3.25.3 \(\int \frac {24+194 x+196 x^2-e x^2+(144+168 x) \log (x)}{2 e x} \, dx\)

Optimal. Leaf size=28 \[ -\frac {x^2}{4}+\frac {-4-x+(1+x+6 (x+\log (x)))^2}{e} \]

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Rubi [A]  time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.64, number of steps used = 9, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6, 12, 14, 2346, 2301, 2295} \begin {gather*} \frac {(196-e) x^2}{4 e}+\frac {13 x}{e}+\frac {36 \log ^2(x)}{e}+\frac {84 x \log (x)}{e}+\frac {12 \log (x)}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(24 + 194*x + 196*x^2 - E*x^2 + (144 + 168*x)*Log[x])/(2*E*x),x]

[Out]

(13*x)/E + ((196 - E)*x^2)/(4*E) + (12*Log[x])/E + (84*x*Log[x])/E + (36*Log[x]^2)/E

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2346

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[((d
+ e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {24+194 x+(196-e) x^2+(144+168 x) \log (x)}{2 e x} \, dx\\ &=\frac {\int \frac {24+194 x+(196-e) x^2+(144+168 x) \log (x)}{x} \, dx}{2 e}\\ &=\frac {\int \left (\frac {24+194 x+(196-e) x^2}{x}+\frac {24 (6+7 x) \log (x)}{x}\right ) \, dx}{2 e}\\ &=\frac {\int \frac {24+194 x+(196-e) x^2}{x} \, dx}{2 e}+\frac {12 \int \frac {(6+7 x) \log (x)}{x} \, dx}{e}\\ &=\frac {\int \left (194+\frac {24}{x}+(196-e) x\right ) \, dx}{2 e}+\frac {72 \int \frac {\log (x)}{x} \, dx}{e}+\frac {84 \int \log (x) \, dx}{e}\\ &=\frac {13 x}{e}+\frac {(196-e) x^2}{4 e}+\frac {12 \log (x)}{e}+\frac {84 x \log (x)}{e}+\frac {36 \log ^2(x)}{e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 38, normalized size = 1.36 \begin {gather*} \frac {26 x+\frac {1}{2} (196-e) x^2+168 x \log (x)+2 (1+6 \log (x))^2}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24 + 194*x + 196*x^2 - E*x^2 + (144 + 168*x)*Log[x])/(2*E*x),x]

[Out]

(26*x + ((196 - E)*x^2)/2 + 168*x*Log[x] + 2*(1 + 6*Log[x])^2)/(2*E)

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fricas [A]  time = 0.77, size = 34, normalized size = 1.21 \begin {gather*} -\frac {1}{4} \, {\left (x^{2} e - 196 \, x^{2} - 48 \, {\left (7 \, x + 1\right )} \log \relax (x) - 144 \, \log \relax (x)^{2} - 52 \, x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((168*x+144)*log(x)-x^2*exp(1)+196*x^2+194*x+24)/x/exp(1),x, algorithm="fricas")

[Out]

-1/4*(x^2*e - 196*x^2 - 48*(7*x + 1)*log(x) - 144*log(x)^2 - 52*x)*e^(-1)

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giac [A]  time = 0.19, size = 34, normalized size = 1.21 \begin {gather*} -\frac {1}{4} \, {\left (x^{2} e - 196 \, x^{2} - 336 \, x \log \relax (x) - 144 \, \log \relax (x)^{2} - 52 \, x - 48 \, \log \relax (x)\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((168*x+144)*log(x)-x^2*exp(1)+196*x^2+194*x+24)/x/exp(1),x, algorithm="giac")

[Out]

-1/4*(x^2*e - 196*x^2 - 336*x*log(x) - 144*log(x)^2 - 52*x - 48*log(x))*e^(-1)

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maple [A]  time = 0.09, size = 38, normalized size = 1.36

method	result	size
default	\(\frac {{\mathrm e}^{-1} \left (-\frac {x^{2} {\mathrm e}}{2}+168 x \ln \relax (x )+26 x +98 x^{2}+72 \ln \relax (x )^{2}+24 \ln \relax (x )\right )}{2}\)	\(38\)
risch	\(-\frac {x^{2}}{4}+84 x \,{\mathrm e}^{-1} \ln \relax (x )+13 \,{\mathrm e}^{-1} x +49 x^{2} {\mathrm e}^{-1}+36 \,{\mathrm e}^{-1} \ln \relax (x )^{2}+12 \ln \relax (x ) {\mathrm e}^{-1}\)	\(40\)
norman	\(12 \ln \relax (x ) {\mathrm e}^{-1}+13 \,{\mathrm e}^{-1} x +36 \,{\mathrm e}^{-1} \ln \relax (x )^{2}+84 x \,{\mathrm e}^{-1} \ln \relax (x )-\frac {{\mathrm e}^{-1} \left ({\mathrm e}-196\right ) x^{2}}{4}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((168*x+144)*ln(x)-x^2*exp(1)+196*x^2+194*x+24)/x/exp(1),x,method=_RETURNVERBOSE)

[Out]

1/2/exp(1)*(-1/2*x^2*exp(1)+168*x*ln(x)+26*x+98*x^2+72*ln(x)^2+24*ln(x))

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maxima [A]  time = 0.49, size = 34, normalized size = 1.21 \begin {gather*} -\frac {1}{4} \, {\left (x^{2} e - 196 \, x^{2} - 336 \, x \log \relax (x) - 144 \, \log \relax (x)^{2} - 52 \, x - 48 \, \log \relax (x)\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((168*x+144)*log(x)-x^2*exp(1)+196*x^2+194*x+24)/x/exp(1),x, algorithm="maxima")

[Out]

-1/4*(x^2*e - 196*x^2 - 336*x*log(x) - 144*log(x)^2 - 52*x - 48*log(x))*e^(-1)

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mupad [B]  time = 1.47, size = 35, normalized size = 1.25 \begin {gather*} \frac {{\mathrm {e}}^{-1}\,\left (52\,x+48\,\ln \relax (x)+144\,{\ln \relax (x)}^2-x^2\,\mathrm {e}+336\,x\,\ln \relax (x)+196\,x^2\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-1)*(97*x + (log(x)*(168*x + 144))/2 - (x^2*exp(1))/2 + 98*x^2 + 12))/x,x)

[Out]

(exp(-1)*(52*x + 48*log(x) + 144*log(x)^2 - x^2*exp(1) + 336*x*log(x) + 196*x^2))/4

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sympy [A]  time = 0.16, size = 42, normalized size = 1.50 \begin {gather*} \frac {84 x \log {\relax (x )}}{e} + \frac {x^{2} \left (98 - \frac {e}{2}\right ) + 26 x + 24 \log {\relax (x )}}{2 e} + \frac {36 \log {\relax (x )}^{2}}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((168*x+144)*ln(x)-x**2*exp(1)+196*x**2+194*x+24)/x/exp(1),x)

[Out]

84*x*exp(-1)*log(x) + (x**2*(98 - E/2) + 26*x + 24*log(x))*exp(-1)/2 + 36*exp(-1)*log(x)**2

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