3.2.33 \(\int \frac {10 e^x x^2+e^{\frac {-1-e+2 e^5 x+2 x^2}{2 x}} (1+e+2 x^2)}{10 x^2} \, dx\)

Optimal. Leaf size=30 \[ 5+e^x+\frac {1}{5} \left (e^4+e^{e^5-\frac {1+e}{2 x}+x}\right ) \]

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Rubi [F]  time = 0.15, 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 {10 e^x x^2+e^{\frac {-1-e+2 e^5 x+2 x^2}{2 x}} \left (1+e+2 x^2\right )}{10 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10*E^x*x^2 + E^((-1 - E + 2*E^5*x + 2*x^2)/(2*x))*(1 + E + 2*x^2))/(10*x^2),x]

[Out]

E^x + Defer[Int][E^(E^5 - (1 + E)/(2*x) + x), x]/5 + ((1 + E)*Defer[Int][E^(E^5 - (1 + E)/(2*x) + x)/x^2, x])/
10

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 27, normalized size = 0.90 \begin {gather*} e^x+\frac {1}{5} e^{e^5+\frac {-1-e}{2 x}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*E^x*x^2 + E^((-1 - E + 2*E^5*x + 2*x^2)/(2*x))*(1 + E + 2*x^2))/(10*x^2),x]

[Out]

E^x + E^(E^5 + (-1 - E)/(2*x) + x)/5

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fricas [A]  time = 0.61, size = 27, normalized size = 0.90 \begin {gather*} e^{x} + \frac {1}{5} \, e^{\left (\frac {2 \, x^{2} + 2 \, x e^{5} - e - 1}{2 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*((exp(1)+2*x^2+1)*exp(1/2*(2*x*exp(5)-exp(1)+2*x^2-1)/x)+10*exp(x)*x^2)/x^2,x, algorithm="frica
s")

[Out]

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

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giac [A]  time = 0.46, size = 27, normalized size = 0.90 \begin {gather*} e^{x} + \frac {1}{5} \, e^{\left (\frac {2 \, x^{2} + 2 \, x e^{5} - e - 1}{2 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*((exp(1)+2*x^2+1)*exp(1/2*(2*x*exp(5)-exp(1)+2*x^2-1)/x)+10*exp(x)*x^2)/x^2,x, algorithm="giac"
)

[Out]

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

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maple [A]  time = 0.09, size = 26, normalized size = 0.87

method	result	size
risch	\({\mathrm e}^{x}+\frac {{\mathrm e}^{-\frac {-2 x \,{\mathrm e}^{5}+{\mathrm e}-2 x^{2}+1}{2 x}}}{5}\)	\(26\)
norman	\(\frac {{\mathrm e}^{x} x +\frac {x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}-{\mathrm e}+2 x^{2}-1}{2 x}}}{5}}{x}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/10*((exp(1)+2*x^2+1)*exp(1/2*(2*x*exp(5)-exp(1)+2*x^2-1)/x)+10*exp(x)*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(x)+1/5*exp(-1/2*(-2*x*exp(5)+exp(1)-2*x^2+1)/x)

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maxima [A]  time = 0.74, size = 22, normalized size = 0.73 \begin {gather*} \frac {1}{5} \, e^{\left (x - \frac {e}{2 \, x} - \frac {1}{2 \, x} + e^{5}\right )} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*((exp(1)+2*x^2+1)*exp(1/2*(2*x*exp(5)-exp(1)+2*x^2-1)/x)+10*exp(x)*x^2)/x^2,x, algorithm="maxim
a")

[Out]

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

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mupad [B]  time = 0.39, size = 22, normalized size = 0.73 \begin {gather*} \frac {{\mathrm {e}}^{x+{\mathrm {e}}^5-\frac {\mathrm {e}}{2\,x}-\frac {1}{2\,x}}}{5}+{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(x) + (exp(-(exp(1)/2 - x*exp(5) - x^2 + 1/2)/x)*(exp(1) + 2*x^2 + 1))/10)/x^2,x)

[Out]

exp(x + exp(5) - exp(1)/(2*x) - 1/(2*x))/5 + exp(x)

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sympy [A]  time = 0.23, size = 24, normalized size = 0.80 \begin {gather*} e^{x} + \frac {e^{\frac {x^{2} + x e^{5} - \frac {e}{2} - \frac {1}{2}}{x}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*((exp(1)+2*x**2+1)*exp(1/2*(2*x*exp(5)-exp(1)+2*x**2-1)/x)+10*exp(x)*x**2)/x**2,x)

[Out]

exp(x) + exp((x**2 + x*exp(5) - E/2 - 1/2)/x)/5

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