3.94.71 \(\int \frac {-x^2+2 x^3+e^{\frac {2 (30-40 x^2+x^3+x^2 \log (x))}{x}} (-60-78 x^2+4 x^3+2 x^2 \log (x))+e^{\frac {30-40 x^2+x^3+x^2 \log (x)}{x}} (-60 x+2 x^2-78 x^3+4 x^4+2 x^3 \log (x))}{x^2} \, dx\)

Optimal. Leaf size=29 \[ -x+\left (e^{\frac {10 \left (3-4 x^2\right )}{x}+x (x+\log (x))}+x\right )^2 \]

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Rubi [F]  time = 1.53, 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 {-x^2+2 x^3+e^{\frac {2 \left (30-40 x^2+x^3+x^2 \log (x)\right )}{x}} \left (-60-78 x^2+4 x^3+2 x^2 \log (x)\right )+e^{\frac {30-40 x^2+x^3+x^2 \log (x)}{x}} \left (-60 x+2 x^2-78 x^3+4 x^4+2 x^3 \log (x)\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^2 + 2*x^3 + E^((2*(30 - 40*x^2 + x^3 + x^2*Log[x]))/x)*(-60 - 78*x^2 + 4*x^3 + 2*x^2*Log[x]) + E^((30
- 40*x^2 + x^3 + x^2*Log[x])/x)*(-60*x + 2*x^2 - 78*x^3 + 4*x^4 + 2*x^3*Log[x]))/x^2,x]

[Out]

-x + x^2 - 60*Defer[Int][E^(30/x - 40*x + x^2)*x^(-1 + x), x] + 2*Defer[Int][E^(30/x - 40*x + x^2)*x^x, x] - 7
8*Defer[Int][E^((2*(30 - 40*x^2 + x^3))/x)*x^(2*x), x] + 2*Log[x]*Defer[Int][E^((2*(30 - 40*x^2 + x^3))/x)*x^(
2*x), x] - 78*Defer[Int][E^(30/x - 40*x + x^2)*x^(1 + x), x] + 2*Log[x]*Defer[Int][E^(30/x - 40*x + x^2)*x^(1
+ x), x] + 4*Defer[Int][E^(30/x - 40*x + x^2)*x^(2 + x), x] - 60*Defer[Int][E^((2*(30 - 40*x^2 + x^3))/x)*x^(-
2 + 2*x), x] + 4*Defer[Int][E^((2*(30 - 40*x^2 + x^3))/x)*x^(1 + 2*x), x] - 2*Defer[Int][Defer[Int][E^((2*(30
- 40*x^2 + x^3))/x)*x^(2*x), x]/x, x] - 2*Defer[Int][Defer[Int][E^(30/x - 40*x + x^2)*x^(1 + x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+2 x+2 e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{-2+2 x} \left (-30-39 x^2+2 x^3+x^2 \log (x)\right )+2 e^{\frac {30}{x}-40 x+x^2} x^{-1+x} \left (-30+x-39 x^2+2 x^3+x^2 \log (x)\right )\right ) \, dx\\ &=-x+x^2+2 \int e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{-2+2 x} \left (-30-39 x^2+2 x^3+x^2 \log (x)\right ) \, dx+2 \int e^{\frac {30}{x}-40 x+x^2} x^{-1+x} \left (-30+x-39 x^2+2 x^3+x^2 \log (x)\right ) \, dx\\ &=-x+x^2+2 \int \left (-39 e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{2 x}-30 e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{-2+2 x}+2 e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{1+2 x}+e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{2 x} \log (x)\right ) \, dx+2 \int \left (-30 e^{\frac {30}{x}-40 x+x^2} x^{-1+x}+e^{\frac {30}{x}-40 x+x^2} x^x-39 e^{\frac {30}{x}-40 x+x^2} x^{1+x}+2 e^{\frac {30}{x}-40 x+x^2} x^{2+x}+e^{\frac {30}{x}-40 x+x^2} x^{1+x} \log (x)\right ) \, dx\\ &=-x+x^2+2 \int e^{\frac {30}{x}-40 x+x^2} x^x \, dx+2 \int e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{2 x} \log (x) \, dx+2 \int e^{\frac {30}{x}-40 x+x^2} x^{1+x} \log (x) \, dx+4 \int e^{\frac {30}{x}-40 x+x^2} x^{2+x} \, dx+4 \int e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{1+2 x} \, dx-60 \int e^{\frac {30}{x}-40 x+x^2} x^{-1+x} \, dx-60 \int e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{-2+2 x} \, dx-78 \int e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{2 x} \, dx-78 \int e^{\frac {30}{x}-40 x+x^2} x^{1+x} \, dx\\ &=-x+x^2+2 \int e^{\frac {30}{x}-40 x+x^2} x^x \, dx-2 \int \frac {\int e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{2 x} \, dx}{x} \, dx-2 \int \frac {\int e^{\frac {30}{x}-40 x+x^2} x^{1+x} \, dx}{x} \, dx+4 \int e^{\frac {30}{x}-40 x+x^2} x^{2+x} \, dx+4 \int e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{1+2 x} \, dx-60 \int e^{\frac {30}{x}-40 x+x^2} x^{-1+x} \, dx-60 \int e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{-2+2 x} \, dx-78 \int e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{2 x} \, dx-78 \int e^{\frac {30}{x}-40 x+x^2} x^{1+x} \, dx+(2 \log (x)) \int e^{\frac {2 \left (30-40 x^2+x^3\right )}{x}} x^{2 x} \, dx+(2 \log (x)) \int e^{\frac {30}{x}-40 x+x^2} x^{1+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.05, size = 50, normalized size = 1.72 \begin {gather*} -x+x^2+e^{\frac {60}{x}-80 x+2 x^2} x^{2 x}+2 e^{\frac {30}{x}-40 x+x^2} x^{1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + 2*x^3 + E^((2*(30 - 40*x^2 + x^3 + x^2*Log[x]))/x)*(-60 - 78*x^2 + 4*x^3 + 2*x^2*Log[x]) + E
^((30 - 40*x^2 + x^3 + x^2*Log[x])/x)*(-60*x + 2*x^2 - 78*x^3 + 4*x^4 + 2*x^3*Log[x]))/x^2,x]

