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

Optimal. Leaf size=24 \[ 3+e^{-6+5 e^{x+x^2}+\frac {e^{2 x}}{x^2}} \]

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Rubi [A]  time = 0.52, antiderivative size = 29, normalized size of antiderivative = 1.21, number of steps used = 1, number of rules used = 1, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6706} \begin {gather*} e^{\frac {5 e^{x^2+x} x^2-6 x^2+e^{2 x}}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[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 6706

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 {gather*} \begin {aligned} \text {integral} &=e^{\frac {e^{2 x}-6 x^2+5 e^{x+x^2} x^2}{x^2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.51, size = 22, normalized size = 0.92 \begin {gather*} e^{-6+5 e^{x+x^2}+\frac {e^{2 x}}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.61, size = 26, normalized size = 1.08 \begin {gather*} e^{\left (\frac {5 \, x^{2} e^{\left (x^{2} + x\right )} - 6 \, x^{2} + e^{\left (2 \, x\right )}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.44, size = 19, normalized size = 0.79 \begin {gather*} e^{\left (\frac {e^{\left (2 \, x\right )}}{x^{2}} + 5 \, e^{\left (x^{2} + x\right )} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(e^(2*x)/x^2 + 5*e^(x^2 + x) - 6)

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maple [A]  time = 0.05, size = 27, normalized size = 1.12

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.59, size = 19, normalized size = 0.79 \begin {gather*} e^{\left (\frac {e^{\left (2 \, x\right )}}{x^{2}} + 5 \, e^{\left (x^{2} + x\right )} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(e^(2*x)/x^2 + 5*e^(x^2 + x) - 6)

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mupad [B]  time = 1.07, size = 21, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^{-6}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x}}{x^2}}\,{\mathrm {e}}^{5\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.40, size = 26, normalized size = 1.08 \begin {gather*} e^{\frac {5 x^{2} e^{x^{2} + x} - 6 x^{2} + e^{2 x}}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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