3.84.70 \(\int \frac {2 x^5+e^{\frac {36 x^4+12 x^5+x^6+4 x^{2 x}+x^x (24 x^2+4 x^3)}{4 x^4}} (6 x^5+x^6+x^{2 x} (-8+4 x+4 x \log (x))+x^x (-24 x^2+10 x^3+2 x^4+(12 x^3+2 x^4) \log (x)))}{2 x^5} \, dx\)

Optimal. Leaf size=18 \[ e^{\left (3+\frac {x}{2}+x^{-2+x}\right )^2}+x \]

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Rubi [F]  time = 3.95, 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 x^5+\exp \left (\frac {36 x^4+12 x^5+x^6+4 x^{2 x}+x^x \left (24 x^2+4 x^3\right )}{4 x^4}\right ) \left (6 x^5+x^6+x^{2 x} (-8+4 x+4 x \log (x))+x^x \left (-24 x^2+10 x^3+2 x^4+\left (12 x^3+2 x^4\right ) \log (x)\right )\right )}{2 x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x^5 + E^((36*x^4 + 12*x^5 + x^6 + 4*x^(2*x) + x^x*(24*x^2 + 4*x^3))/(4*x^4))*(6*x^5 + x^6 + x^(2*x)*(-8
+ 4*x + 4*x*Log[x]) + x^x*(-24*x^2 + 10*x^3 + 2*x^4 + (12*x^3 + 2*x^4)*Log[x])))/(2*x^5),x]

[Out]

x + 3*Defer[Int][E^((6*x^2 + x^3 + 2*x^x)^2/(4*x^4)), x] + Defer[Int][E^((6*x^2 + x^3 + 2*x^x)^2/(4*x^4))*x, x
]/2 - 12*Defer[Int][E^((6*x^2 + x^3 + 2*x^x)^2/(4*x^4))*x^(-3 + x), x] + 5*Defer[Int][E^((6*x^2 + x^3 + 2*x^x)
^2/(4*x^4))*x^(-2 + x), x] + 6*Log[x]*Defer[Int][E^((6*x^2 + x^3 + 2*x^x)^2/(4*x^4))*x^(-2 + x), x] + Defer[In
t][E^((6*x^2 + x^3 + 2*x^x)^2/(4*x^4))*x^(-1 + x), x] + Log[x]*Defer[Int][E^((6*x^2 + x^3 + 2*x^x)^2/(4*x^4))*
x^(-1 + x), x] - 4*Defer[Int][E^((6*x^2 + x^3 + 2*x^x)^2/(4*x^4))*x^(-5 + 2*x), x] + 2*Defer[Int][E^((6*x^2 +
x^3 + 2*x^x)^2/(4*x^4))*x^(-4 + 2*x), x] + 2*Log[x]*Defer[Int][E^((6*x^2 + x^3 + 2*x^x)^2/(4*x^4))*x^(-4 + 2*x
), x] - 6*Defer[Int][Defer[Int][E^((6*x^2 + x^3 + 2*x^x)^2/(4*x^4))*x^(-2 + x), x]/x, x] - Defer[Int][Defer[In
t][E^((6*x^2 + x^3 + 2*x^x)^2/(4*x^4))*x^(-1 + x), x]/x, x] - 2*Defer[Int][Defer[Int][E^((6*x^2 + x^3 + 2*x^x)
^2/(4*x^4))*x^(-4 + 2*x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {2 x^5+\exp \left (\frac {36 x^4+12 x^5+x^6+4 x^{2 x}+x^x \left (24 x^2+4 x^3\right )}{4 x^4}\right ) \left (6 x^5+x^6+x^{2 x} (-8+4 x+4 x \log (x))+x^x \left (-24 x^2+10 x^3+2 x^4+\left (12 x^3+2 x^4\right ) \log (x)\right )\right )}{x^5} \, dx\\ &=\frac {1}{2} \int \left (2+6 e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}}+e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x+4 e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-5+2 x} (-2+x+x \log (x))+2 e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-3+x} \left (-12+5 x+x^2+6 x \log (x)+x^2 \log (x)\right )\right ) \, dx\\ &=x+\frac {1}{2} \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x \, dx+2 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-5+2 x} (-2+x+x \log (x)) \, dx+3 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} \, dx+\int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-3+x} \left (-12+5 x+x^2+6 x \log (x)+x^2 \log (x)\right ) \, dx\\ &=x+\frac {1}{2} \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x \, dx+2 \int \left (-2 e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-5+2 x}+e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-4+2 x}+e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-4+2 x} \log (x)\right ) \, dx+3 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} \, dx+\int \left (-12 e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-3+x}+5 e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-2+x}+e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-1+x}+6 e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-2+x} \log (x)+e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-1+x} \log (x)\right ) \, dx\\ &=x+\frac {1}{2} \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x \, dx+2 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-4+2 x} \, dx+2 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-4+2 x} \log (x) \, dx+3 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} \, dx-4 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-5+2 x} \, dx+5 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-2+x} \, dx+6 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-2+x} \log (x) \, dx-12 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-3+x} \, dx+\int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-1+x} \, dx+\int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-1+x} \log (x) \, dx\\ &=x+\frac {1}{2} \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x \, dx+2 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-4+2 x} \, dx-2 \int \frac {\int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-4+2 x} \, dx}{x} \, dx+3 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} \, dx-4 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-5+2 x} \, dx+5 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-2+x} \, dx-6 \int \frac {\int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-2+x} \, dx}{x} \, dx-12 \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-3+x} \, dx+\log (x) \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-1+x} \, dx+(2 \log (x)) \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-4+2 x} \, dx+(6 \log (x)) \int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-2+x} \, dx+\int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-1+x} \, dx-\int \frac {\int e^{\frac {\left (6 x^2+x^3+2 x^x\right )^2}{4 x^4}} x^{-1+x} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.16, size = 32, normalized size = 1.78 \begin {gather*} e^{9+3 x+\frac {x^2}{4}+x^{-4+2 x}+x^{-2+x} (6+x)}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^5 + E^((36*x^4 + 12*x^5 + x^6 + 4*x^(2*x) + x^x*(24*x^2 + 4*x^3))/(4*x^4))*(6*x^5 + x^6 + x^(2*
x)*(-8 + 4*x + 4*x*Log[x]) + x^x*(-24*x^2 + 10*x^3 + 2*x^4 + (12*x^3 + 2*x^4)*Log[x])))/(2*x^5),x]