[Out]

-x + x^2 + E^(60/x - 80*x + 2*x^2)*x^(2*x) + 2*E^(30/x - 40*x + x^2)*x^(1 + x)

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fricas [A]  time = 0.82, size = 53, normalized size = 1.83 \begin {gather*} x^{2} + 2 \, x e^{\left (\frac {x^{3} + x^{2} \log \relax (x) - 40 \, x^{2} + 30}{x}\right )} - x + e^{\left (\frac {2 \, {\left (x^{3} + x^{2} \log \relax (x) - 40 \, x^{2} + 30\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(x)+4*x^3-78*x^2-60)*exp((x^2*log(x)+x^3-40*x^2+30)/x)^2+(2*x^3*log(x)+4*x^4-78*x^3+2*x^2
-60*x)*exp((x^2*log(x)+x^3-40*x^2+30)/x)+2*x^3-x^2)/x^2,x, algorithm="fricas")

[Out]

x^2 + 2*x*e^((x^3 + x^2*log(x) - 40*x^2 + 30)/x) - x + e^(2*(x^3 + x^2*log(x) - 40*x^2 + 30)/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{3} - x^{2} + 2 \, {\left (2 \, x^{3} + x^{2} \log \relax (x) - 39 \, x^{2} - 30\right )} e^{\left (\frac {2 \, {\left (x^{3} + x^{2} \log \relax (x) - 40 \, x^{2} + 30\right )}}{x}\right )} + 2 \, {\left (2 \, x^{4} + x^{3} \log \relax (x) - 39 \, x^{3} + x^{2} - 30 \, x\right )} e^{\left (\frac {x^{3} + x^{2} \log \relax (x) - 40 \, x^{2} + 30}{x}\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(x)+4*x^3-78*x^2-60)*exp((x^2*log(x)+x^3-40*x^2+30)/x)^2+(2*x^3*log(x)+4*x^4-78*x^3+2*x^2
-60*x)*exp((x^2*log(x)+x^3-40*x^2+30)/x)+2*x^3-x^2)/x^2,x, algorithm="giac")

