3.62.99 \(\int \frac {2 x^2+e^{\frac {6+11 x+3 e^x x+3 x^2}{3 x}} (-2+x^2+e^x x^2)}{x^2} \, dx\)

Optimal. Leaf size=19 \[ e^{\frac {11}{3}+e^x+\frac {2}{x}+x}+2 x \]

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Rubi [F]  time = 0.37, 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 {2 x^2+e^{\frac {6+11 x+3 e^x x+3 x^2}{3 x}} \left (-2+x^2+e^x x^2\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x^2 + E^((6 + 11*x + 3*E^x*x + 3*x^2)/(3*x))*(-2 + x^2 + E^x*x^2))/x^2,x]

[Out]

2*x + Defer[Int][E^(11/3 + E^x + 2/x + x), x] + Defer[Int][E^(11/3 + E^x + 2/x + 2*x), x] - 2*Defer[Int][E^(11
/3 + E^x + 2/x + x)/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+e^{\frac {11}{3}+e^x+\frac {2}{x}+2 x}+\frac {e^{\frac {11}{3}+e^x+\frac {2}{x}+x} \left (-2+x^2\right )}{x^2}\right ) \, dx\\ &=2 x+\int e^{\frac {11}{3}+e^x+\frac {2}{x}+2 x} \, dx+\int \frac {e^{\frac {11}{3}+e^x+\frac {2}{x}+x} \left (-2+x^2\right )}{x^2} \, dx\\ &=2 x+\int e^{\frac {11}{3}+e^x+\frac {2}{x}+2 x} \, dx+\int \left (e^{\frac {11}{3}+e^x+\frac {2}{x}+x}-\frac {2 e^{\frac {11}{3}+e^x+\frac {2}{x}+x}}{x^2}\right ) \, dx\\ &=2 x-2 \int \frac {e^{\frac {11}{3}+e^x+\frac {2}{x}+x}}{x^2} \, dx+\int e^{\frac {11}{3}+e^x+\frac {2}{x}+x} \, dx+\int e^{\frac {11}{3}+e^x+\frac {2}{x}+2 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 19, normalized size = 1.00 \begin {gather*} e^{\frac {11}{3}+e^x+\frac {2}{x}+x}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^2 + E^((6 + 11*x + 3*E^x*x + 3*x^2)/(3*x))*(-2 + x^2 + E^x*x^2))/x^2,x]

[Out]

E^(11/3 + E^x + 2/x + x) + 2*x

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fricas [A]  time = 0.76, size = 25, normalized size = 1.32 \begin {gather*} 2 \, x + e^{\left (\frac {3 \, x^{2} + 3 \, x e^{x} + 11 \, x + 6}{3 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x^2+x^2-2)*exp(1/3*(3*exp(x)*x+3*x^2+11*x+6)/x)+2*x^2)/x^2,x, algorithm="fricas")

[Out]

2*x + e^(1/3*(3*x^2 + 3*x*e^x + 11*x + 6)/x)

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giac [A]  time = 0.17, size = 15, normalized size = 0.79 \begin {gather*} 2 \, x + e^{\left (x + \frac {2}{x} + e^{x} + \frac {11}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x^2+x^2-2)*exp(1/3*(3*exp(x)*x+3*x^2+11*x+6)/x)+2*x^2)/x^2,x, algorithm="giac")

[Out]

2*x + e^(x + 2/x + e^x + 11/3)

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maple [A]  time = 0.22, size = 26, normalized size = 1.37

method	result	size
risch	\(2 x +{\mathrm e}^{\frac {3 \,{\mathrm e}^{x} x +3 x^{2}+11 x +6}{3 x}}\)	\(26\)
norman	\(\frac {x \,{\mathrm e}^{\frac {3 \,{\mathrm e}^{x} x +3 x^{2}+11 x +6}{3 x}}+2 x^{2}}{x}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)*x^2+x^2-2)*exp(1/3*(3*exp(x)*x+3*x^2+11*x+6)/x)+2*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

2*x+exp(1/3*(3*exp(x)*x+3*x^2+11*x+6)/x)

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maxima [A]  time = 0.42, size = 15, normalized size = 0.79 \begin {gather*} 2 \, x + e^{\left (x + \frac {2}{x} + e^{x} + \frac {11}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x^2+x^2-2)*exp(1/3*(3*exp(x)*x+3*x^2+11*x+6)/x)+2*x^2)/x^2,x, algorithm="maxima")

[Out]

2*x + e^(x + 2/x + e^x + 11/3)

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mupad [B]  time = 4.12, size = 18, normalized size = 0.95 \begin {gather*} 2\,x+{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{11/3}\,{\mathrm {e}}^{2/x}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(((11*x)/3 + x*exp(x) + x^2 + 2)/x)*(x^2*exp(x) + x^2 - 2) + 2*x^2)/x^2,x)

[Out]

2*x + exp(exp(x))*exp(11/3)*exp(2/x)*exp(x)

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sympy [A]  time = 0.18, size = 20, normalized size = 1.05 \begin {gather*} 2 x + e^{\frac {x^{2} + x e^{x} + \frac {11 x}{3} + 2}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x**2+x**2-2)*exp(1/3*(3*exp(x)*x+3*x**2+11*x+6)/x)+2*x**2)/x**2,x)

[Out]

2*x + exp((x**2 + x*exp(x) + 11*x/3 + 2)/x)

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