3.46.19 \(\int \frac {1+2 x-6 x^2+\frac {1}{2} e^{-5+e^4+x} (2 x+7 x^2+2 x^3)}{x+x^2-2 x^3+\frac {1}{2} e^{-5+e^4+x} (x^2+2 x^3)} \, dx\)

Optimal. Leaf size=26 \[ \log \left (x (1+2 x) \left (1-x+\frac {1}{2} e^{-5+e^4+x} x\right )\right ) \]

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Rubi [F]  time = 0.84, 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 {1+2 x-6 x^2+\frac {1}{2} e^{-5+e^4+x} \left (2 x+7 x^2+2 x^3\right )}{x+x^2-2 x^3+\frac {1}{2} e^{-5+e^4+x} \left (x^2+2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + 2*x - 6*x^2 + (E^(-5 + E^4 + x)*(2*x + 7*x^2 + 2*x^3))/2)/(x + x^2 - 2*x^3 + (E^(-5 + E^4 + x)*(x^2 +
2*x^3))/2),x]

[Out]

x + 2*Log[x] + Log[1 + 2*x] + 2*E^5*Defer[Int][(-2*E^5 + 2*E^5*x - E^(E^4 + x)*x)^(-1), x] + 2*E^5*Defer[Int][
1/(x*(-2*E^5 + 2*E^5*x - E^(E^4 + x)*x)), x] - 2*E^5*Defer[Int][x/(-2*E^5 + 2*E^5*x - E^(E^4 + x)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 e^5 \left (-1-x+x^2\right )}{x \left (-2 e^5+2 e^5 x-e^{e^4+x} x\right )}+\frac {2+7 x+2 x^2}{x (1+2 x)}\right ) \, dx\\ &=-\left (\left (2 e^5\right ) \int \frac {-1-x+x^2}{x \left (-2 e^5+2 e^5 x-e^{e^4+x} x\right )} \, dx\right )+\int \frac {2+7 x+2 x^2}{x (1+2 x)} \, dx\\ &=-\left (\left (2 e^5\right ) \int \left (-\frac {1}{-2 e^5+2 e^5 x-e^{e^4+x} x}-\frac {1}{x \left (-2 e^5+2 e^5 x-e^{e^4+x} x\right )}+\frac {x}{-2 e^5+2 e^5 x-e^{e^4+x} x}\right ) \, dx\right )+\int \left (1+\frac {2}{x}+\frac {2}{1+2 x}\right ) \, dx\\ &=x+2 \log (x)+\log (1+2 x)+\left (2 e^5\right ) \int \frac {1}{-2 e^5+2 e^5 x-e^{e^4+x} x} \, dx+\left (2 e^5\right ) \int \frac {1}{x \left (-2 e^5+2 e^5 x-e^{e^4+x} x\right )} \, dx-\left (2 e^5\right ) \int \frac {x}{-2 e^5+2 e^5 x-e^{e^4+x} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.41, size = 29, normalized size = 1.12 \begin {gather*} \log (x)+\log (1+2 x)+\log \left (2 e^5 (-1+x)-e^{e^4+x} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x - 6*x^2 + (E^(-5 + E^4 + x)*(2*x + 7*x^2 + 2*x^3))/2)/(x + x^2 - 2*x^3 + (E^(-5 + E^4 + x)*
(x^2 + 2*x^3))/2),x]

[Out]

Log[x] + Log[1 + 2*x] + Log[2*E^5*(-1 + x) - E^(E^4 + x)*x]

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fricas [A]  time = 0.80, size = 33, normalized size = 1.27 \begin {gather*} \log \left (2 \, x + 1\right ) + 2 \, \log \relax (x) + \log \left (\frac {x e^{\left (x + e^{4} - \log \relax (2) - 5\right )} - x + 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+7*x^2+2*x)*exp(-log(2)+exp(4)+x-5)-6*x^2+2*x+1)/((2*x^3+x^2)*exp(-log(2)+exp(4)+x-5)-2*x^3+x
^2+x),x, algorithm="fricas")

