3.6.37 \(\int \frac {1}{25} e^{-2+\frac {-50 e^2+25 e^{2+x}+x^4}{25 e^2}} (450 e^2 x+225 e^{2+x} x^2+36 x^5) \, dx\)

Optimal. Leaf size=22 \[ 9 e^{-2+e^x+\frac {x^4}{25 e^2}} x^2 \]

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Rubi [B]  time = 0.09, antiderivative size = 60, normalized size of antiderivative = 2.73, number of steps used = 2, number of rules used = 2, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 2288} \begin {gather*} \frac {9 e^{-\frac {-x^4-25 e^{x+2}+50 e^2}{25 e^2}} \left (4 x^5+25 e^{x+2} x^2\right )}{4 x^3+25 e^{x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-2 + (-50*E^2 + 25*E^(2 + x) + x^4)/(25*E^2))*(450*E^2*x + 225*E^(2 + x)*x^2 + 36*x^5))/25,x]

[Out]

(9*(25*E^(2 + x)*x^2 + 4*x^5))/(E^((50*E^2 - 25*E^(2 + x) - x^4)/(25*E^2))*(25*E^(2 + x) + 4*x^3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int e^{-2+\frac {-50 e^2+25 e^{2+x}+x^4}{25 e^2}} \left (450 e^2 x+225 e^{2+x} x^2+36 x^5\right ) \, dx\\ &=\frac {9 e^{-\frac {50 e^2-25 e^{2+x}-x^4}{25 e^2}} \left (25 e^{2+x} x^2+4 x^5\right )}{25 e^{2+x}+4 x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 22, normalized size = 1.00 \begin {gather*} 9 e^{-2+e^x+\frac {x^4}{25 e^2}} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-2 + (-50*E^2 + 25*E^(2 + x) + x^4)/(25*E^2))*(450*E^2*x + 225*E^(2 + x)*x^2 + 36*x^5))/25,x]

[Out]

9*E^(-2 + E^x + x^4/(25*E^2))*x^2

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fricas [A]  time = 0.77, size = 26, normalized size = 1.18 \begin {gather*} 9 \, x^{2} e^{\left (\frac {1}{25} \, {\left (x^{4} - 100 \, e^{2} + 25 \, e^{\left (x + 2\right )}\right )} e^{\left (-2\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(225*x^2*exp(1)^2*exp(x)+450*x*exp(1)^2+36*x^5)*exp(1/25*(25*exp(1)^2*exp(x)-50*exp(1)^2+x^4)/e
xp(1)^2)/exp(1)^2,x, algorithm="fricas")

[Out]

9*x^2*e^(1/25*(x^4 - 100*e^2 + 25*e^(x + 2))*e^(-2) + 2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {9}{25} \, {\left (4 \, x^{5} + 25 \, x^{2} e^{\left (x + 2\right )} + 50 \, x e^{2}\right )} e^{\left (\frac {1}{25} \, {\left (x^{4} - 50 \, e^{2} + 25 \, e^{\left (x + 2\right )}\right )} e^{\left (-2\right )} - 2\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(225*x^2*exp(1)^2*exp(x)+450*x*exp(1)^2+36*x^5)*exp(1/25*(25*exp(1)^2*exp(x)-50*exp(1)^2+x^4)/e
xp(1)^2)/exp(1)^2,x, algorithm="giac")

[Out]

integrate(9/25*(4*x^5 + 25*x^2*e^(x + 2) + 50*x*e^2)*e^(1/25*(x^4 - 50*e^2 + 25*e^(x + 2))*e^(-2) - 2), x)

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maple [A]  time = 0.14, size = 27, normalized size = 1.23

method	result	size
risch	\(9 x^{2} {\mathrm e}^{-\frac {\left (-x^{4}+50 \,{\mathrm e}^{2}-25 \,{\mathrm e}^{2+x}\right ) {\mathrm e}^{-2}}{25}}\)	\(27\)
norman	\(9 x^{2} {\mathrm e}^{\frac {\left (25 \,{\mathrm e}^{2} {\mathrm e}^{x}-50 \,{\mathrm e}^{2}+x^{4}\right ) {\mathrm e}^{-2}}{25}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*(225*x^2*exp(1)^2*exp(x)+450*x*exp(1)^2+36*x^5)*exp(1/25*(25*exp(1)^2*exp(x)-50*exp(1)^2+x^4)/exp(1)^
2)/exp(1)^2,x,method=_RETURNVERBOSE)

[Out]

9*x^2*exp(-1/25*(-x^4+50*exp(2)-25*exp(2+x))*exp(-2))

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maxima [A]  time = 0.86, size = 17, normalized size = 0.77 \begin {gather*} 9 \, x^{2} e^{\left (\frac {1}{25} \, x^{4} e^{\left (-2\right )} + e^{x} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(225*x^2*exp(1)^2*exp(x)+450*x*exp(1)^2+36*x^5)*exp(1/25*(25*exp(1)^2*exp(x)-50*exp(1)^2+x^4)/e
xp(1)^2)/exp(1)^2,x, algorithm="maxima")

[Out]

9*x^2*e^(1/25*x^4*e^(-2) + e^x - 2)

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mupad [B]  time = 0.61, size = 18, normalized size = 0.82 \begin {gather*} 9\,x^2\,{\mathrm {e}}^{\frac {x^4\,{\mathrm {e}}^{-2}}{25}}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(-2)*(exp(2)*exp(x) - 2*exp(2) + x^4/25))*exp(-2)*(450*x*exp(2) + 36*x^5 + 225*x^2*exp(2)*exp(x)))
/25,x)

[Out]

9*x^2*exp((x^4*exp(-2))/25)*exp(exp(x))*exp(-2)

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sympy [A]  time = 0.19, size = 26, normalized size = 1.18 \begin {gather*} 9 x^{2} e^{\frac {\frac {x^{4}}{25} + e^{2} e^{x} - 2 e^{2}}{e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(225*x**2*exp(1)**2*exp(x)+450*x*exp(1)**2+36*x**5)*exp(1/25*(25*exp(1)**2*exp(x)-50*exp(1)**2+
x**4)/exp(1)**2)/exp(1)**2,x)

[Out]

9*x**2*exp((x**4/25 + exp(2)*exp(x) - 2*exp(2))*exp(-2))

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