3.84.15 \(\int (5 e^{3 x^2-e^{3+2 x} x^2} (24 e^{10} x+e^{13+2 x} (-8 x-8 x^2))+25 e^{6 x^2-2 e^{3+2 x} x^2} (12 e^{10} x+e^{13+2 x} (-4 x-4 x^2))) \, dx\)

Optimal. Leaf size=27 \[ e^{10} \left (2+5 e^{\left (3-e^{3+2 x}\right ) x^2}\right )^2 \]

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Rubi [A]  time = 0.21, antiderivative size = 47, normalized size of antiderivative = 1.74, number of steps used = 3, number of rules used = 1, integrand size = 93, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6706} \begin {gather*} 25 e^{-2 e^{2 x+3} x^2+6 x^2+10}+20 e^{-e^{2 x+3} x^2+3 x^2+10} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[5*E^(3*x^2 - E^(3 + 2*x)*x^2)*(24*E^10*x + E^(13 + 2*x)*(-8*x - 8*x^2)) + 25*E^(6*x^2 - 2*E^(3 + 2*x)*x^2)
*(12*E^10*x + E^(13 + 2*x)*(-4*x - 4*x^2)),x]

[Out]

25*E^(10 + 6*x^2 - 2*E^(3 + 2*x)*x^2) + 20*E^(10 + 3*x^2 - E^(3 + 2*x)*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} &=5 \int e^{3 x^2-e^{3+2 x} x^2} \left (24 e^{10} x+e^{13+2 x} \left (-8 x-8 x^2\right )\right ) \, dx+25 \int e^{6 x^2-2 e^{3+2 x} x^2} \left (12 e^{10} x+e^{13+2 x} \left (-4 x-4 x^2\right )\right ) \, dx\\ &=25 e^{10+6 x^2-2 e^{3+2 x} x^2}+20 e^{10+3 x^2-e^{3+2 x} x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.99, size = 51, normalized size = 1.89 \begin {gather*} -20 \left (-\frac {5}{4} e^{10+6 x^2-2 e^{3+2 x} x^2}-e^{10+3 x^2-e^{3+2 x} x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[5*E^(3*x^2 - E^(3 + 2*x)*x^2)*(24*E^10*x + E^(13 + 2*x)*(-8*x - 8*x^2)) + 25*E^(6*x^2 - 2*E^(3 + 2*x
)*x^2)*(12*E^10*x + E^(13 + 2*x)*(-4*x - 4*x^2)),x]

[Out]

-20*((-5*E^(10 + 6*x^2 - 2*E^(3 + 2*x)*x^2))/4 - E^(10 + 3*x^2 - E^(3 + 2*x)*x^2))

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fricas [B]  time = 0.96, size = 64, normalized size = 2.37 \begin {gather*} e^{\left (2 \, {\left (3 \, x^{2} e^{10} - x^{2} e^{\left (2 \, x + 13\right )} + e^{10} \log \relax (5)\right )} e^{\left (-10\right )} + 10\right )} + 4 \, e^{\left ({\left (3 \, x^{2} e^{10} - x^{2} e^{\left (2 \, x + 13\right )} + e^{10} \log \relax (5)\right )} e^{\left (-10\right )} + 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-4*x)*exp(5)^2*exp(2*x+3)+12*x*exp(5)^2)*exp(-x^2*exp(2*x+3)+log(5)+3*x^2)^2+((-8*x^2-8*x)*e
xp(5)^2*exp(2*x+3)+24*x*exp(5)^2)*exp(-x^2*exp(2*x+3)+log(5)+3*x^2),x, algorithm="fricas")

[Out]

e^(2*(3*x^2*e^10 - x^2*e^(2*x + 13) + e^10*log(5))*e^(-10) + 10) + 4*e^((3*x^2*e^10 - x^2*e^(2*x + 13) + e^10*
log(5))*e^(-10) + 10)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int 8 \, {\left (3 \, x e^{10} - {\left (x^{2} + x\right )} e^{\left (2 \, x + 13\right )}\right )} e^{\left (-x^{2} e^{\left (2 \, x + 3\right )} + 3 \, x^{2} + \log \relax (5)\right )} + 4 \, {\left (3 \, x e^{10} - {\left (x^{2} + x\right )} e^{\left (2 \, x + 13\right )}\right )} e^{\left (-2 \, x^{2} e^{\left (2 \, x + 3\right )} + 6 \, x^{2} + 2 \, \log \relax (5)\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-4*x)*exp(5)^2*exp(2*x+3)+12*x*exp(5)^2)*exp(-x^2*exp(2*x+3)+log(5)+3*x^2)^2+((-8*x^2-8*x)*e
xp(5)^2*exp(2*x+3)+24*x*exp(5)^2)*exp(-x^2*exp(2*x+3)+log(5)+3*x^2),x, algorithm="giac")

