3.49.34 \(\int \frac {e^{e^5} (e^2 (-36-12 x-x^2)+e^{x^2} (-36-12 x-73 x^2-24 x^3-2 x^4))+x^{\frac {1}{6+x}} (e^{e^5} (42+13 x+x^2)-e^{e^5} x \log (x))}{36+12 x+x^2} \, dx\)

Optimal. Leaf size=27 \[ e^{e^5} x \left (-e^2-e^{x^2}+x^{\frac {1}{6+x}}\right ) \]

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Rubi [F]  time = 0.88, 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^5} \left (e^2 \left (-36-12 x-x^2\right )+e^{x^2} \left (-36-12 x-73 x^2-24 x^3-2 x^4\right )\right )+x^{\frac {1}{6+x}} \left (e^{e^5} \left (42+13 x+x^2\right )-e^{e^5} x \log (x)\right )}{36+12 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^E^5*(E^2*(-36 - 12*x - x^2) + E^x^2*(-36 - 12*x - 73*x^2 - 24*x^3 - 2*x^4)) + x^(6 + x)^(-1)*(E^E^5*(42
+ 13*x + x^2) - E^E^5*x*Log[x]))/(36 + 12*x + x^2),x]

[Out]

-(E^(2 + E^5)*x) - E^(E^5 + x^2)*x + (E^E^5*x^(1 + (6 + x)^(-1))*Hypergeometric2F1[1, 1 + (6 + x)^(-1), 2 + (6
+ x)^(-1), -1/6*x])/(6*(1 + (6 + x)^(-1))) + E^E^5*Defer[Int][x^(6 + x)^(-1), x] - E^E^5*Defer[Int][(x^((7 +
x)/(6 + x))*Log[x])/(6 + x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{e^5} \left (e^2 \left (-36-12 x-x^2\right )+e^{x^2} \left (-36-12 x-73 x^2-24 x^3-2 x^4\right )\right )+x^{\frac {1}{6+x}} \left (e^{e^5} \left (42+13 x+x^2\right )-e^{e^5} x \log (x)\right )}{(6+x)^2} \, dx\\ &=\int \left (-e^{e^5} \left (e^2+e^{x^2}+2 e^{x^2} x^2\right )+\frac {e^{e^5} x^{\frac {1}{6+x}} \left (42+13 x+x^2-x \log (x)\right )}{(6+x)^2}\right ) \, dx\\ &=-\left (e^{e^5} \int \left (e^2+e^{x^2}+2 e^{x^2} x^2\right ) \, dx\right )+e^{e^5} \int \frac {x^{\frac {1}{6+x}} \left (42+13 x+x^2-x \log (x)\right )}{(6+x)^2} \, dx\\ &=-e^{2+e^5} x-e^{e^5} \int e^{x^2} \, dx+e^{e^5} \int \left (\frac {x^{\frac {1}{6+x}} (7+x)}{6+x}-\frac {x^{1+\frac {1}{6+x}} \log (x)}{(6+x)^2}\right ) \, dx-\left (2 e^{e^5}\right ) \int e^{x^2} x^2 \, dx\\ &=-e^{2+e^5} x-e^{e^5+x^2} x-\frac {1}{2} e^{e^5} \sqrt {\pi } \text {erfi}(x)+e^{e^5} \int e^{x^2} \, dx+e^{e^5} \int \frac {x^{\frac {1}{6+x}} (7+x)}{6+x} \, dx-e^{e^5} \int \frac {x^{1+\frac {1}{6+x}} \log (x)}{(6+x)^2} \, dx\\ &=-e^{2+e^5} x-e^{e^5+x^2} x+e^{e^5} \int \left (x^{\frac {1}{6+x}}+\frac {x^{\frac {1}{6+x}}}{6+x}\right ) \, dx-e^{e^5} \int \frac {x^{\frac {7+x}{6+x}} \log (x)}{(6+x)^2} \, dx\\ &=-e^{2+e^5} x-e^{e^5+x^2} x+e^{e^5} \int x^{\frac {1}{6+x}} \, dx+e^{e^5} \int \frac {x^{\frac {1}{6+x}}}{6+x} \, dx-e^{e^5} \int \frac {x^{\frac {7+x}{6+x}} \log (x)}{(6+x)^2} \, dx\\ &=-e^{2+e^5} x-e^{e^5+x^2} x+\frac {e^{e^5} x^{1+\frac {1}{6+x}} \, _2F_1\left (1,1+\frac {1}{6+x};2+\frac {1}{6+x};-\frac {x}{6}\right )}{6 \left (1+\frac {1}{6+x}\right )}+e^{e^5} \int x^{\frac {1}{6+x}} \, dx-e^{e^5} \int \frac {x^{\frac {7+x}{6+x}} \log (x)}{(6+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 26, normalized size = 0.96 \begin {gather*} -e^{e^5} x \left (e^2+e^{x^2}-x^{\frac {1}{6+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^5*(E^2*(-36 - 12*x - x^2) + E^x^2*(-36 - 12*x - 73*x^2 - 24*x^3 - 2*x^4)) + x^(6 + x)^(-1)*(E^E
^5*(42 + 13*x + x^2) - E^E^5*x*Log[x]))/(36 + 12*x + x^2),x]

[Out]

-(E^E^5*x*(E^2 + E^x^2 - x^(6 + x)^(-1)))