[Out]

E^(9 + 3*x + x^2/4 + x^(-4 + 2*x) + x^(-2 + x)*(6 + x)) + x

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fricas [B]  time = 0.83, size = 43, normalized size = 2.39 \begin {gather*} x + e^{\left (\frac {x^{6} + 12 \, x^{5} + 36 \, x^{4} + 4 \, {\left (x^{3} + 6 \, x^{2}\right )} x^{x} + 4 \, x^{2 \, x}}{4 \, x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((4*x*log(x)+4*x-8)*exp(x*log(x))^2+((2*x^4+12*x^3)*log(x)+2*x^4+10*x^3-24*x^2)*exp(x*log(x))+x
^6+6*x^5)*exp(1/4*(4*exp(x*log(x))^2+(4*x^3+24*x^2)*exp(x*log(x))+x^6+12*x^5+36*x^4)/x^4)+2*x^5)/x^5,x, algori
thm="fricas")

[Out]

x + e^(1/4*(x^6 + 12*x^5 + 36*x^4 + 4*(x^3 + 6*x^2)*x^x + 4*x^(2*x))/x^4)

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giac [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(1/2*(((4*x*log(x)+4*x-8)*exp(x*log(x))^2+((2*x^4+12*x^3)*log(x)+2*x^4+10*x^3-24*x^2)*exp(x*log(x))+x
^6+6*x^5)*exp(1/4*(4*exp(x*log(x))^2+(4*x^3+24*x^2)*exp(x*log(x))+x^6+12*x^5+36*x^4)/x^4)+2*x^5)/x^5,x, algori
thm="giac")

