3.5.27 \(\int \frac {1}{96} e^{\frac {1}{24} (96+720 x+48 x^2+x \log ^2(9))} (720+96 x+\log ^2(9)) \, dx\)

Optimal. Leaf size=28 \[ \frac {1}{4} e^{2 x \left (\frac {2}{x}+x+3 \left (5+\frac {\log ^2(9)}{144}\right )\right )} \]

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Rubi [A]  time = 0.13, antiderivative size = 24, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 2244, 2236} \begin {gather*} \frac {1}{4} e^{2 x^2+\frac {1}{24} x \left (720+\log ^2(9)\right )+4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((96 + 720*x + 48*x^2 + x*Log[9]^2)/24)*(720 + 96*x + Log[9]^2))/96,x]

[Out]

E^(4 + 2*x^2 + (x*(720 + Log[9]^2))/24)/4

Rule 12

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

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{96} \int e^{\frac {1}{24} \left (96+720 x+48 x^2+x \log ^2(9)\right )} \left (720+96 x+\log ^2(9)\right ) \, dx\\ &=\frac {1}{96} \int e^{4+2 x^2+\frac {1}{24} x \left (720+\log ^2(9)\right )} \left (720+96 x+\log ^2(9)\right ) \, dx\\ &=\frac {1}{4} e^{4+2 x^2+\frac {1}{24} x \left (720+\log ^2(9)\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{4} e^{4+2 x^2+\frac {1}{24} x \left (720+\log ^2(9)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((96 + 720*x + 48*x^2 + x*Log[9]^2)/24)*(720 + 96*x + Log[9]^2))/96,x]

[Out]

E^(4 + 2*x^2 + (x*(720 + Log[9]^2))/24)/4

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fricas [A]  time = 0.60, size = 20, normalized size = 0.71 \begin {gather*} \frac {1}{4} \, e^{\left (\frac {1}{6} \, x \log \relax (3)^{2} + 2 \, x^{2} + 30 \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/96*(4*log(3)^2+96*x+720)*exp(1/6*x*log(3)^2+2*x^2+30*x+4),x, algorithm="fricas")

[Out]

1/4*e^(1/6*x*log(3)^2 + 2*x^2 + 30*x + 4)

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giac [A]  time = 0.27, size = 20, normalized size = 0.71 \begin {gather*} \frac {1}{4} \, e^{\left (\frac {1}{6} \, x \log \relax (3)^{2} + 2 \, x^{2} + 30 \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/96*(4*log(3)^2+96*x+720)*exp(1/6*x*log(3)^2+2*x^2+30*x+4),x, algorithm="giac")

[Out]

1/4*e^(1/6*x*log(3)^2 + 2*x^2 + 30*x + 4)

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maple [A]  time = 0.06, size = 21, normalized size = 0.75

method	result	size
gosper	\(\frac {{\mathrm e}^{\frac {x \ln \relax (3)^{2}}{6}+2 x^{2}+30 x +4}}{4}\)	\(21\)
default	\(\frac {{\mathrm e}^{4+2 x^{2}+\left (\frac {\ln \relax (3)^{2}}{6}+30\right ) x}}{4}\)	\(21\)
norman	\(\frac {{\mathrm e}^{\frac {x \ln \relax (3)^{2}}{6}+2 x^{2}+30 x +4}}{4}\)	\(21\)
risch	\(\frac {{\mathrm e}^{\frac {x \ln \relax (3)^{2}}{6}+2 x^{2}+30 x +4}}{4}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/96*(4*ln(3)^2+96*x+720)*exp(1/6*x*ln(3)^2+2*x^2+30*x+4),x,method=_RETURNVERBOSE)

[Out]

1/4*exp(1/6*x*ln(3)^2+2*x^2+30*x+4)

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maxima [A]  time = 0.49, size = 20, normalized size = 0.71 \begin {gather*} \frac {1}{4} \, e^{\left (\frac {1}{6} \, x \log \relax (3)^{2} + 2 \, x^{2} + 30 \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/96*(4*log(3)^2+96*x+720)*exp(1/6*x*log(3)^2+2*x^2+30*x+4),x, algorithm="maxima")

[Out]

1/4*e^(1/6*x*log(3)^2 + 2*x^2 + 30*x + 4)

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mupad [B]  time = 0.52, size = 22, normalized size = 0.79 \begin {gather*} \frac {{\mathrm {e}}^{\frac {x\,{\ln \relax (3)}^2}{6}}\,{\mathrm {e}}^{30\,x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{2\,x^2}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(30*x + (x*log(3)^2)/6 + 2*x^2 + 4)*(96*x + 4*log(3)^2 + 720))/96,x)

[Out]

(exp((x*log(3)^2)/6)*exp(30*x)*exp(4)*exp(2*x^2))/4

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sympy [A]  time = 0.12, size = 20, normalized size = 0.71 \begin {gather*} \frac {e^{2 x^{2} + \frac {x \log {\relax (3 )}^{2}}{6} + 30 x + 4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/96*(4*ln(3)**2+96*x+720)*exp(1/6*x*ln(3)**2+2*x**2+30*x+4),x)

[Out]

exp(2*x**2 + x*log(3)**2/6 + 30*x + 4)/4

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