3.70.68 \(\int \frac {2+3 x-2 x^2+6 x^3+e^{2 x} (2+8 x)+e^x (2+13 x+14 x^2)}{x+e^{2 x} x-x^2+x^3+e^x (x+2 x^2)} \, dx\)

Optimal. Leaf size=30 \[ 6 x+\log \left (\frac {1}{4} x^2 \left (5+5 \left (e^x-x+\left (e^x+x\right )^2\right )\right )\right ) \]

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Rubi [F]  time = 0.87, 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+3 x-2 x^2+6 x^3+e^{2 x} (2+8 x)+e^x \left (2+13 x+14 x^2\right )}{x+e^{2 x} x-x^2+x^3+e^x \left (x+2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + 3*x - 2*x^2 + 6*x^3 + E^(2*x)*(2 + 8*x) + E^x*(2 + 13*x + 14*x^2))/(x + E^(2*x)*x - x^2 + x^3 + E^x*(
x + 2*x^2)),x]

[Out]

8*x + 2*Log[x] - 3*Defer[Int][(1 + E^x + E^(2*x) - x + 2*E^x*x + x^2)^(-1), x] + Defer[Int][E^x/(1 + E^x + E^(
2*x) - x + 2*E^x*x + x^2), x] + 4*Defer[Int][x/(1 + E^x + E^(2*x) - x + 2*E^x*x + x^2), x] - 2*Defer[Int][(E^x
*x)/(1 + E^x + E^(2*x) - x + 2*E^x*x + x^2), x] - 2*Defer[Int][x^2/(1 + E^x + E^(2*x) - x + 2*E^x*x + x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 (1+4 x)}{x}-\frac {3-e^x-4 x+2 e^x x+2 x^2}{1+e^x+e^{2 x}-x+2 e^x x+x^2}\right ) \, dx\\ &=2 \int \frac {1+4 x}{x} \, dx-\int \frac {3-e^x-4 x+2 e^x x+2 x^2}{1+e^x+e^{2 x}-x+2 e^x x+x^2} \, dx\\ &=2 \int \left (4+\frac {1}{x}\right ) \, dx-\int \left (\frac {3}{1+e^x+e^{2 x}-x+2 e^x x+x^2}-\frac {e^x}{1+e^x+e^{2 x}-x+2 e^x x+x^2}-\frac {4 x}{1+e^x+e^{2 x}-x+2 e^x x+x^2}+\frac {2 e^x x}{1+e^x+e^{2 x}-x+2 e^x x+x^2}+\frac {2 x^2}{1+e^x+e^{2 x}-x+2 e^x x+x^2}\right ) \, dx\\ &=8 x+2 \log (x)-2 \int \frac {e^x x}{1+e^x+e^{2 x}-x+2 e^x x+x^2} \, dx-2 \int \frac {x^2}{1+e^x+e^{2 x}-x+2 e^x x+x^2} \, dx-3 \int \frac {1}{1+e^x+e^{2 x}-x+2 e^x x+x^2} \, dx+4 \int \frac {x}{1+e^x+e^{2 x}-x+2 e^x x+x^2} \, dx+\int \frac {e^x}{1+e^x+e^{2 x}-x+2 e^x x+x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 31, normalized size = 1.03 \begin {gather*} 6 x+2 \log (x)+\log \left (1+e^x+e^{2 x}-x+2 e^x x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x - 2*x^2 + 6*x^3 + E^(2*x)*(2 + 8*x) + E^x*(2 + 13*x + 14*x^2))/(x + E^(2*x)*x - x^2 + x^3 +
E^x*(x + 2*x^2)),x]

[Out]

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

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fricas [A]  time = 0.74, size = 29, normalized size = 0.97 \begin {gather*} 6 \, x + \log \left (x^{2} + {\left (2 \, x + 1\right )} e^{x} - x + e^{\left (2 \, x\right )} + 1\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+2)*exp(x)^2+(14*x^2+13*x+2)*exp(x)+6*x^3-2*x^2+3*x+2)/(x*exp(x)^2+(2*x^2+x)*exp(x)+x^3-x^2+x),
x, algorithm="fricas")

[Out]

6*x + log(x^2 + (2*x + 1)*e^x - x + e^(2*x) + 1) + 2*log(x)

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giac [A]  time = 0.43, size = 28, normalized size = 0.93 \begin {gather*} 6 \, x + \log \left (x^{2} + 2 \, x e^{x} - x + e^{\left (2 \, x\right )} + e^{x} + 1\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+2)*exp(x)^2+(14*x^2+13*x+2)*exp(x)+6*x^3-2*x^2+3*x+2)/(x*exp(x)^2+(2*x^2+x)*exp(x)+x^3-x^2+x),
x, algorithm="giac")

[Out]

6*x + log(x^2 + 2*x*e^x - x + e^(2*x) + e^x + 1) + 2*log(x)

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maple [A]  time = 0.04, size = 29, normalized size = 0.97

method	result	size
norman	\(6 x +2 \ln \relax (x )+\ln \left ({\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}+{\mathrm e}^{x}-x +1\right )\)	\(29\)
risch	\(6 x +2 \ln \relax (x )+\ln \left ({\mathrm e}^{2 x}+\left (2 x +1\right ) {\mathrm e}^{x}+x^{2}-x +1\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x+2)*exp(x)^2+(14*x^2+13*x+2)*exp(x)+6*x^3-2*x^2+3*x+2)/(x*exp(x)^2+(2*x^2+x)*exp(x)+x^3-x^2+x),x,meth
od=_RETURNVERBOSE)

[Out]

6*x+2*ln(x)+ln(exp(x)^2+2*exp(x)*x+x^2+exp(x)-x+1)

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maxima [A]  time = 0.43, size = 29, normalized size = 0.97 \begin {gather*} 6 \, x + \log \left (x^{2} + {\left (2 \, x + 1\right )} e^{x} - x + e^{\left (2 \, x\right )} + 1\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+2)*exp(x)^2+(14*x^2+13*x+2)*exp(x)+6*x^3-2*x^2+3*x+2)/(x*exp(x)^2+(2*x^2+x)*exp(x)+x^3-x^2+x),
x, algorithm="maxima")

[Out]

6*x + log(x^2 + (2*x + 1)*e^x - x + e^(2*x) + 1) + 2*log(x)

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mupad [B]  time = 0.15, size = 28, normalized size = 0.93 \begin {gather*} 6\,x+\ln \left ({\mathrm {e}}^{2\,x}-x+{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^x+x^2+1\right )+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + exp(x)*(13*x + 14*x^2 + 2) + exp(2*x)*(8*x + 2) - 2*x^2 + 6*x^3 + 2)/(x + x*exp(2*x) + exp(x)*(x +
2*x^2) - x^2 + x^3),x)

[Out]

6*x + log(exp(2*x) - x + exp(x) + 2*x*exp(x) + x^2 + 1) + 2*log(x)

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sympy [A]  time = 0.16, size = 29, normalized size = 0.97 \begin {gather*} 6 x + 2 \log {\relax (x )} + \log {\left (x^{2} - x + \left (2 x + 1\right ) e^{x} + e^{2 x} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+2)*exp(x)**2+(14*x**2+13*x+2)*exp(x)+6*x**3-2*x**2+3*x+2)/(x*exp(x)**2+(2*x**2+x)*exp(x)+x**3-
x**2+x),x)

[Out]

6*x + 2*log(x) + log(x**2 - x + (2*x + 1)*exp(x) + exp(2*x) + 1)

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