3.89.32 \(\int \frac {-e^3 x+2 x^2-12 \log (5)-72 \log (x)}{x} \, dx\)

Optimal. Leaf size=24 \[ \frac {23}{3}-e^3 x+x^2-(\log (5)+6 \log (x))^2 \]

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Rubi [A]  time = 0.03, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {14, 2301} \begin {gather*} x^2-e^3 x-36 \log ^2(x)-12 \log (5) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-(E^3*x) + 2*x^2 - 12*Log[5] - 72*Log[x])/x,x]

[Out]

-(E^3*x) + x^2 - 12*Log[5]*Log[x] - 36*Log[x]^2

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 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-e^3 x+2 x^2-12 \log (5)}{x}-\frac {72 \log (x)}{x}\right ) \, dx\\ &=-\left (72 \int \frac {\log (x)}{x} \, dx\right )+\int \frac {-e^3 x+2 x^2-12 \log (5)}{x} \, dx\\ &=-36 \log ^2(x)+\int \left (-e^3+2 x-\frac {12 \log (5)}{x}\right ) \, dx\\ &=-e^3 x+x^2-12 \log (5) \log (x)-36 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 0.92 \begin {gather*} -e^3 x+x^2-12 \log (5) \log (x)-36 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(E^3*x) + 2*x^2 - 12*Log[5] - 72*Log[x])/x,x]

[Out]

-(E^3*x) + x^2 - 12*Log[5]*Log[x] - 36*Log[x]^2

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fricas [A]  time = 0.53, size = 21, normalized size = 0.88 \begin {gather*} x^{2} - x e^{3} - 12 \, \log \relax (5) \log \relax (x) - 36 \, \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*log(x)-12*log(5)-x*exp(3)+2*x^2)/x,x, algorithm="fricas")

[Out]

x^2 - x*e^3 - 12*log(5)*log(x) - 36*log(x)^2

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giac [A]  time = 0.19, size = 21, normalized size = 0.88 \begin {gather*} x^{2} - x e^{3} - 12 \, \log \relax (5) \log \relax (x) - 36 \, \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*log(x)-12*log(5)-x*exp(3)+2*x^2)/x,x, algorithm="giac")

[Out]

x^2 - x*e^3 - 12*log(5)*log(x) - 36*log(x)^2

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maple [A]  time = 0.04, size = 22, normalized size = 0.92

method	result	size
default	\(-x \,{\mathrm e}^{3}+x^{2}-36 \ln \relax (x )^{2}-12 \ln \relax (5) \ln \relax (x )\)	\(22\)
norman	\(-x \,{\mathrm e}^{3}+x^{2}-36 \ln \relax (x )^{2}-12 \ln \relax (5) \ln \relax (x )\)	\(22\)
risch	\(-x \,{\mathrm e}^{3}+x^{2}-36 \ln \relax (x )^{2}-12 \ln \relax (5) \ln \relax (x )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-72*ln(x)-12*ln(5)-x*exp(3)+2*x^2)/x,x,method=_RETURNVERBOSE)

[Out]

-x*exp(3)+x^2-36*ln(x)^2-12*ln(5)*ln(x)

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maxima [A]  time = 0.36, size = 21, normalized size = 0.88 \begin {gather*} x^{2} - x e^{3} - 12 \, \log \relax (5) \log \relax (x) - 36 \, \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*log(x)-12*log(5)-x*exp(3)+2*x^2)/x,x, algorithm="maxima")

[Out]

x^2 - x*e^3 - 12*log(5)*log(x) - 36*log(x)^2

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mupad [B]  time = 5.30, size = 21, normalized size = 0.88 \begin {gather*} x^2-{\mathrm {e}}^3\,x-36\,{\ln \relax (x)}^2-12\,\ln \relax (5)\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12*log(5) + 72*log(x) + x*exp(3) - 2*x^2)/x,x)

[Out]

x^2 - 36*log(x)^2 - 12*log(5)*log(x) - x*exp(3)

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sympy [A]  time = 0.13, size = 22, normalized size = 0.92 \begin {gather*} x^{2} - x e^{3} - 36 \log {\relax (x )}^{2} - 12 \log {\relax (5 )} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*ln(x)-12*ln(5)-x*exp(3)+2*x**2)/x,x)

[Out]

x**2 - x*exp(3) - 36*log(x)**2 - 12*log(5)*log(x)

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