3.6.43 \(\int e^{-32+2 x^2+24 e^2 x^2 \log (x)} (4 x+24 e^2 x+48 e^2 x \log (x)) \, dx\)

Optimal. Leaf size=22 \[ e^{2 x^2-8 \left (4-3 e^2 x^2 \log (x)\right )} \]

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Rubi [A]  time = 0.20, antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6, 2561, 6706} \begin {gather*} e^{2 x^2-32} x^{24 e^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-32 + 2*x^2 + 24*E^2*x^2*Log[x])*(4*x + 24*E^2*x + 48*E^2*x*Log[x]),x]

[Out]

E^(-32 + 2*x^2)*x^(24*E^2*x^2)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

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} &=\int e^{-32+2 x^2+24 e^2 x^2 \log (x)} \left (\left (4+24 e^2\right ) x+48 e^2 x \log (x)\right ) \, dx\\ &=\int e^{-32+2 x^2+24 e^2 x^2 \log (x)} x \left (4+24 e^2+48 e^2 \log (x)\right ) \, dx\\ &=e^{-32+2 x^2} x^{24 e^2 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.91 \begin {gather*} e^{-32+2 x^2} x^{24 e^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-32 + 2*x^2 + 24*E^2*x^2*Log[x])*(4*x + 24*E^2*x + 48*E^2*x*Log[x]),x]

[Out]

E^(-32 + 2*x^2)*x^(24*E^2*x^2)

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fricas [A]  time = 0.62, size = 17, normalized size = 0.77 \begin {gather*} e^{\left (24 \, x^{2} e^{2} \log \relax (x) + 2 \, x^{2} - 32\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x*exp(2)*log(x)+24*exp(2)*x+4*x)*exp(12*x^2*exp(2)*log(x)+x^2-16)^2,x, algorithm="fricas")

[Out]

e^(24*x^2*e^2*log(x) + 2*x^2 - 32)

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giac [A]  time = 0.62, size = 17, normalized size = 0.77 \begin {gather*} e^{\left (24 \, x^{2} e^{2} \log \relax (x) + 2 \, x^{2} - 32\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x*exp(2)*log(x)+24*exp(2)*x+4*x)*exp(12*x^2*exp(2)*log(x)+x^2-16)^2,x, algorithm="giac")

[Out]

e^(24*x^2*e^2*log(x) + 2*x^2 - 32)

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maple [A]  time = 0.07, size = 18, normalized size = 0.82

method	result	size
norman	\({\mathrm e}^{24 x^{2} {\mathrm e}^{2} \ln \relax (x )+2 x^{2}-32}\)	\(18\)
risch	\(x^{24 x^{2} {\mathrm e}^{2}} {\mathrm e}^{2 \left (x -4\right ) \left (4+x \right )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((48*x*exp(2)*ln(x)+24*exp(2)*x+4*x)*exp(12*x^2*exp(2)*ln(x)+x^2-16)^2,x,method=_RETURNVERBOSE)

[Out]

exp(12*x^2*exp(2)*ln(x)+x^2-16)^2

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maxima [A]  time = 0.51, size = 17, normalized size = 0.77 \begin {gather*} e^{\left (24 \, x^{2} e^{2} \log \relax (x) + 2 \, x^{2} - 32\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x*exp(2)*log(x)+24*exp(2)*x+4*x)*exp(12*x^2*exp(2)*log(x)+x^2-16)^2,x, algorithm="maxima")

[Out]

e^(24*x^2*e^2*log(x) + 2*x^2 - 32)

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mupad [B]  time = 0.57, size = 18, normalized size = 0.82 \begin {gather*} x^{24\,x^2\,{\mathrm {e}}^2}\,{\mathrm {e}}^{-32}\,{\mathrm {e}}^{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x^2 + 24*x^2*exp(2)*log(x) - 32)*(4*x + 24*x*exp(2) + 48*x*exp(2)*log(x)),x)

[Out]

x^(24*x^2*exp(2))*exp(-32)*exp(2*x^2)

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sympy [A]  time = 0.31, size = 19, normalized size = 0.86 \begin {gather*} e^{24 x^{2} e^{2} \log {\relax (x )} + 2 x^{2} - 32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x*exp(2)*ln(x)+24*exp(2)*x+4*x)*exp(12*x**2*exp(2)*ln(x)+x**2-16)**2,x)

[Out]

exp(24*x**2*exp(2)*log(x) + 2*x**2 - 32)

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