[Out]

Timed out

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maple [A]  time = 0.15, size = 25, normalized size = 1.39

method	result	size
risch	\(x +{\mathrm e}^{\frac {\left (x^{3}+6 x^{2}+2 x^{x}\right )^{2}}{4 x^{4}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(((4*x*ln(x)+4*x-8)*exp(x*ln(x))^2+((2*x^4+12*x^3)*ln(x)+2*x^4+10*x^3-24*x^2)*exp(x*ln(x))+x^6+6*x^5)*
exp(1/4*(4*exp(x*ln(x))^2+(4*x^3+24*x^2)*exp(x*ln(x))+x^6+12*x^5+36*x^4)/x^4)+2*x^5)/x^5,x,method=_RETURNVERBO
SE)

[Out]

x+exp(1/4*(x^3+6*x^2+2*x^x)^2/x^4)

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maxima [B]  time = 0.50, size = 37, normalized size = 2.06 \begin {gather*} x + e^{\left (\frac {1}{4} \, x^{2} + 3 \, x + \frac {x^{x}}{x} + \frac {6 \, x^{x}}{x^{2}} + \frac {x^{2 \, x}}{x^{4}} + 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((4*x*log(x)+4*x-8)*exp(x*log(x))^2+((2*x^4+12*x^3)*log(x)+2*x^4+10*x^3-24*x^2)*exp(x*log(x))+x
^6+6*x^5)*exp(1/4*(4*exp(x*log(x))^2+(4*x^3+24*x^2)*exp(x*log(x))+x^6+12*x^5+36*x^4)/x^4)+2*x^5)/x^5,x, algori
thm="maxima")

[Out]

x + e^(1/4*x^2 + 3*x + x^x/x + 6*x^x/x^2 + x^(2*x)/x^4 + 9)

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mupad [B]  time = 5.64, size = 42, normalized size = 2.33 \begin {gather*} x+{\left ({\mathrm {e}}^{x^2}\right )}^{1/4}\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{\frac {x^{2\,x}}{x^4}}\,{\mathrm {e}}^9\,{\mathrm {e}}^{\frac {x^x}{x}}\,{\mathrm {e}}^{\frac {6\,x^x}{x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5 + (exp((exp(2*x*log(x)) + (exp(x*log(x))*(24*x^2 + 4*x^3))/4 + 9*x^4 + 3*x^5 + x^6/4)/x^4)*(exp(2*x*l
og(x))*(4*x + 4*x*log(x) - 8) + exp(x*log(x))*(log(x)*(12*x^3 + 2*x^4) - 24*x^2 + 10*x^3 + 2*x^4) + 6*x^5 + x^
6))/2)/x^5,x)

[Out]

x + exp(x^2)^(1/4)*exp(3*x)*exp(x^(2*x)/x^4)*exp(9)*exp(x^x/x)*exp((6*x^x)/x^2)

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sympy [B]  time = 0.98, size = 48, normalized size = 2.67 \begin {gather*} x + e^{\frac {\frac {x^{6}}{4} + 3 x^{5} + 9 x^{4} + \frac {\left (4 x^{3} + 24 x^{2}\right ) e^{x \log {\relax (x )}}}{4} + e^{2 x \log {\relax (x )}}}{x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((4*x*ln(x)+4*x-8)*exp(x*ln(x))**2+((2*x**4+12*x**3)*ln(x)+2*x**4+10*x**3-24*x**2)*exp(x*ln(x))
+x**6+6*x**5)*exp(1/4*(4*exp(x*ln(x))**2+(4*x**3+24*x**2)*exp(x*ln(x))+x**6+12*x**5+36*x**4)/x**4)+2*x**5)/x**
5,x)

[Out]

x + exp((x**6/4 + 3*x**5 + 9*x**4 + (4*x**3 + 24*x**2)*exp(x*log(x))/4 + exp(2*x*log(x)))/x**4)

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