3.91.18 \(\int e^{x^2+(-6 e^2 x+24 e x^2-24 x^3) \log (5)+(9 e^4-72 e^3 x+216 e^2 x^2-288 e x^3+144 x^4) \log ^2(5)} (2 x+(-6 e^2+48 e x-72 x^2) \log (5)+(-72 e^3+432 e^2 x-864 e x^2+576 x^3) \log ^2(5)) \, dx\)

Optimal. Leaf size=21 \[ e^{\left (-x+3 (-e+2 x)^2 \log (5)\right )^2} \]

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Rubi [B]  time = 0.34, antiderivative size = 62, normalized size of antiderivative = 2.95, number of steps used = 1, number of rules used = 1, integrand size = 114, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {6706} \begin {gather*} 5^{-24 x^3+24 e x^2-6 e^2 x} \exp \left (x^2+9 \left (16 x^4-32 e x^3+24 e^2 x^2-8 e^3 x+e^4\right ) \log ^2(5)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(x^2 + (-6*E^2*x + 24*E*x^2 - 24*x^3)*Log[5] + (9*E^4 - 72*E^3*x + 216*E^2*x^2 - 288*E*x^3 + 144*x^4)*Lo
g[5]^2)*(2*x + (-6*E^2 + 48*E*x - 72*x^2)*Log[5] + (-72*E^3 + 432*E^2*x - 864*E*x^2 + 576*x^3)*Log[5]^2),x]

[Out]

5^(-6*E^2*x + 24*E*x^2 - 24*x^3)*E^(x^2 + 9*(E^4 - 8*E^3*x + 24*E^2*x^2 - 32*E*x^3 + 16*x^4)*Log[5]^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} &=5^{-6 e^2 x+24 e x^2-24 x^3} \exp \left (x^2+9 \left (e^4-8 e^3 x+24 e^2 x^2-32 e x^3+16 x^4\right ) \log ^2(5)\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.91, size = 51, normalized size = 2.43 \begin {gather*} 5^{-6 (e-2 x)^2 x} e^{\frac {1}{4} \left (e^2-2 e (e-2 x)+(e-2 x)^2+36 (e-2 x)^4 \log ^2(5)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(x^2 + (-6*E^2*x + 24*E*x^2 - 24*x^3)*Log[5] + (9*E^4 - 72*E^3*x + 216*E^2*x^2 - 288*E*x^3 + 144*x
^4)*Log[5]^2)*(2*x + (-6*E^2 + 48*E*x - 72*x^2)*Log[5] + (-72*E^3 + 432*E^2*x - 864*E*x^2 + 576*x^3)*Log[5]^2)
,x]

[Out]

E^((E^2 - 2*E*(E - 2*x) + (E - 2*x)^2 + 36*(E - 2*x)^4*Log[5]^2)/4)/5^(6*(E - 2*x)^2*x)

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fricas [B]  time = 0.48, size = 59, normalized size = 2.81 \begin {gather*} e^{\left (9 \, {\left (16 \, x^{4} - 32 \, x^{3} e + 24 \, x^{2} e^{2} - 8 \, x e^{3} + e^{4}\right )} \log \relax (5)^{2} + x^{2} - 6 \, {\left (4 \, x^{3} - 4 \, x^{2} e + x e^{2}\right )} \log \relax (5)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*exp(1)^3+432*x*exp(1)^2-864*x^2*exp(1)+576*x^3)*log(5)^2+(-6*exp(1)^2+48*x*exp(1)-72*x^2)*log(
5)+2*x)*exp((9*exp(1)^4-72*x*exp(1)^3+216*x^2*exp(1)^2-288*x^3*exp(1)+144*x^4)*log(5)^2+(-6*x*exp(1)^2+24*x^2*
exp(1)-24*x^3)*log(5)+x^2),x, algorithm="fricas")

[Out]

e^(9*(16*x^4 - 32*x^3*e + 24*x^2*e^2 - 8*x*e^3 + e^4)*log(5)^2 + x^2 - 6*(4*x^3 - 4*x^2*e + x*e^2)*log(5))

