3.51.43 \(\int \frac {4 x^3+2 x^4+e^{\frac {-x^3+e^{x^2} (25-10 x+x^2)}{x^2}} (-x^3+e^{x^2} (-50+10 x+50 x^2-20 x^3+2 x^4))}{x^3} \, dx\)

Optimal. Leaf size=31 \[ e^{e^{x^2} \left (2+\frac {-5-x}{x}\right )^2-x}+(2+x)^2 \]

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Rubi [F]  time = 3.63, 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 {4 x^3+2 x^4+e^{\frac {-x^3+e^{x^2} \left (25-10 x+x^2\right )}{x^2}} \left (-x^3+e^{x^2} \left (-50+10 x+50 x^2-20 x^3+2 x^4\right )\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x^3 + 2*x^4 + E^((-x^3 + E^x^2*(25 - 10*x + x^2))/x^2)*(-x^3 + E^x^2*(-50 + 10*x + 50*x^2 - 20*x^3 + 2*
x^4)))/x^3,x]

[Out]

(2 + x)^2 - Defer[Int][E^((E^x^2*(-5 + x)^2)/x^2 - x), x] - 20*Defer[Int][E^((E^x^2*(-5 + x)^2)/x^2 - x + x^2)
, x] - 50*Defer[Int][E^((E^x^2*(-5 + x)^2)/x^2 - x + x^2)/x^3, x] + 10*Defer[Int][E^((E^x^2*(-5 + x)^2)/x^2 -
x + x^2)/x^2, x] + 50*Defer[Int][E^((E^x^2*(-5 + x)^2)/x^2 - x + x^2)/x, x] + 2*Defer[Int][E^((E^x^2*(-5 + x)^
2)/x^2 - x + x^2)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 (2+x)+\frac {e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x} \left (-50 e^{x^2}+10 e^{x^2} x+50 e^{x^2} x^2-x^3-20 e^{x^2} x^3+2 e^{x^2} x^4\right )}{x^3}\right ) \, dx\\ &=(2+x)^2+\int \frac {e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x} \left (-50 e^{x^2}+10 e^{x^2} x+50 e^{x^2} x^2-x^3-20 e^{x^2} x^3+2 e^{x^2} x^4\right )}{x^3} \, dx\\ &=(2+x)^2+\int \frac {e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x} \left (-x^3+2 e^{x^2} \left (-25+5 x+25 x^2-10 x^3+x^4\right )\right )}{x^3} \, dx\\ &=(2+x)^2+\int \left (-e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x}+\frac {2 e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2} (-5+x) \left (5-5 x^2+x^3\right )}{x^3}\right ) \, dx\\ &=(2+x)^2+2 \int \frac {e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2} (-5+x) \left (5-5 x^2+x^3\right )}{x^3} \, dx-\int e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x} \, dx\\ &=(2+x)^2+2 \int \left (-10 e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2}-\frac {25 e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2}}{x^3}+\frac {5 e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2}}{x^2}+\frac {25 e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2}}{x}+e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2} x\right ) \, dx-\int e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x} \, dx\\ &=(2+x)^2+2 \int e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2} x \, dx+10 \int \frac {e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2}}{x^2} \, dx-20 \int e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2} \, dx-50 \int \frac {e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2}}{x^3} \, dx+50 \int \frac {e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x+x^2}}{x} \, dx-\int e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.52, size = 27, normalized size = 0.87 \begin {gather*} e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x}+4 x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^3 + 2*x^4 + E^((-x^3 + E^x^2*(25 - 10*x + x^2))/x^2)*(-x^3 + E^x^2*(-50 + 10*x + 50*x^2 - 20*x^
3 + 2*x^4)))/x^3,x]

[Out]

E^((E^x^2*(-5 + x)^2)/x^2 - x) + 4*x + x^2

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fricas [A]  time = 0.92, size = 31, normalized size = 1.00 \begin {gather*} x^{2} + 4 \, x + e^{\left (-\frac {x^{3} - {\left (x^{2} - 10 \, x + 25\right )} e^{\left (x^{2}\right )}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^4-20*x^3+50*x^2+10*x-50)*exp(x^2)-x^3)*exp(((x^2-10*x+25)*exp(x^2)-x^3)/x^2)+2*x^4+4*x^3)/x^3
,x, algorithm="fricas")

