3.72.47 \(\int \frac {e (-9+18 x+3 x^2)+e x^2 \log (x^2)}{x^2} \, dx\)

Optimal. Leaf size=20 \[ e \left (5-x+(9+x) \left (2+\frac {1}{x}+\log \left (x^2\right )\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14, 2295} \begin {gather*} e x \log \left (x^2\right )+e x+\frac {9 e}{x}+18 e \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E*(-9 + 18*x + 3*x^2) + E*x^2*Log[x^2])/x^2,x]

[Out]

(9*E)/x + E*x + 18*E*Log[x] + E*x*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 2295

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3 e \left (-3+6 x+x^2\right )}{x^2}+e \log \left (x^2\right )\right ) \, dx\\ &=e \int \log \left (x^2\right ) \, dx+(3 e) \int \frac {-3+6 x+x^2}{x^2} \, dx\\ &=-2 e x+e x \log \left (x^2\right )+(3 e) \int \left (1-\frac {3}{x^2}+\frac {6}{x}\right ) \, dx\\ &=\frac {9 e}{x}+e x+18 e \log (x)+e x \log \left (x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.10 \begin {gather*} \frac {9 e}{x}+e x+18 e \log (x)+e x \log \left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E*(-9 + 18*x + 3*x^2) + E*x^2*Log[x^2])/x^2,x]

[Out]

(9*E)/x + E*x + 18*E*Log[x] + E*x*Log[x^2]

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fricas [A]  time = 1.13, size = 27, normalized size = 1.35 \begin {gather*} \frac {{\left (x^{2} + 9 \, x\right )} e \log \left (x^{2}\right ) + {\left (x^{2} + 9\right )} e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(1)*log(x^2)+(3*x^2+18*x-9)*exp(1))/x^2,x, algorithm="fricas")

[Out]

((x^2 + 9*x)*e*log(x^2) + (x^2 + 9)*e)/x

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giac [A]  time = 0.20, size = 32, normalized size = 1.60 \begin {gather*} \frac {x^{2} e \log \left (x^{2}\right ) + x^{2} e + 18 \, x e \log \relax (x) + 9 \, e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(1)*log(x^2)+(3*x^2+18*x-9)*exp(1))/x^2,x, algorithm="giac")

[Out]

(x^2*e*log(x^2) + x^2*e + 18*x*e*log(x) + 9*e)/x

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maple [A]  time = 0.03, size = 26, normalized size = 1.30

method	result	size
risch	\(x \,{\mathrm e} \ln \left (x^{2}\right )+\frac {{\mathrm e} \left (18 x \ln \relax (x )+x^{2}+9\right )}{x}\)	\(26\)
default	\(x \,{\mathrm e} \ln \left (x^{2}\right )+x \,{\mathrm e}+18 \,{\mathrm e} \ln \relax (x )+\frac {9 \,{\mathrm e}}{x}\)	\(27\)
norman	\(\frac {x^{2} {\mathrm e}+x^{2} {\mathrm e} \ln \left (x^{2}\right )+9 x \,{\mathrm e} \ln \left (x^{2}\right )+9 \,{\mathrm e}}{x}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(1)*ln(x^2)+(3*x^2+18*x-9)*exp(1))/x^2,x,method=_RETURNVERBOSE)

[Out]

x*exp(1)*ln(x^2)+exp(1)*(18*x*ln(x)+x^2+9)/x

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maxima [A]  time = 0.35, size = 32, normalized size = 1.60 \begin {gather*} {\left (x \log \left (x^{2}\right ) - 2 \, x\right )} e + 3 \, x e + 18 \, e \log \relax (x) + \frac {9 \, e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(1)*log(x^2)+(3*x^2+18*x-9)*exp(1))/x^2,x, algorithm="maxima")

[Out]

(x*log(x^2) - 2*x)*e + 3*x*e + 18*e*log(x) + 9*e/x

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mupad [B]  time = 4.17, size = 26, normalized size = 1.30 \begin {gather*} 9\,\ln \left (x^2\right )\,\mathrm {e}+\frac {9\,\mathrm {e}}{x}+x\,\mathrm {e}\,\left (\ln \left (x^2\right )+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1)*(18*x + 3*x^2 - 9) + x^2*log(x^2)*exp(1))/x^2,x)

[Out]

9*log(x^2)*exp(1) + (9*exp(1))/x + x*exp(1)*(log(x^2) + 1)

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sympy [A]  time = 0.19, size = 29, normalized size = 1.45 \begin {gather*} e x \log {\left (x^{2} \right )} + e x + 18 e \log {\relax (x )} + \frac {9 e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*exp(1)*ln(x**2)+(3*x**2+18*x-9)*exp(1))/x**2,x)

[Out]

E*x*log(x**2) + E*x + 18*E*log(x) + 9*E/x

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