3.52.47 \(\int \frac {-7 x^2+e^x (2+x+x^2)}{7 x^2+e^x (-2 x+3 x^2+e^3 x^2+x^3)+e^x x^2 \log (x)} \, dx\)

Optimal. Leaf size=29 \[ \log \left (e^2+\frac {4+7 e^{-x}+x-\frac {2+x}{x}+\log (x)}{e}\right ) \]

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Rubi [F]  time = 6.73, 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 {-7 x^2+e^x \left (2+x+x^2\right )}{7 x^2+e^x \left (-2 x+3 x^2+e^3 x^2+x^3\right )+e^x x^2 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-7*x^2 + E^x*(2 + x + x^2))/(7*x^2 + E^x*(-2*x + 3*x^2 + E^3*x^2 + x^3) + E^x*x^2*Log[x]),x]

[Out]

Defer[Int][(-2 + 3*(1 + E^3/3)*x + x^2 + x*Log[x])^(-1), x] + 2*Defer[Int][1/(x*(-2 + 3*(1 + E^3/3)*x + x^2 +
x*Log[x])), x] + Defer[Int][x/(-2 + 3*(1 + E^3/3)*x + x^2 + x*Log[x]), x] + 14*Defer[Int][1/((-2 + 3*(1 + E^3/
3)*x + x^2 + x*Log[x])*(2*E^x - 7*x - 3*E^x*(1 + E^3/3)*x - E^x*x^2 - E^x*x*Log[x])), x] + 7*Defer[Int][x^3/((
-2 + 3*(1 + E^3/3)*x + x^2 + x*Log[x])*(2*E^x - 7*x - 3*E^x*(1 + E^3/3)*x - E^x*x^2 - E^x*x*Log[x])), x] + 7*D
efer[Int][(x^2*Log[x])/((-2 + 3*(1 + E^3/3)*x + x^2 + x*Log[x])*(2*E^x - 7*x - 3*E^x*(1 + E^3/3)*x - E^x*x^2 -
E^x*x*Log[x])), x] + 7*Defer[Int][x/((-2 + (3 + E^3)*x + x^2 + x*Log[x])*(7*x + E^(3 + x)*x + E^x*(-2 + 3*x +
x^2) + E^x*x*Log[x])), x] - 7*(4 + E^3)*Defer[Int][x^2/((-2 + (3 + E^3)*x + x^2 + x*Log[x])*(7*x + E^(3 + x)*
x + E^x*(-2 + 3*x + x^2) + E^x*x*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-2-x-x^2}{x \left (2-3 \left (1+\frac {e^3}{3}\right ) x-x^2-x \log (x)\right )}+\frac {7 \left (-2+x-4 \left (1+\frac {e^3}{4}\right ) x^2-x^3-x^2 \log (x)\right )}{\left (2-3 \left (1+\frac {e^3}{3}\right ) x-x^2-x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )}\right ) \, dx\\ &=7 \int \frac {-2+x-4 \left (1+\frac {e^3}{4}\right ) x^2-x^3-x^2 \log (x)}{\left (2-3 \left (1+\frac {e^3}{3}\right ) x-x^2-x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )} \, dx+\int \frac {-2-x-x^2}{x \left (2-3 \left (1+\frac {e^3}{3}\right ) x-x^2-x \log (x)\right )} \, dx\\ &=7 \int \left (\frac {x}{\left (2-3 \left (1+\frac {e^3}{3}\right ) x-x^2-x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )}+\frac {\left (-4-e^3\right ) x^2}{\left (2-3 \left (1+\frac {e^3}{3}\right ) x-x^2-x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )}+\frac {2}{\left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )}+\frac {x^3}{\left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )}+\frac {x^2 \log (x)}{\left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )}\right ) \, dx+\int \left (\frac {1}{-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)}+\frac {2}{x \left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right )}+\frac {x}{-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)}\right ) \, dx\\ &=2 \int \frac {1}{x \left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right )} \, dx+7 \int \frac {x}{\left (2-3 \left (1+\frac {e^3}{3}\right ) x-x^2-x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )} \, dx+7 \int \frac {x^3}{\left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )} \, dx+7 \int \frac {x^2 \log (x)}{\left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )} \, dx+14 \int \frac {1}{\left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )} \, dx-\left (7 \left (4+e^3\right )\right ) \int \frac {x^2}{\left (2-3 \left (1+\frac {e^3}{3}\right ) x-x^2-x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )} \, dx+\int \frac {1}{-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)} \, dx+\int \frac {x}{-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)} \, dx\\ &=2 \int \frac {1}{x \left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right )} \, dx+7 \int \frac {x^3}{\left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )} \, dx+7 \int \frac {x^2 \log (x)}{\left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )} \, dx+7 \int \frac {x}{\left (-2+\left (3+e^3\right ) x+x^2+x \log (x)\right ) \left (7 x+e^{3+x} x+e^x \left (-2+3 x+x^2\right )+e^x x \log (x)\right )} \, dx+14 \int \frac {1}{\left (-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)\right ) \left (2 e^x-7 x-3 e^x \left (1+\frac {e^3}{3}\right ) x-e^x x^2-e^x x \log (x)\right )} \, dx-\left (7 \left (4+e^3\right )\right ) \int \frac {x^2}{\left (-2+\left (3+e^3\right ) x+x^2+x \log (x)\right ) \left (7 x+e^{3+x} x+e^x \left (-2+3 x+x^2\right )+e^x x \log (x)\right )} \, dx+\int \frac {1}{-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)} \, dx+\int \frac {x}{-2+3 \left (1+\frac {e^3}{3}\right ) x+x^2+x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.98, size = 45, normalized size = 1.55 \begin {gather*} -x-\log (x)+\log \left (-2 e^x+7 x+3 e^x x+e^{3+x} x+e^x x^2+e^x x \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7*x^2 + E^x*(2 + x + x^2))/(7*x^2 + E^x*(-2*x + 3*x^2 + E^3*x^2 + x^3) + E^x*x^2*Log[x]),x]