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fricas [A]  time = 0.73, size = 29, normalized size = 1.07 \begin {gather*} x x^{\left (\frac {1}{x + 6}\right )} e^{\left (e^{5}\right )} - {\left (x e^{2} + x e^{\left (x^{2}\right )}\right )} e^{\left (e^{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(exp(5))*log(x)+(x^2+13*x+42)*exp(exp(5)))*exp(log(x)/(x+6))+((-2*x^4-24*x^3-73*x^2-12*x-36)
*exp(x^2)+(-x^2-12*x-36)*exp(2))*exp(exp(5)))/(x^2+12*x+36),x, algorithm="fricas")

[Out]

x*x^(1/(x + 6))*e^(e^5) - (x*e^2 + x*e^(x^2))*e^(e^5)

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giac [A]  time = 0.31, size = 41, normalized size = 1.52 \begin {gather*} x^{\frac {x}{x + 6}} x^{\frac {7}{x + 6}} e^{\left (e^{5}\right )} - x e^{\left (x^{2} + e^{5}\right )} - x e^{\left (e^{5} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(exp(5))*log(x)+(x^2+13*x+42)*exp(exp(5)))*exp(log(x)/(x+6))+((-2*x^4-24*x^3-73*x^2-12*x-36)
*exp(x^2)+(-x^2-12*x-36)*exp(2))*exp(exp(5)))/(x^2+12*x+36),x, algorithm="giac")

[Out]

x^(x/(x + 6))*x^(7/(x + 6))*e^(e^5) - x*e^(x^2 + e^5) - x*e^(e^5 + 2)

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maple [A]  time = 0.24, size = 32, normalized size = 1.19

method	result	size
risch	\(-x \,{\mathrm e}^{{\mathrm e}^{5}+2}-x \,{\mathrm e}^{x^{2}+{\mathrm e}^{5}}+{\mathrm e}^{{\mathrm e}^{5}} x \,x^{\frac {1}{x +6}}\)	\(32\)
default	\(\frac {{\mathrm e}^{{\mathrm e}^{5}} x^{3} {\mathrm e}^{\frac {\ln \relax (x )}{x +6}}+36 x \,{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{\frac {\ln \relax (x )}{x +6}}+12 x^{2} {\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{\frac {\ln \relax (x )}{x +6}}}{\left (x +6\right )^{2}}-{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{2} x -{\mathrm e}^{{\mathrm e}^{5}} x \,{\mathrm e}^{x^{2}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x*exp(exp(5))*ln(x)+(x^2+13*x+42)*exp(exp(5)))*exp(ln(x)/(x+6))+((-2*x^4-24*x^3-73*x^2-12*x-36)*exp(x^2
)+(-x^2-12*x-36)*exp(2))*exp(exp(5)))/(x^2+12*x+36),x,method=_RETURNVERBOSE)

[Out]

-x*exp(exp(5)+2)-x*exp(x^2+exp(5))+exp(exp(5))*x*x^(1/(x+6))

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maxima [B]  time = 0.41, size = 78, normalized size = 2.89 \begin {gather*} -x e^{\left (x^{2} + e^{5}\right )} + x e^{\left (\frac {\log \relax (x)}{x + 6} + e^{5}\right )} - {\left (x - \frac {36}{x + 6} - 12 \, \log \left (x + 6\right )\right )} e^{\left (e^{5} + 2\right )} - 12 \, {\left (\frac {6}{x + 6} + \log \left (x + 6\right )\right )} e^{\left (e^{5} + 2\right )} + \frac {36 \, e^{\left (e^{5} + 2\right )}}{x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(exp(5))*log(x)+(x^2+13*x+42)*exp(exp(5)))*exp(log(x)/(x+6))+((-2*x^4-24*x^3-73*x^2-12*x-36)
*exp(x^2)+(-x^2-12*x-36)*exp(2))*exp(exp(5)))/(x^2+12*x+36),x, algorithm="maxima")

[Out]

-x*e^(x^2 + e^5) + x*e^(log(x)/(x + 6) + e^5) - (x - 36/(x + 6) - 12*log(x + 6))*e^(e^5 + 2) - 12*(6/(x + 6) +
log(x + 6))*e^(e^5 + 2) + 36*e^(e^5 + 2)/(x + 6)

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mupad [B]  time = 4.17, size = 22, normalized size = 0.81 \begin {gather*} -x\,{\mathrm {e}}^{{\mathrm {e}}^5}\,\left ({\mathrm {e}}^{x^2}+{\mathrm {e}}^2-x^{\frac {1}{x+6}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(x)/(x + 6))*(exp(exp(5))*(13*x + x^2 + 42) - x*exp(exp(5))*log(x)) - exp(exp(5))*(exp(x^2)*(12*x
+ 73*x^2 + 24*x^3 + 2*x^4 + 36) + exp(2)*(12*x + x^2 + 36)))/(12*x + x^2 + 36),x)

[Out]

-x*exp(exp(5))*(exp(x^2) + exp(2) - x^(1/(x + 6)))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(exp(5))*ln(x)+(x**2+13*x+42)*exp(exp(5)))*exp(ln(x)/(x+6))+((-2*x**4-24*x**3-73*x**2-12*x-3
6)*exp(x**2)+(-x**2-12*x-36)*exp(2))*exp(exp(5)))/(x**2+12*x+36),x)

[Out]

Timed out

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