3.62.89 \(\int \frac {e^{e^x x} (5+e^x+x) (1+e^{2 x} (1+x)+e^x (6+6 x+x^2))+2 e^x x \log ^2(7)+(10 x+2 x^2) \log ^2(7)}{5+e^x+x} \, dx\)

Optimal. Leaf size=23 \[ e^{e^x x} \left (5+e^x+x\right )+x^2 \log ^2(7) \]

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Rubi [F]  time = 1.12, 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 {e^{e^x x} \left (5+e^x+x\right ) \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right )+2 e^x x \log ^2(7)+\left (10 x+2 x^2\right ) \log ^2(7)}{5+e^x+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^x*x)*(5 + E^x + x)*(1 + E^(2*x)*(1 + x) + E^x*(6 + 6*x + x^2)) + 2*E^x*x*Log[7]^2 + (10*x + 2*x^2)*L
og[7]^2)/(5 + E^x + x),x]

[Out]

x^2*Log[7]^2 + Defer[Int][E^(E^x*x), x] + 6*Defer[Int][E^(x + E^x*x), x] + Defer[Int][E^(2*x + E^x*x), x] + 6*
Defer[Int][E^(x + E^x*x)*x, x] + Defer[Int][E^(2*x + E^x*x)*x, x] + Defer[Int][E^(x + E^x*x)*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{e^x x} \left (1+6 e^x+e^{2 x}+6 e^x x+e^{2 x} x+e^x x^2\right )+2 x \log ^2(7)\right ) \, dx\\ &=x^2 \log ^2(7)+\int e^{e^x x} \left (1+6 e^x+e^{2 x}+6 e^x x+e^{2 x} x+e^x x^2\right ) \, dx\\ &=x^2 \log ^2(7)+\int e^{e^x x} \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right ) \, dx\\ &=x^2 \log ^2(7)+\int \left (e^{e^x x}+e^{2 x+e^x x} (1+x)+e^{x+e^x x} \left (6+6 x+x^2\right )\right ) \, dx\\ &=x^2 \log ^2(7)+\int e^{e^x x} \, dx+\int e^{2 x+e^x x} (1+x) \, dx+\int e^{x+e^x x} \left (6+6 x+x^2\right ) \, dx\\ &=x^2 \log ^2(7)+\int e^{e^x x} \, dx+\int \left (e^{2 x+e^x x}+e^{2 x+e^x x} x\right ) \, dx+\int \left (6 e^{x+e^x x}+6 e^{x+e^x x} x+e^{x+e^x x} x^2\right ) \, dx\\ &=x^2 \log ^2(7)+6 \int e^{x+e^x x} \, dx+6 \int e^{x+e^x x} x \, dx+\int e^{e^x x} \, dx+\int e^{2 x+e^x x} \, dx+\int e^{2 x+e^x x} x \, dx+\int e^{x+e^x x} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 29, normalized size = 1.26 \begin {gather*} e^{x+e^x x}+e^{e^x x} (5+x)+x^2 \log ^2(7) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^x*x)*(5 + E^x + x)*(1 + E^(2*x)*(1 + x) + E^x*(6 + 6*x + x^2)) + 2*E^x*x*Log[7]^2 + (10*x + 2*
x^2)*Log[7]^2)/(5 + E^x + x),x]

[Out]

E^(x + E^x*x) + E^(E^x*x)*(5 + x) + x^2*Log[7]^2

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fricas [A]  time = 1.05, size = 21, normalized size = 0.91 \begin {gather*} x^{2} \log \relax (7)^{2} + e^{\left (x e^{x} + \log \left (x + e^{x} + 5\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+1)*exp(x)^2+(x^2+6*x+6)*exp(x)+1)*exp(log(exp(x)+5+x)+exp(x)*x)+2*x*log(7)^2*exp(x)+(2*x^2+10*x
)*log(7)^2)/(exp(x)+5+x),x, algorithm="fricas")

[Out]

x^2*log(7)^2 + e^(x*e^x + log(x + e^x + 5))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x e^{x} \log \relax (7)^{2} + 2 \, {\left (x^{2} + 5 \, x\right )} \log \relax (7)^{2} + {\left ({\left (x + 1\right )} e^{\left (2 \, x\right )} + {\left (x^{2} + 6 \, x + 6\right )} e^{x} + 1\right )} e^{\left (x e^{x} + \log \left (x + e^{x} + 5\right )\right )}}{x + e^{x} + 5}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+1)*exp(x)^2+(x^2+6*x+6)*exp(x)+1)*exp(log(exp(x)+5+x)+exp(x)*x)+2*x*log(7)^2*exp(x)+(2*x^2+10*x
)*log(7)^2)/(exp(x)+5+x),x, algorithm="giac")

[Out]

integrate((2*x*e^x*log(7)^2 + 2*(x^2 + 5*x)*log(7)^2 + ((x + 1)*e^(2*x) + (x^2 + 6*x + 6)*e^x + 1)*e^(x*e^x +
log(x + e^x + 5)))/(x + e^x + 5), x)

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maple [A]  time = 0.07, size = 21, normalized size = 0.91

method	result	size
risch	\(\left ({\mathrm e}^{x}+5+x \right ) {\mathrm e}^{{\mathrm e}^{x} x}+x^{2} \ln \relax (7)^{2}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x+1)*exp(x)^2+(x^2+6*x+6)*exp(x)+1)*exp(ln(exp(x)+5+x)+exp(x)*x)+2*x*ln(7)^2*exp(x)+(2*x^2+10*x)*ln(7)^
2)/(exp(x)+5+x),x,method=_RETURNVERBOSE)

[Out]

(exp(x)+5+x)*exp(exp(x)*x)+x^2*ln(7)^2

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maxima [A]  time = 0.46, size = 20, normalized size = 0.87 \begin {gather*} x^{2} \log \relax (7)^{2} + {\left (x + e^{x} + 5\right )} e^{\left (x e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+1)*exp(x)^2+(x^2+6*x+6)*exp(x)+1)*exp(log(exp(x)+5+x)+exp(x)*x)+2*x*log(7)^2*exp(x)+(2*x^2+10*x
)*log(7)^2)/(exp(x)+5+x),x, algorithm="maxima")

[Out]

x^2*log(7)^2 + (x + e^x + 5)*e^(x*e^x)

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mupad [B]  time = 4.50, size = 30, normalized size = 1.30 \begin {gather*} 5\,{\mathrm {e}}^{x\,{\mathrm {e}}^x}+{\mathrm {e}}^{x+x\,{\mathrm {e}}^x}+x^2\,{\ln \relax (7)}^2+x\,{\mathrm {e}}^{x\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(x + exp(x) + 5) + x*exp(x))*(exp(x)*(6*x + x^2 + 6) + exp(2*x)*(x + 1) + 1) + log(7)^2*(10*x + 2*
x^2) + 2*x*exp(x)*log(7)^2)/(x + exp(x) + 5),x)

[Out]

5*exp(x*exp(x)) + exp(x + x*exp(x)) + x^2*log(7)^2 + x*exp(x*exp(x))

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sympy [A]  time = 0.24, size = 20, normalized size = 0.87 \begin {gather*} x^{2} \log {\relax (7 )}^{2} + \left (x + e^{x} + 5\right ) e^{x e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+1)*exp(x)**2+(x**2+6*x+6)*exp(x)+1)*exp(ln(exp(x)+5+x)+exp(x)*x)+2*x*ln(7)**2*exp(x)+(2*x**2+10
*x)*ln(7)**2)/(exp(x)+5+x),x)

[Out]

x**2*log(7)**2 + (x + exp(x) + 5)*exp(x*exp(x))

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