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giac [B]  time = 0.18, size = 76, normalized size = 3.62 \begin {gather*} e^{\left (144 \, x^{4} \log \relax (5)^{2} - 288 \, x^{3} e \log \relax (5)^{2} + 216 \, x^{2} e^{2} \log \relax (5)^{2} - 24 \, x^{3} \log \relax (5) + 24 \, x^{2} e \log \relax (5) - 72 \, x e^{3} \log \relax (5)^{2} - 6 \, x e^{2} \log \relax (5) + 9 \, e^{4} \log \relax (5)^{2} + x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*exp(1)^3+432*x*exp(1)^2-864*x^2*exp(1)+576*x^3)*log(5)^2+(-6*exp(1)^2+48*x*exp(1)-72*x^2)*log(
5)+2*x)*exp((9*exp(1)^4-72*x*exp(1)^3+216*x^2*exp(1)^2-288*x^3*exp(1)+144*x^4)*log(5)^2+(-6*x*exp(1)^2+24*x^2*
exp(1)-24*x^3)*log(5)+x^2),x, algorithm="giac")

[Out]

e^(144*x^4*log(5)^2 - 288*x^3*e*log(5)^2 + 216*x^2*e^2*log(5)^2 - 24*x^3*log(5) + 24*x^2*e*log(5) - 72*x*e^3*l
og(5)^2 - 6*x*e^2*log(5) + 9*e^4*log(5)^2 + x^2)

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maple [B]  time = 0.08, size = 69, normalized size = 3.29

method	result	size
derivativedivides	\({\mathrm e}^{\left (9 \,{\mathrm e}^{4}-72 x \,{\mathrm e}^{3}+216 x^{2} {\mathrm e}^{2}-288 x^{3} {\mathrm e}+144 x^{4}\right ) \ln \relax (5)^{2}+\left (-6 \,{\mathrm e}^{2} x +24 x^{2} {\mathrm e}-24 x^{3}\right ) \ln \relax (5)+x^{2}}\)	\(69\)
default	\({\mathrm e}^{\left (9 \,{\mathrm e}^{4}-72 x \,{\mathrm e}^{3}+216 x^{2} {\mathrm e}^{2}-288 x^{3} {\mathrm e}+144 x^{4}\right ) \ln \relax (5)^{2}+\left (-6 \,{\mathrm e}^{2} x +24 x^{2} {\mathrm e}-24 x^{3}\right ) \ln \relax (5)+x^{2}}\)	\(69\)
norman	\({\mathrm e}^{\left (9 \,{\mathrm e}^{4}-72 x \,{\mathrm e}^{3}+216 x^{2} {\mathrm e}^{2}-288 x^{3} {\mathrm e}+144 x^{4}\right ) \ln \relax (5)^{2}+\left (-6 \,{\mathrm e}^{2} x +24 x^{2} {\mathrm e}-24 x^{3}\right ) \ln \relax (5)+x^{2}}\)	\(69\)
risch	\(15625^{-\left (-4 x \,{\mathrm e}+4 x^{2}+{\mathrm e}^{2}\right ) x} {\mathrm e}^{-288 \ln \relax (5)^{2} {\mathrm e} x^{3}+144 x^{4} \ln \relax (5)^{2}+216 \,{\mathrm e}^{2} \ln \relax (5)^{2} x^{2}-72 \ln \relax (5)^{2} {\mathrm e}^{3} x +9 \ln \relax (5)^{2} {\mathrm e}^{4}+x^{2}}\)	\(73\)
gosper	\({\mathrm e}^{9 \ln \relax (5)^{2} {\mathrm e}^{4}-72 \ln \relax (5)^{2} {\mathrm e}^{3} x +216 \,{\mathrm e}^{2} \ln \relax (5)^{2} x^{2}-288 \ln \relax (5)^{2} {\mathrm e} x^{3}+144 x^{4} \ln \relax (5)^{2}-6 x \,{\mathrm e}^{2} \ln \relax (5)+24 \ln \relax (5) {\mathrm e} x^{2}-24 x^{3} \ln \relax (5)+x^{2}}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-72*exp(1)^3+432*x*exp(1)^2-864*x^2*exp(1)+576*x^3)*ln(5)^2+(-6*exp(1)^2+48*x*exp(1)-72*x^2)*ln(5)+2*x)*
exp((9*exp(1)^4-72*x*exp(1)^3+216*x^2*exp(1)^2-288*x^3*exp(1)+144*x^4)*ln(5)^2+(-6*x*exp(1)^2+24*x^2*exp(1)-24
*x^3)*ln(5)+x^2),x,method=_RETURNVERBOSE)

[Out]

exp((9*exp(1)^4-72*x*exp(1)^3+216*x^2*exp(1)^2-288*x^3*exp(1)+144*x^4)*ln(5)^2+(-6*x*exp(1)^2+24*x^2*exp(1)-24
*x^3)*ln(5)+x^2)

