm e}+6561\right ) {\mathrm e}^{-1} x^{4}}{6561}}{x^{2}}\)	\(39\)
default	\(\frac {{\mathrm e}^{-1} \left (6400 x^{2} {\mathrm e}-6561 x \,{\mathrm e}+6561 x^{2}-2 \,{\mathrm e} \left (-\frac {\ln \relax (x )^{2}}{2 x^{2}}-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )-160 \,{\mathrm e} \ln \relax (x )+2 \,{\mathrm e} \left (-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )\right )}{6561}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/6561*(-2*exp(1)*ln(x)^2+2*exp(1)*ln(x)+(12800*x^4-6561*x^3-160*x^2)*exp(1)+13122*x^4)/x^3/exp(1),x,metho
d=_RETURNVERBOSE)

[Out]

1/6561*ln(x)^2/x^2+6400/6561*x^2-x+x^2*exp(-1)-160/6561*ln(x)

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maxima [B]  time = 0.35, size = 61, normalized size = 2.18 \begin {gather*} \frac {1}{13122} \, {\left (12800 \, x^{2} e + 13122 \, x^{2} - 13122 \, x e - {\left (\frac {2 \, \log \relax (x)}{x^{2}} + \frac {1}{x^{2}}\right )} e - 320 \, e \log \relax (x) + \frac {{\left (2 \, \log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )} e}{x^{2}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6561*(-2*exp(1)*log(x)^2+2*exp(1)*log(x)+(12800*x^4-6561*x^3-160*x^2)*exp(1)+13122*x^4)/x^3/exp(1)
,x, algorithm="maxima")

[Out]

1/13122*(12800*x^2*e + 13122*x^2 - 13122*x*e - (2*log(x)/x^2 + 1/x^2)*e - 320*e*log(x) + (2*log(x)^2 + 2*log(x
) + 1)*e/x^2)*e^(-1)

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mupad [B]  time = 7.45, size = 25, normalized size = 0.89 \begin {gather*} \frac {{\ln \relax (x)}^2}{6561\,x^2}-\frac {160\,\ln \relax (x)}{6561}-x+x^2\,\left ({\mathrm {e}}^{-1}+\frac {6400}{6561}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-1)*((2*exp(1)*log(x)^2)/6561 - (2*exp(1)*log(x))/6561 + (exp(1)*(160*x^2 + 6561*x^3 - 12800*x^4))/6
561 - 2*x^4))/x^3,x)

[Out]

log(x)^2/(6561*x^2) - (160*log(x))/6561 - x + x^2*(exp(-1) + 6400/6561)

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sympy [A]  time = 0.17, size = 39, normalized size = 1.39 \begin {gather*} \frac {x^{2} \left (6561 + 6400 e\right ) - 6561 e x - 160 e \log {\relax (x )}}{6561 e} + \frac {\log {\relax (x )}^{2}}{6561 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6561*(-2*exp(1)*ln(x)**2+2*exp(1)*ln(x)+(12800*x**4-6561*x**3-160*x**2)*exp(1)+13122*x**4)/x**3/ex
p(1),x)

[Out]

(x**2*(6561 + 6400*E) - 6561*E*x - 160*E*log(x))*exp(-1)/6561 + log(x)**2/(6561*x**2)

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