3.36.72 \(\int \frac {e^4 (-1-10 x)-x+2 x^2+e^4 x \log (x)}{x} \, dx\)

Optimal. Leaf size=17 \[ (-1+x) \left (x-e^4 (11-\log (x))\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 34, normalized size of antiderivative = 2.00, number of steps used = 5, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {14, 2295} \begin {gather*} x^2-\left (1+10 e^4\right ) x-e^4 x+e^4 x \log (x)-e^4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^4*(-1 - 10*x) - x + 2*x^2 + E^4*x*Log[x])/x,x]

[Out]

-(E^4*x) - (1 + 10*E^4)*x + x^2 - E^4*Log[x] + E^4*x*Log[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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-e^4-\left (1+10 e^4\right ) x+2 x^2}{x}+e^4 \log (x)\right ) \, dx\\ &=e^4 \int \log (x) \, dx+\int \frac {-e^4-\left (1+10 e^4\right ) x+2 x^2}{x} \, dx\\ &=-e^4 x+e^4 x \log (x)+\int \left (-1-10 e^4-\frac {e^4}{x}+2 x\right ) \, dx\\ &=-e^4 x-\left (1+10 e^4\right ) x+x^2-e^4 \log (x)+e^4 x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.59 \begin {gather*} -x-11 e^4 x+x^2-e^4 \log (x)+e^4 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(-1 - 10*x) - x + 2*x^2 + E^4*x*Log[x])/x,x]

[Out]

-x - 11*E^4*x + x^2 - E^4*Log[x] + E^4*x*Log[x]

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fricas [A]  time = 1.15, size = 20, normalized size = 1.18 \begin {gather*} {\left (x - 1\right )} e^{4} \log \relax (x) + x^{2} - 11 \, x e^{4} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(4)*log(x)+(-10*x-1)*exp(4)+2*x^2-x)/x,x, algorithm="fricas")

[Out]

(x - 1)*e^4*log(x) + x^2 - 11*x*e^4 - x

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giac [A]  time = 0.14, size = 24, normalized size = 1.41 \begin {gather*} x e^{4} \log \relax (x) + x^{2} - 11 \, x e^{4} - e^{4} \log \relax (x) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(4)*log(x)+(-10*x-1)*exp(4)+2*x^2-x)/x,x, algorithm="giac")

[Out]

x*e^4*log(x) + x^2 - 11*x*e^4 - e^4*log(x) - x

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maple [A]  time = 0.03, size = 25, normalized size = 1.47

method	result	size
norman	\(x^{2}+\left (-1-11 \,{\mathrm e}^{4}\right ) x -{\mathrm e}^{4} \ln \relax (x )+x \,{\mathrm e}^{4} \ln \relax (x )\)	\(25\)
risch	\(x \,{\mathrm e}^{4} \ln \relax (x )-11 x \,{\mathrm e}^{4}+x^{2}-x -{\mathrm e}^{4} \ln \relax (x )\)	\(25\)
default	\({\mathrm e}^{4} \left (x \ln \relax (x )-x \right )-10 x \,{\mathrm e}^{4}+x^{2}-{\mathrm e}^{4} \ln \relax (x )-x\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(4)*ln(x)+(-10*x-1)*exp(4)+2*x^2-x)/x,x,method=_RETURNVERBOSE)

[Out]

x^2+(-1-11*exp(4))*x-exp(4)*ln(x)+x*exp(4)*ln(x)

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maxima [B]  time = 0.37, size = 29, normalized size = 1.71 \begin {gather*} x^{2} + {\left (x \log \relax (x) - x\right )} e^{4} - 10 \, x e^{4} - e^{4} \log \relax (x) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(4)*log(x)+(-10*x-1)*exp(4)+2*x^2-x)/x,x, algorithm="maxima")

[Out]

x^2 + (x*log(x) - x)*e^4 - 10*x*e^4 - e^4*log(x) - x

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mupad [B]  time = 2.11, size = 24, normalized size = 1.41 \begin {gather*} x^2-11\,x\,{\mathrm {e}}^4-{\mathrm {e}}^4\,\ln \relax (x)-x+x\,{\mathrm {e}}^4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 2*x^2 + exp(4)*(10*x + 1) - x*exp(4)*log(x))/x,x)

[Out]

x^2 - 11*x*exp(4) - exp(4)*log(x) - x + x*exp(4)*log(x)

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sympy [B]  time = 0.14, size = 27, normalized size = 1.59 \begin {gather*} x^{2} + x e^{4} \log {\relax (x )} + x \left (- 11 e^{4} - 1\right ) - e^{4} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(4)*ln(x)+(-10*x-1)*exp(4)+2*x**2-x)/x,x)

[Out]

x**2 + x*exp(4)*log(x) + x*(-11*exp(4) - 1) - exp(4)*log(x)

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