[Out]

integrate((2*x^3 - x^2 + 2*(2*x^3 + x^2*log(x) - 39*x^2 - 30)*e^(2*(x^3 + x^2*log(x) - 40*x^2 + 30)/x) + 2*(2*
x^4 + x^3*log(x) - 39*x^3 + x^2 - 30*x)*e^((x^3 + x^2*log(x) - 40*x^2 + 30)/x))/x^2, x)

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maple [A]  time = 0.09, size = 51, normalized size = 1.76

method	result	size
risch	\(x^{2}+2 x^{x} {\mathrm e}^{\frac {x^{3}-40 x^{2}+30}{x}} x +x^{2 x} {\mathrm e}^{\frac {2 x^{3}-80 x^{2}+60}{x}}-x\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2*ln(x)+4*x^3-78*x^2-60)*exp((x^2*ln(x)+x^3-40*x^2+30)/x)^2+(2*x^3*ln(x)+4*x^4-78*x^3+2*x^2-60*x)*ex
p((x^2*ln(x)+x^3-40*x^2+30)/x)+2*x^3-x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

x^2+2*x^x*exp((x^3-40*x^2+30)/x)*x+(x^x)^2*exp(2*(x^3-40*x^2+30)/x)-x

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maxima [A]  time = 0.43, size = 50, normalized size = 1.72 \begin {gather*} x^{2} + {\left (2 \, x e^{\left (x^{2} + x \log \relax (x) + 40 \, x + \frac {30}{x}\right )} + e^{\left (2 \, x^{2} + 2 \, x \log \relax (x) + \frac {60}{x}\right )}\right )} e^{\left (-80 \, x\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(x)+4*x^3-78*x^2-60)*exp((x^2*log(x)+x^3-40*x^2+30)/x)^2+(2*x^3*log(x)+4*x^4-78*x^3+2*x^2
-60*x)*exp((x^2*log(x)+x^3-40*x^2+30)/x)+2*x^3-x^2)/x^2,x, algorithm="maxima")

[Out]

x^2 + (2*x*e^(x^2 + x*log(x) + 40*x + 30/x) + e^(2*x^2 + 2*x*log(x) + 60/x))*e^(-80*x) - x

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mupad [B]  time = 9.27, size = 47, normalized size = 1.62 \begin {gather*} x^{2\,x}\,{\mathrm {e}}^{\frac {60}{x}-80\,x+2\,x^2}-x+x^2+2\,x\,x^x\,{\mathrm {e}}^{\frac {30}{x}-40\,x+x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*(x^2*log(x) - 40*x^2 + x^3 + 30))/x)*(2*x^2*log(x) - 78*x^2 + 4*x^3 - 60) + exp((x^2*log(x) - 40*x
^2 + x^3 + 30)/x)*(2*x^3*log(x) - 60*x + 2*x^2 - 78*x^3 + 4*x^4) - x^2 + 2*x^3)/x^2,x)

[Out]

x^(2*x)*exp(60/x - 80*x + 2*x^2) - x + x^2 + 2*x*x^x*exp(30/x - 40*x + x^2)

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sympy [B]  time = 0.43, size = 51, normalized size = 1.76 \begin {gather*} x^{2} + 2 x e^{\frac {x^{3} + x^{2} \log {\relax (x )} - 40 x^{2} + 30}{x}} - x + e^{\frac {2 \left (x^{3} + x^{2} \log {\relax (x )} - 40 x^{2} + 30\right )}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2*ln(x)+4*x**3-78*x**2-60)*exp((x**2*ln(x)+x**3-40*x**2+30)/x)**2+(2*x**3*ln(x)+4*x**4-78*x**
3+2*x**2-60*x)*exp((x**2*ln(x)+x**3-40*x**2+30)/x)+2*x**3-x**2)/x**2,x)

[Out]

x**2 + 2*x*exp((x**3 + x**2*log(x) - 40*x**2 + 30)/x) - x + exp(2*(x**3 + x**2*log(x) - 40*x**2 + 30)/x)

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