[Out]

integrate(8*(3*x*e^10 - (x^2 + x)*e^(2*x + 13))*e^(-x^2*e^(2*x + 3) + 3*x^2 + log(5)) + 4*(3*x*e^10 - (x^2 + x
)*e^(2*x + 13))*e^(-2*x^2*e^(2*x + 3) + 6*x^2 + 2*log(5)), x)

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maple [A]  time = 0.19, size = 44, normalized size = 1.63

method	result	size
risch	\(20 \,{\mathrm e}^{-x^{2} {\mathrm e}^{2 x +3}+3 x^{2}+10}+25 \,{\mathrm e}^{-2 x^{2} {\mathrm e}^{2 x +3}+6 x^{2}+10}\)	\(44\)
default	\(4 \,{\mathrm e}^{10} {\mathrm e}^{-x^{2} {\mathrm e}^{2 x +3}+\ln \relax (5)+3 x^{2}}+25 \,{\mathrm e}^{10} {\mathrm e}^{-2 x^{2} {\mathrm e}^{2 x +3}+6 x^{2}}\)	\(55\)
norman	\(4 \,{\mathrm e}^{10} {\mathrm e}^{-x^{2} {\mathrm e}^{2 x +3}+\ln \relax (5)+3 x^{2}}+25 \,{\mathrm e}^{10} {\mathrm e}^{-2 x^{2} {\mathrm e}^{2 x +3}+6 x^{2}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^2-4*x)*exp(5)^2*exp(2*x+3)+12*x*exp(5)^2)*exp(-x^2*exp(2*x+3)+ln(5)+3*x^2)^2+((-8*x^2-8*x)*exp(5)^2
*exp(2*x+3)+24*x*exp(5)^2)*exp(-x^2*exp(2*x+3)+ln(5)+3*x^2),x,method=_RETURNVERBOSE)

[Out]

20*exp(-x^2*exp(2*x+3)+3*x^2+10)+25*exp(-2*x^2*exp(2*x+3)+6*x^2+10)

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maxima [A]  time = 0.45, size = 43, normalized size = 1.59 \begin {gather*} 20 \, e^{\left (-x^{2} e^{\left (2 \, x + 3\right )} + 3 \, x^{2} + 10\right )} + 25 \, e^{\left (-2 \, x^{2} e^{\left (2 \, x + 3\right )} + 6 \, x^{2} + 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-4*x)*exp(5)^2*exp(2*x+3)+12*x*exp(5)^2)*exp(-x^2*exp(2*x+3)+log(5)+3*x^2)^2+((-8*x^2-8*x)*e
xp(5)^2*exp(2*x+3)+24*x*exp(5)^2)*exp(-x^2*exp(2*x+3)+log(5)+3*x^2),x, algorithm="maxima")

[Out]

20*e^(-x^2*e^(2*x + 3) + 3*x^2 + 10) + 25*e^(-2*x^2*e^(2*x + 3) + 6*x^2 + 10)

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mupad [B]  time = 5.18, size = 44, normalized size = 1.63 \begin {gather*} 5\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^3}\,{\mathrm {e}}^{10}\,{\mathrm {e}}^{3\,x^2}\,\left (4\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^3}+5\,{\mathrm {e}}^{3\,x^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(log(5) - x^2*exp(2*x + 3) + 3*x^2)*(24*x*exp(10) - exp(10)*exp(2*x + 3)*(8*x + 8*x^2)) + exp(2*log(5)
- 2*x^2*exp(2*x + 3) + 6*x^2)*(12*x*exp(10) - exp(10)*exp(2*x + 3)*(4*x + 4*x^2)),x)

[Out]

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

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sympy [B]  time = 0.29, size = 44, normalized size = 1.63 \begin {gather*} 25 e^{10} e^{- 2 x^{2} e^{2 x + 3} + 6 x^{2}} + 20 e^{10} e^{- x^{2} e^{2 x + 3} + 3 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**2-4*x)*exp(5)**2*exp(2*x+3)+12*x*exp(5)**2)*exp(-x**2*exp(2*x+3)+ln(5)+3*x**2)**2+((-8*x**2-
8*x)*exp(5)**2*exp(2*x+3)+24*x*exp(5)**2)*exp(-x**2*exp(2*x+3)+ln(5)+3*x**2),x)

[Out]

25*exp(10)*exp(-2*x**2*exp(2*x + 3) + 6*x**2) + 20*exp(10)*exp(-x**2*exp(2*x + 3) + 3*x**2)

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