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maxima [B]  time = 0.66, size = 76, normalized size = 3.62 \begin {gather*} e^{\left (144 \, x^{4} \log \relax (5)^{2} - 288 \, x^{3} e \log \relax (5)^{2} + 216 \, x^{2} e^{2} \log \relax (5)^{2} - 24 \, x^{3} \log \relax (5) + 24 \, x^{2} e \log \relax (5) - 72 \, x e^{3} \log \relax (5)^{2} - 6 \, x e^{2} \log \relax (5) + 9 \, e^{4} \log \relax (5)^{2} + x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*exp(1)^3+432*x*exp(1)^2-864*x^2*exp(1)+576*x^3)*log(5)^2+(-6*exp(1)^2+48*x*exp(1)-72*x^2)*log(
5)+2*x)*exp((9*exp(1)^4-72*x*exp(1)^3+216*x^2*exp(1)^2-288*x^3*exp(1)+144*x^4)*log(5)^2+(-6*x*exp(1)^2+24*x^2*
exp(1)-24*x^3)*log(5)+x^2),x, algorithm="maxima")

[Out]

e^(144*x^4*log(5)^2 - 288*x^3*e*log(5)^2 + 216*x^2*e^2*log(5)^2 - 24*x^3*log(5) + 24*x^2*e*log(5) - 72*x*e^3*l
og(5)^2 - 6*x*e^2*log(5) + 9*e^4*log(5)^2 + x^2)

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mupad [B]  time = 7.49, size = 85, normalized size = 4.05 \begin {gather*} \frac {5^{24\,x^2\,\mathrm {e}}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{144\,x^4\,{\ln \relax (5)}^2}\,{\mathrm {e}}^{-72\,x\,{\mathrm {e}}^3\,{\ln \relax (5)}^2}\,{\mathrm {e}}^{216\,x^2\,{\mathrm {e}}^2\,{\ln \relax (5)}^2}\,{\mathrm {e}}^{-288\,x^3\,\mathrm {e}\,{\ln \relax (5)}^2}\,{\mathrm {e}}^{9\,{\mathrm {e}}^4\,{\ln \relax (5)}^2}}{5^{24\,x^3}\,5^{6\,x\,{\mathrm {e}}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x^2 - log(5)*(6*x*exp(2) - 24*x^2*exp(1) + 24*x^3) + log(5)^2*(9*exp(4) - 72*x*exp(3) + 216*x^2*exp(2
) - 288*x^3*exp(1) + 144*x^4))*(log(5)^2*(72*exp(3) - 432*x*exp(2) + 864*x^2*exp(1) - 576*x^3) - 2*x + log(5)*
(6*exp(2) - 48*x*exp(1) + 72*x^2)),x)

[Out]

(5^(24*x^2*exp(1))*exp(x^2)*exp(144*x^4*log(5)^2)*exp(-72*x*exp(3)*log(5)^2)*exp(216*x^2*exp(2)*log(5)^2)*exp(
-288*x^3*exp(1)*log(5)^2)*exp(9*exp(4)*log(5)^2))/(5^(24*x^3)*5^(6*x*exp(2)))

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sympy [B]  time = 0.25, size = 66, normalized size = 3.14 \begin {gather*} e^{x^{2} + \left (- 24 x^{3} + 24 e x^{2} - 6 x e^{2}\right ) \log {\relax (5 )} + \left (144 x^{4} - 288 e x^{3} + 216 x^{2} e^{2} - 72 x e^{3} + 9 e^{4}\right ) \log {\relax (5 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*exp(1)**3+432*x*exp(1)**2-864*x**2*exp(1)+576*x**3)*ln(5)**2+(-6*exp(1)**2+48*x*exp(1)-72*x**2
)*ln(5)+2*x)*exp((9*exp(1)**4-72*x*exp(1)**3+216*x**2*exp(1)**2-288*x**3*exp(1)+144*x**4)*ln(5)**2+(-6*x*exp(1
)**2+24*x**2*exp(1)-24*x**3)*ln(5)+x**2),x)

[Out]

exp(x**2 + (-24*x**3 + 24*E*x**2 - 6*x*exp(2))*log(5) + (144*x**4 - 288*E*x**3 + 216*x**2*exp(2) - 72*x*exp(3)
+ 9*exp(4))*log(5)**2)

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