[Out]

log(2*x + 1) + 2*log(x) + log((x*e^(x + e^4 - log(2) - 5) - x + 1)/x)

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giac [A]  time = 0.26, size = 23, normalized size = 0.88 \begin {gather*} \log \left (x e^{\left (x + e^{4} - 5\right )} - 2 \, x + 2\right ) + \log \left (2 \, x + 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+7*x^2+2*x)*exp(-log(2)+exp(4)+x-5)-6*x^2+2*x+1)/((2*x^3+x^2)*exp(-log(2)+exp(4)+x-5)-2*x^3+x
^2+x),x, algorithm="giac")

[Out]

log(x*e^(x + e^4 - 5) - 2*x + 2) + log(2*x + 1) + log(x)

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maple [A]  time = 0.63, size = 28, normalized size = 1.08

method	result	size
norman	\(\ln \relax (x )+\ln \left (2 x +1\right )+\ln \left (x \,{\mathrm e}^{-\ln \relax (2)+{\mathrm e}^{4}+x -5}-x +1\right )\)	\(28\)
risch	\(2 \ln \relax (x )+\ln \left (2 x +1\right )+\ln \relax (2)-{\mathrm e}^{4}+5+\ln \left (\frac {{\mathrm e}^{-5+{\mathrm e}^{4}+x}}{2}-\frac {x -1}{x}\right )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+7*x^2+2*x)*exp(-ln(2)+exp(4)+x-5)-6*x^2+2*x+1)/((2*x^3+x^2)*exp(-ln(2)+exp(4)+x-5)-2*x^3+x^2+x),x,
method=_RETURNVERBOSE)

[Out]

ln(x)+ln(2*x+1)+ln(x*exp(-ln(2)+exp(4)+x-5)-x+1)

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maxima [A]  time = 0.43, size = 40, normalized size = 1.54 \begin {gather*} \log \left (2 \, x + 1\right ) + 2 \, \log \relax (x) + \log \left (-\frac {{\left (2 \, x e^{5} - x e^{\left (x + e^{4}\right )} - 2 \, e^{5}\right )} e^{\left (-e^{4}\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+7*x^2+2*x)*exp(-log(2)+exp(4)+x-5)-6*x^2+2*x+1)/((2*x^3+x^2)*exp(-log(2)+exp(4)+x-5)-2*x^3+x
^2+x),x, algorithm="maxima")

[Out]

log(2*x + 1) + 2*log(x) + log(-(2*x*e^5 - x*e^(x + e^4) - 2*e^5)*e^(-e^4)/x)

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mupad [B]  time = 3.16, size = 23, normalized size = 0.88 \begin {gather*} \ln \left (x+\frac {1}{2}\right )+\ln \left (\frac {x\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{{\mathrm {e}}^4}\,{\mathrm {e}}^x}{2}-x+1\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + exp(x + exp(4) - log(2) - 5)*(2*x + 7*x^2 + 2*x^3) - 6*x^2 + 1)/(x + exp(x + exp(4) - log(2) - 5)*(
x^2 + 2*x^3) + x^2 - 2*x^3),x)

[Out]

log(x + 1/2) + log((x*exp(-5)*exp(exp(4))*exp(x))/2 - x + 1) + log(x)

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sympy [A]  time = 0.24, size = 27, normalized size = 1.04 \begin {gather*} 2 \log {\relax (x )} + \log {\left (x + \frac {1}{2} \right )} + \log {\left (e^{x - 5 + e^{4}} + \frac {2 - 2 x}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+7*x**2+2*x)*exp(-ln(2)+exp(4)+x-5)-6*x**2+2*x+1)/((2*x**3+x**2)*exp(-ln(2)+exp(4)+x-5)-2*x*
*3+x**2+x),x)

[Out]

2*log(x) + log(x + 1/2) + log(exp(x - 5 + exp(4)) + (2 - 2*x)/x)

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