3.52.93 \(\int \frac {2 x+x^2+x^3+e^x (2+2 x^2)}{e^x x^2+x^3} \, dx\)

Optimal. Leaf size=23 \[ \log \left (\frac {1}{2} e^{\frac {-2+(-5+x) x}{x}} \left (e^x+x\right )\right ) \]

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Rubi [F]  time = 0.17, 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+x^2+x^3+e^x \left (2+2 x^2\right )}{e^x x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 13, normalized size = 0.57 \begin {gather*} -\frac {2}{x}+x+\log \left (e^x+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/x + x + Log[E^x + x]

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fricas [A]  time = 0.55, size = 16, normalized size = 0.70 \begin {gather*} \frac {x^{2} + x \log \left (x + e^{x}\right ) - 2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 + x*log(x + e^x) - 2)/x

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giac [A]  time = 0.14, size = 16, normalized size = 0.70 \begin {gather*} \frac {x^{2} + x \log \left (x + e^{x}\right ) - 2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 + x*log(x + e^x) - 2)/x

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maple [A]  time = 0.03, size = 13, normalized size = 0.57

method	result	size
risch	\(x -\frac {2}{x}+\ln \left ({\mathrm e}^{x}+x \right )\)	\(13\)
norman	\(\frac {x^{2}-2}{x}+\ln \left ({\mathrm e}^{x}+x \right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+2)*exp(x)+x^3+x^2+2*x)/(exp(x)*x^2+x^3),x,method=_RETURNVERBOSE)

[Out]

x-2/x+ln(exp(x)+x)

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maxima [A]  time = 0.37, size = 15, normalized size = 0.65 \begin {gather*} \frac {x^{2} - 2}{x} + \log \left (x + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 - 2)/x + log(x + e^x)

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mupad [B]  time = 0.08, size = 15, normalized size = 0.65 \begin {gather*} \ln \left (x+{\mathrm {e}}^x\right )+\frac {x^2-2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + exp(x)) + (x^2 - 2)/x

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sympy [A]  time = 0.11, size = 10, normalized size = 0.43 \begin {gather*} x + \log {\left (x + e^{x} \right )} - \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+2)*exp(x)+x**3+x**2+2*x)/(exp(x)*x**2+x**3),x)

[Out]

x + log(x + exp(x)) - 2/x

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