3.90.52 \(\int \frac {e^{\frac {e^{3+x} (25-x)+e^{x^2} x}{x}} (2 e^{x^2} x^3+e^{3+x} (-25+25 x-x^2))}{x^2} \, dx\)

Optimal. Leaf size=21 \[ e^{e^{x^2}-\frac {e^{3+x} (-25+x)}{x}} \]

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Rubi [F]  time = 3.18, 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^{\frac {e^{3+x} (25-x)+e^{x^2} x}{x}} \left (2 e^{x^2} x^3+e^{3+x} \left (-25+25 x-x^2\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((E^(3 + x)*(25 - x) + E^x^2*x)/x)*(2*E^x^2*x^3 + E^(3 + x)*(-25 + 25*x - x^2)))/x^2,x]

[Out]

-Defer[Int][E^(3 + E^x^2 + (E^(3 + x)*(25 - x))/x + x), x] - 25*Defer[Int][E^(3 + E^x^2 + (E^(3 + x)*(25 - x))
/x + x)/x^2, x] + 25*Defer[Int][E^(3 + E^x^2 + (E^(3 + x)*(25 - x))/x + x)/x, x] + 2*Defer[Int][E^(E^x^2 + (E^
(3 + x)*(25 - x))/x + x^2)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{e^{x^2}+\frac {e^{3+x} (25-x)}{x}} \left (2 e^{x^2} x^3+e^{3+x} \left (-25+25 x-x^2\right )\right )}{x^2} \, dx\\ &=\int \left (2 e^{e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x^2} x-\frac {e^{3+e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x} \left (25-25 x+x^2\right )}{x^2}\right ) \, dx\\ &=2 \int e^{e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x^2} x \, dx-\int \frac {e^{3+e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x} \left (25-25 x+x^2\right )}{x^2} \, dx\\ &=2 \int e^{e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x^2} x \, dx-\int \left (e^{3+e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x}+\frac {25 e^{3+e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x}}{x^2}-\frac {25 e^{3+e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x}}{x}\right ) \, dx\\ &=2 \int e^{e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x^2} x \, dx-25 \int \frac {e^{3+e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x}}{x^2} \, dx+25 \int \frac {e^{3+e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x}}{x} \, dx-\int e^{3+e^{x^2}+\frac {e^{3+x} (25-x)}{x}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.49, size = 21, normalized size = 1.00 \begin {gather*} e^{e^{x^2}-\frac {e^{3+x} (-25+x)}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((E^(3 + x)*(25 - x) + E^x^2*x)/x)*(2*E^x^2*x^3 + E^(3 + x)*(-25 + 25*x - x^2)))/x^2,x]

[Out]

E^(E^x^2 - (E^(3 + x)*(-25 + x))/x)

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fricas [A]  time = 0.77, size = 21, normalized size = 1.00 \begin {gather*} e^{\left (\frac {x e^{\left (x^{2}\right )} - {\left (x - 25\right )} e^{\left (x + 3\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3*exp(x^2)+(-x^2+25*x-25)*exp(3)*exp(x))*exp((exp(x^2)*x+(-x+25)*exp(3)*exp(x))/x)/x^2,x, algor
ithm="fricas")

[Out]

e^((x*e^(x^2) - (x - 25)*e^(x + 3))/x)

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giac [A]  time = 0.16, size = 21, normalized size = 1.00 \begin {gather*} e^{\left (\frac {25 \, e^{\left (x + 3\right )}}{x} + e^{\left (x^{2}\right )} - e^{\left (x + 3\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3*exp(x^2)+(-x^2+25*x-25)*exp(3)*exp(x))*exp((exp(x^2)*x+(-x+25)*exp(3)*exp(x))/x)/x^2,x, algor
ithm="giac")

[Out]

e^(25*e^(x + 3)/x + e^(x^2) - e^(x + 3))

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maple [A]  time = 0.07, size = 27, normalized size = 1.29

method	result	size
risch	\({\mathrm e}^{-\frac {-{\mathrm e}^{x^{2}} x +{\mathrm e}^{3+x} x -25 \,{\mathrm e}^{3+x}}{x}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3*exp(x^2)+(-x^2+25*x-25)*exp(3)*exp(x))*exp((exp(x^2)*x+(-x+25)*exp(3)*exp(x))/x)/x^2,x,method=_RETU
RNVERBOSE)

[Out]

exp(-(-exp(x^2)*x+exp(3+x)*x-25*exp(3+x))/x)

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maxima [A]  time = 0.45, size = 21, normalized size = 1.00 \begin {gather*} e^{\left (\frac {25 \, e^{\left (x + 3\right )}}{x} + e^{\left (x^{2}\right )} - e^{\left (x + 3\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3*exp(x^2)+(-x^2+25*x-25)*exp(3)*exp(x))*exp((exp(x^2)*x+(-x+25)*exp(3)*exp(x))/x)/x^2,x, algor
ithm="maxima")

[Out]

e^(25*e^(x + 3)/x + e^(x^2) - e^(x + 3))

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mupad [B]  time = 6.42, size = 23, normalized size = 1.10 \begin {gather*} {\mathrm {e}}^{-{\mathrm {e}}^3\,{\mathrm {e}}^x}\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{\frac {25\,{\mathrm {e}}^3\,{\mathrm {e}}^x}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x*exp(x^2) - exp(3)*exp(x)*(x - 25))/x)*(2*x^3*exp(x^2) - exp(3)*exp(x)*(x^2 - 25*x + 25)))/x^2,x)

[Out]

exp(-exp(3)*exp(x))*exp(exp(x^2))*exp((25*exp(3)*exp(x))/x)

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sympy [A]  time = 0.37, size = 19, normalized size = 0.90 \begin {gather*} e^{\frac {x e^{x^{2}} + \left (25 - x\right ) e^{3} e^{x}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**3*exp(x**2)+(-x**2+25*x-25)*exp(3)*exp(x))*exp((exp(x**2)*x+(-x+25)*exp(3)*exp(x))/x)/x**2,x)

[Out]

exp((x*exp(x**2) + (25 - x)*exp(3)*exp(x))/x)

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