\(\int \frac {e^{\frac {e^{2 x}-6 x^2+5 e^{x+x^2} x^2}{x^2}} (e^{2 x} (-2+2 x)+e^{x+x^2} (5 x^3+10 x^4))}{x^3} \, dx\) [1425]

Optimal result

Integrand size = 64, antiderivative size = 24 \[ \int \frac {e^{\frac {e^{2 x}-6 x^2+5 e^{x+x^2} x^2}{x^2}} \left (e^{2 x} (-2+2 x)+e^{x+x^2} \left (5 x^3+10 x^4\right )\right )}{x^3} \, dx=3+e^{-6+5 e^{x+x^2}+\frac {e^{2 x}}{x^2}} \]

[Out]

exp(exp(x)^2/x^2-6+5*exp(x^2+x))+3

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6838} \[ \int \frac {e^{\frac {e^{2 x}-6 x^2+5 e^{x+x^2} x^2}{x^2}} \left (e^{2 x} (-2+2 x)+e^{x+x^2} \left (5 x^3+10 x^4\right )\right )}{x^3} \, dx=e^{\frac {5 e^{x^2+x} x^2-6 x^2+e^{2 x}}{x^2}} \]

[In]

Int[(E^((E^(2*x) - 6*x^2 + 5*E^(x + x^2)*x^2)/x^2)*(E^(2*x)*(-2 + 2*x) + E^(x + x^2)*(5*x^3 + 10*x^4)))/x^3,x]

[Out]

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

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = e^{\frac {e^{2 x}-6 x^2+5 e^{x+x^2} x^2}{x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {e^{2 x}-6 x^2+5 e^{x+x^2} x^2}{x^2}} \left (e^{2 x} (-2+2 x)+e^{x+x^2} \left (5 x^3+10 x^4\right )\right )}{x^3} \, dx=e^{-6+5 e^{x+x^2}+\frac {e^{2 x}}{x^2}} \]

[In]

Integrate[(E^((E^(2*x) - 6*x^2 + 5*E^(x + x^2)*x^2)/x^2)*(E^(2*x)*(-2 + 2*x) + E^(x + x^2)*(5*x^3 + 10*x^4)))/
x^3,x]

[Out]

E^(-6 + 5*E^(x + x^2) + E^(2*x)/x^2)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12

method	result	size
risch	\({\mathrm e}^{\frac {5 x^{2} {\mathrm e}^{\left (1+x \right ) x}+{\mathrm e}^{2 x}-6 x^{2}}{x^{2}}}\)	\(27\)
parallelrisch	\({\mathrm e}^{\frac {5 x^{2} {\mathrm e}^{x^{2}+x}+{\mathrm e}^{2 x}-6 x^{2}}{x^{2}}}\)	\(27\)

[In]

int(((10*x^4+5*x^3)*exp(x^2+x)+(-2+2*x)*exp(x)^2)*exp((5*x^2*exp(x^2+x)+exp(x)^2-6*x^2)/x^2)/x^3,x,method=_RET
URNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {e^{2 x}-6 x^2+5 e^{x+x^2} x^2}{x^2}} \left (e^{2 x} (-2+2 x)+e^{x+x^2} \left (5 x^3+10 x^4\right )\right )}{x^3} \, dx=e^{\left (\frac {5 \, x^{2} e^{\left (x^{2} + x\right )} - 6 \, x^{2} + e^{\left (2 \, x\right )}}{x^{2}}\right )} \]

[In]

integrate(((10*x^4+5*x^3)*exp(x^2+x)+(-2+2*x)*exp(x)^2)*exp((5*x^2*exp(x^2+x)+exp(x)^2-6*x^2)/x^2)/x^3,x, algo
rithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {e^{2 x}-6 x^2+5 e^{x+x^2} x^2}{x^2}} \left (e^{2 x} (-2+2 x)+e^{x+x^2} \left (5 x^3+10 x^4\right )\right )}{x^3} \, dx=e^{\frac {5 x^{2} e^{x^{2} + x} - 6 x^{2} + e^{2 x}}{x^{2}}} \]

[In]

integrate(((10*x**4+5*x**3)*exp(x**2+x)+(-2+2*x)*exp(x)**2)*exp((5*x**2*exp(x**2+x)+exp(x)**2-6*x**2)/x**2)/x*
*3,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {e^{2 x}-6 x^2+5 e^{x+x^2} x^2}{x^2}} \left (e^{2 x} (-2+2 x)+e^{x+x^2} \left (5 x^3+10 x^4\right )\right )}{x^3} \, dx=e^{\left (\frac {e^{\left (2 \, x\right )}}{x^{2}} + 5 \, e^{\left (x^{2} + x\right )} - 6\right )} \]

[In]

integrate(((10*x^4+5*x^3)*exp(x^2+x)+(-2+2*x)*exp(x)^2)*exp((5*x^2*exp(x^2+x)+exp(x)^2-6*x^2)/x^2)/x^3,x, algo
rithm="maxima")

[Out]

e^(e^(2*x)/x^2 + 5*e^(x^2 + x) - 6)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {e^{2 x}-6 x^2+5 e^{x+x^2} x^2}{x^2}} \left (e^{2 x} (-2+2 x)+e^{x+x^2} \left (5 x^3+10 x^4\right )\right )}{x^3} \, dx=e^{\left (\frac {e^{\left (2 \, x\right )}}{x^{2}} + 5 \, e^{\left (x^{2} + x\right )} - 6\right )} \]

[In]

integrate(((10*x^4+5*x^3)*exp(x^2+x)+(-2+2*x)*exp(x)^2)*exp((5*x^2*exp(x^2+x)+exp(x)^2-6*x^2)/x^2)/x^3,x, algo
rithm="giac")

[Out]

e^(e^(2*x)/x^2 + 5*e^(x^2 + x) - 6)

Mupad [B] (verification not implemented)

Time = 8.78 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\frac {e^{2 x}-6 x^2+5 e^{x+x^2} x^2}{x^2}} \left (e^{2 x} (-2+2 x)+e^{x+x^2} \left (5 x^3+10 x^4\right )\right )}{x^3} \, dx={\mathrm {e}}^{-6}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x}}{x^2}}\,{\mathrm {e}}^{5\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x} \]

[In]

int((exp((exp(2*x) + 5*x^2*exp(x + x^2) - 6*x^2)/x^2)*(exp(x + x^2)*(5*x^3 + 10*x^4) + exp(2*x)*(2*x - 2)))/x^
3,x)

[Out]

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