3.53.6 \(\int \frac {5 x+5 x^2+5 e^x x^2+e^{x^{2/x}} x^{2/x} (10-10 \log (x))}{x^2} \, dx\)

Optimal. Leaf size=19 \[ 5 \left (-5+e^x+e^{x^{2/x}}+x+\log (x)\right ) \]

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Rubi [F]  time = 0.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {5 x+5 x^2+5 e^x x^2+e^{x^{2/x}} x^{2/x} (10-10 \log (x))}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5*x + 5*x^2 + 5*E^x*x^2 + E^x^(2/x)*x^(2/x)*(10 - 10*Log[x]))/x^2,x]

[Out]

5*E^x + 5*x + 5*Log[x] + 10*Defer[Int][E^x^(2/x)*x^(-2 + 2/x), x] - 10*Log[x]*Defer[Int][E^x^(2/x)*x^(-2 + 2/x
), x] + 10*Defer[Int][Defer[Int][E^x^(2/x)*x^(-2 + 2/x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {5 \left (1+x+e^x x\right )}{x}-10 e^{x^{2/x}} x^{-2+\frac {2}{x}} (-1+\log (x))\right ) \, dx\\ &=5 \int \frac {1+x+e^x x}{x} \, dx-10 \int e^{x^{2/x}} x^{-2+\frac {2}{x}} (-1+\log (x)) \, dx\\ &=5 \int \left (e^x+\frac {1+x}{x}\right ) \, dx-10 \int \left (-e^{x^{2/x}} x^{-2+\frac {2}{x}}+e^{x^{2/x}} x^{-2+\frac {2}{x}} \log (x)\right ) \, dx\\ &=5 \int e^x \, dx+5 \int \frac {1+x}{x} \, dx+10 \int e^{x^{2/x}} x^{-2+\frac {2}{x}} \, dx-10 \int e^{x^{2/x}} x^{-2+\frac {2}{x}} \log (x) \, dx\\ &=5 e^x+5 \int \left (1+\frac {1}{x}\right ) \, dx+10 \int e^{x^{2/x}} x^{-2+\frac {2}{x}} \, dx+10 \int \frac {\int e^{x^{2/x}} x^{-2+\frac {2}{x}} \, dx}{x} \, dx-(10 \log (x)) \int e^{x^{2/x}} x^{-2+\frac {2}{x}} \, dx\\ &=5 e^x+5 x+5 \log (x)+10 \int e^{x^{2/x}} x^{-2+\frac {2}{x}} \, dx+10 \int \frac {\int e^{x^{2/x}} x^{-2+\frac {2}{x}} \, dx}{x} \, dx-(10 \log (x)) \int e^{x^{2/x}} x^{-2+\frac {2}{x}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 18, normalized size = 0.95 \begin {gather*} 5 \left (e^x+e^{x^{2/x}}+x+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5*x + 5*x^2 + 5*E^x*x^2 + E^x^(2/x)*x^(2/x)*(10 - 10*Log[x]))/x^2,x]

[Out]

5*(E^x + E^x^(2/x) + x + Log[x])

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fricas [A]  time = 0.63, size = 22, normalized size = 1.16 \begin {gather*} 5 \, x + 5 \, e^{\left (x^{\frac {2}{x}}\right )} + 5 \, e^{x} + 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*log(x)+10)*exp(2*log(x)/x)*exp(exp(2*log(x)/x))+5*exp(x)*x^2+5*x^2+5*x)/x^2,x, algorithm="fric
as")

[Out]

5*x + 5*e^(x^(2/x)) + 5*e^x + 5*log(x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 \, {\left (x^{2} e^{x} - 2 \, x^{\frac {2}{x}} {\left (\log \relax (x) - 1\right )} e^{\left (x^{\frac {2}{x}}\right )} + x^{2} + x\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*log(x)+10)*exp(2*log(x)/x)*exp(exp(2*log(x)/x))+5*exp(x)*x^2+5*x^2+5*x)/x^2,x, algorithm="giac
")

[Out]

integrate(5*(x^2*e^x - 2*x^(2/x)*(log(x) - 1)*e^(x^(2/x)) + x^2 + x)/x^2, x)

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maple [A]  time = 0.08, size = 23, normalized size = 1.21

method	result	size
risch	\(5 x +5 \ln \relax (x )+5 \,{\mathrm e}^{x}+5 \,{\mathrm e}^{x^{\frac {2}{x}}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*ln(x)+10)*exp(2*ln(x)/x)*exp(exp(2*ln(x)/x))+5*exp(x)*x^2+5*x^2+5*x)/x^2,x,method=_RETURNVERBOSE)

[Out]

5*x+5*ln(x)+5*exp(x)+5*exp(x^(2/x))

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maxima [A]  time = 0.41, size = 22, normalized size = 1.16 \begin {gather*} 5 \, x + 5 \, e^{\left (x^{\frac {2}{x}}\right )} + 5 \, e^{x} + 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*log(x)+10)*exp(2*log(x)/x)*exp(exp(2*log(x)/x))+5*exp(x)*x^2+5*x^2+5*x)/x^2,x, algorithm="maxi
ma")

[Out]

5*x + 5*e^(x^(2/x)) + 5*e^x + 5*log(x)

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mupad [B]  time = 3.47, size = 22, normalized size = 1.16 \begin {gather*} 5\,x+5\,{\mathrm {e}}^{x^{2/x}}+5\,{\mathrm {e}}^x+5\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 5*x^2*exp(x) + 5*x^2 - exp((2*log(x))/x)*exp(exp((2*log(x))/x))*(10*log(x) - 10))/x^2,x)

[Out]

5*x + 5*exp(x^(2/x)) + 5*exp(x) + 5*log(x)

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sympy [A]  time = 0.84, size = 24, normalized size = 1.26 \begin {gather*} 5 x + 5 e^{x} + 5 e^{e^{\frac {2 \log {\relax (x )}}{x}}} + 5 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*ln(x)+10)*exp(2*ln(x)/x)*exp(exp(2*ln(x)/x))+5*exp(x)*x**2+5*x**2+5*x)/x**2,x)

[Out]

5*x + 5*exp(x) + 5*exp(exp(2*log(x)/x)) + 5*log(x)

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