[Out]

-x - Log[x] + Log[-2*E^x + 7*x + 3*E^x*x + E^(3 + x)*x + E^x*x^2 + E^x*x*Log[x]]

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fricas [A]  time = 0.53, size = 34, normalized size = 1.17 \begin {gather*} \log \left (\frac {{\left (x e^{x} \log \relax (x) + {\left (x^{2} + x e^{3} + 3 \, x - 2\right )} e^{x} + 7 \, x\right )} e^{\left (-x\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+x+2)*exp(x)-7*x^2)/(x^2*exp(x)*log(x)+(x^2*exp(1)*exp(2)+x^3+3*x^2-2*x)*exp(x)+7*x^2),x, algor
ithm="fricas")

[Out]

log((x*e^x*log(x) + (x^2 + x*e^3 + 3*x - 2)*e^x + 7*x)*e^(-x)/x)

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giac [A]  time = 1.98, size = 40, normalized size = 1.38 \begin {gather*} -x + \log \left (x^{2} e^{x} + x e^{x} \log \relax (x) + x e^{\left (x + 3\right )} + 3 \, x e^{x} + 7 \, x - 2 \, e^{x}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+x+2)*exp(x)-7*x^2)/(x^2*exp(x)*log(x)+(x^2*exp(1)*exp(2)+x^3+3*x^2-2*x)*exp(x)+7*x^2),x, algor
ithm="giac")

[Out]

-x + log(x^2*e^x + x*e^x*log(x) + x*e^(x + 3) + 3*x*e^x + 7*x - 2*e^x) - log(x)

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maple [A]  time = 0.10, size = 38, normalized size = 1.31

method	result	size
risch	\(\ln \left (\ln \relax (x )+\frac {\left ({\mathrm e}^{3+x} x +{\mathrm e}^{x} x^{2}+3 \,{\mathrm e}^{x} x +7 x -2 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x}\right )\)	\(38\)
norman	\(-\ln \relax (x )-x +\ln \left ({\mathrm e}^{2} {\mathrm e} \,{\mathrm e}^{x} x +x \,{\mathrm e}^{x} \ln \relax (x )+{\mathrm e}^{x} x^{2}+3 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{x}+7 x \right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+x+2)*exp(x)-7*x^2)/(x^2*exp(x)*ln(x)+(x^2*exp(1)*exp(2)+x^3+3*x^2-2*x)*exp(x)+7*x^2),x,method=_RETUR
NVERBOSE)

[Out]

ln(ln(x)+(exp(3+x)*x+exp(x)*x^2+3*exp(x)*x+7*x-2*exp(x))/x*exp(-x))

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maxima [B]  time = 0.40, size = 65, normalized size = 2.24 \begin {gather*} -x + \log \left (\frac {{\left (x^{2} + x {\left (e^{3} + 3\right )} + x \log \relax (x) - 2\right )} e^{x} + 7 \, x}{x^{2} + x {\left (e^{3} + 3\right )} + x \log \relax (x) - 2}\right ) + \log \left (\frac {x^{2} + x {\left (e^{3} + 3\right )} + x \log \relax (x) - 2}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+x+2)*exp(x)-7*x^2)/(x^2*exp(x)*log(x)+(x^2*exp(1)*exp(2)+x^3+3*x^2-2*x)*exp(x)+7*x^2),x, algor
ithm="maxima")

[Out]

-x + log(((x^2 + x*(e^3 + 3) + x*log(x) - 2)*e^x + 7*x)/(x^2 + x*(e^3 + 3) + x*log(x) - 2)) + log((x^2 + x*(e^
3 + 3) + x*log(x) - 2)/x)

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mupad [B]  time = 3.46, size = 19, normalized size = 0.66 \begin {gather*} \ln \left (x+7\,{\mathrm {e}}^{-x}+{\mathrm {e}}^3+\ln \relax (x)-\frac {2}{x}+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(7*x^2 - exp(x)*(x + x^2 + 2))/(exp(x)*(x^2*exp(3) - 2*x + 3*x^2 + x^3) + 7*x^2 + x^2*exp(x)*log(x)),x)

[Out]

log(x + 7*exp(-x) + exp(3) + log(x) - 2/x + 3)

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sympy [A]  time = 2.46, size = 48, normalized size = 1.66 \begin {gather*} - x + \log {\left (\frac {7 x}{x^{2} + x \log {\relax (x )} + 3 x + x e^{3} - 2} + e^{x} \right )} + \log {\left (\log {\relax (x )} + \frac {x^{2} + 3 x + x e^{3} - 2}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(7*x/(x**2 + x*log(x) + 3*x + x*exp(3) - 2) + exp(x)) + log(log(x) + (x**2 + 3*x + x*exp(3) - 2)/x)

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