[Out]

x^2 + 4*x + e^(-(x^3 - (x^2 - 10*x + 25)*e^(x^2))/x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{4} + 4 \, x^{3} - {\left (x^{3} - 2 \, {\left (x^{4} - 10 \, x^{3} + 25 \, x^{2} + 5 \, x - 25\right )} e^{\left (x^{2}\right )}\right )} e^{\left (-\frac {x^{3} - {\left (x^{2} - 10 \, x + 25\right )} e^{\left (x^{2}\right )}}{x^{2}}\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^4-20*x^3+50*x^2+10*x-50)*exp(x^2)-x^3)*exp(((x^2-10*x+25)*exp(x^2)-x^3)/x^2)+2*x^4+4*x^3)/x^3
,x, algorithm="giac")

[Out]

integrate((2*x^4 + 4*x^3 - (x^3 - 2*(x^4 - 10*x^3 + 25*x^2 + 5*x - 25)*e^(x^2))*e^(-(x^3 - (x^2 - 10*x + 25)*e
^(x^2))/x^2))/x^3, x)

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maple [A]  time = 0.08, size = 40, normalized size = 1.29

method	result	size
risch	\(x^{2}+4 x +{\mathrm e}^{-\frac {-x^{2} {\mathrm e}^{x^{2}}+x^{3}+10 \,{\mathrm e}^{x^{2}} x -25 \,{\mathrm e}^{x^{2}}}{x^{2}}}\)	\(40\)
norman	\(\frac {x^{4}+x^{2} {\mathrm e}^{\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}^{x^{2}}-x^{3}}{x^{2}}}+4 x^{3}}{x^{2}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^4-20*x^3+50*x^2+10*x-50)*exp(x^2)-x^3)*exp(((x^2-10*x+25)*exp(x^2)-x^3)/x^2)+2*x^4+4*x^3)/x^3,x,met
hod=_RETURNVERBOSE)

[Out]

x^2+4*x+exp(-(-x^2*exp(x^2)+x^3+10*exp(x^2)*x-25*exp(x^2))/x^2)

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maxima [A]  time = 0.45, size = 34, normalized size = 1.10 \begin {gather*} x^{2} + 4 \, x + e^{\left (-x - \frac {10 \, e^{\left (x^{2}\right )}}{x} + \frac {25 \, e^{\left (x^{2}\right )}}{x^{2}} + e^{\left (x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^4-20*x^3+50*x^2+10*x-50)*exp(x^2)-x^3)*exp(((x^2-10*x+25)*exp(x^2)-x^3)/x^2)+2*x^4+4*x^3)/x^3
,x, algorithm="maxima")

[Out]

x^2 + 4*x + e^(-x - 10*e^(x^2)/x + 25*e^(x^2)/x^2 + e^(x^2))

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mupad [B]  time = 3.59, size = 37, normalized size = 1.19 \begin {gather*} 4\,x+x^2+{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{-\frac {10\,{\mathrm {e}}^{x^2}}{x}}\,{\mathrm {e}}^{\frac {25\,{\mathrm {e}}^{x^2}}{x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((exp(x^2)*(x^2 - 10*x + 25) - x^3)/x^2)*(exp(x^2)*(10*x + 50*x^2 - 20*x^3 + 2*x^4 - 50) - x^3) + 4*x^
3 + 2*x^4)/x^3,x)

[Out]

4*x + x^2 + exp(-x)*exp(exp(x^2))*exp(-(10*exp(x^2))/x)*exp((25*exp(x^2))/x^2)

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sympy [A]  time = 0.39, size = 27, normalized size = 0.87 \begin {gather*} x^{2} + 4 x + e^{\frac {- x^{3} + \left (x^{2} - 10 x + 25\right ) e^{x^{2}}}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**4-20*x**3+50*x**2+10*x-50)*exp(x**2)-x**3)*exp(((x**2-10*x+25)*exp(x**2)-x**3)/x**2)+2*x**4+
4*x**3)/x**3,x)

[Out]

x**2 + 4*x + exp((-x**3 + (x**2 - 10*x + 25)*exp(x**2))/x**2)

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