3.42.90 \(\int e^{\frac {1}{5} (-4+5 x^3+40 x^4+120 x^5+160 x^6+80 x^7-5 x \log (3))} (12 x^2+128 x^3+480 x^4+768 x^5+448 x^6-4 \log (3)) \, dx\)

Optimal. Leaf size=29 \[ 4 e^{-x \left (\frac {4}{5 x}-x^2 (1+2 x)^4+\log (3)\right )} \]

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Rubi [A]  time = 1.06, antiderivative size = 40, normalized size of antiderivative = 1.38, number of steps used = 1, number of rules used = 1, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6706} \begin {gather*} 4\ 3^{-x} \exp \left (\frac {1}{5} \left (80 x^7+160 x^6+120 x^5+40 x^4+5 x^3-4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((-4 + 5*x^3 + 40*x^4 + 120*x^5 + 160*x^6 + 80*x^7 - 5*x*Log[3])/5)*(12*x^2 + 128*x^3 + 480*x^4 + 768*x^
5 + 448*x^6 - 4*Log[3]),x]

[Out]

(4*E^((-4 + 5*x^3 + 40*x^4 + 120*x^5 + 160*x^6 + 80*x^7)/5))/3^x

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} &=4\ 3^{-x} \exp \left (\frac {1}{5} \left (-4+5 x^3+40 x^4+120 x^5+160 x^6+80 x^7\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 36, normalized size = 1.24 \begin {gather*} 4\ 3^{-x} e^{-\frac {4}{5}+x^3+8 x^4+24 x^5+32 x^6+16 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((-4 + 5*x^3 + 40*x^4 + 120*x^5 + 160*x^6 + 80*x^7 - 5*x*Log[3])/5)*(12*x^2 + 128*x^3 + 480*x^4 +
768*x^5 + 448*x^6 - 4*Log[3]),x]

[Out]

(4*E^(-4/5 + x^3 + 8*x^4 + 24*x^5 + 32*x^6 + 16*x^7))/3^x

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fricas [A]  time = 1.55, size = 33, normalized size = 1.14 \begin {gather*} 4 \, e^{\left (16 \, x^{7} + 32 \, x^{6} + 24 \, x^{5} + 8 \, x^{4} + x^{3} - x \log \relax (3) - \frac {4}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(3)+448*x^6+768*x^5+480*x^4+128*x^3+12*x^2)/exp(x*log(3)-16*x^7-32*x^6-24*x^5-8*x^4-x^3+4/5),
x, algorithm="fricas")

[Out]

4*e^(16*x^7 + 32*x^6 + 24*x^5 + 8*x^4 + x^3 - x*log(3) - 4/5)

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giac [A]  time = 0.17, size = 33, normalized size = 1.14 \begin {gather*} 4 \, e^{\left (16 \, x^{7} + 32 \, x^{6} + 24 \, x^{5} + 8 \, x^{4} + x^{3} - x \log \relax (3) - \frac {4}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(3)+448*x^6+768*x^5+480*x^4+128*x^3+12*x^2)/exp(x*log(3)-16*x^7-32*x^6-24*x^5-8*x^4-x^3+4/5),
x, algorithm="giac")

[Out]

4*e^(16*x^7 + 32*x^6 + 24*x^5 + 8*x^4 + x^3 - x*log(3) - 4/5)

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maple [A]  time = 0.07, size = 34, normalized size = 1.17

method	result	size
risch	\(4 \,3^{-x} {\mathrm e}^{-\frac {4}{5}+16 x^{7}+32 x^{6}+24 x^{5}+8 x^{4}+x^{3}}\)	\(34\)
gosper	\(4 \,{\mathrm e}^{-x \ln \relax (3)+16 x^{7}+32 x^{6}+24 x^{5}+8 x^{4}+x^{3}-\frac {4}{5}}\)	\(37\)
norman	\(4 \,{\mathrm e}^{-x \ln \relax (3)+16 x^{7}+32 x^{6}+24 x^{5}+8 x^{4}+x^{3}-\frac {4}{5}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*ln(3)+448*x^6+768*x^5+480*x^4+128*x^3+12*x^2)/exp(x*ln(3)-16*x^7-32*x^6-24*x^5-8*x^4-x^3+4/5),x,method
=_RETURNVERBOSE)

[Out]

4/(3^x)*exp(-4/5+16*x^7+32*x^6+24*x^5+8*x^4+x^3)

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maxima [A]  time = 0.36, size = 33, normalized size = 1.14 \begin {gather*} 4 \, e^{\left (16 \, x^{7} + 32 \, x^{6} + 24 \, x^{5} + 8 \, x^{4} + x^{3} - x \log \relax (3) - \frac {4}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(3)+448*x^6+768*x^5+480*x^4+128*x^3+12*x^2)/exp(x*log(3)-16*x^7-32*x^6-24*x^5-8*x^4-x^3+4/5),
x, algorithm="maxima")

[Out]

4*e^(16*x^7 + 32*x^6 + 24*x^5 + 8*x^4 + x^3 - x*log(3) - 4/5)

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mupad [B]  time = 2.93, size = 37, normalized size = 1.28 \begin {gather*} \frac {4\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{-\frac {4}{5}}\,{\mathrm {e}}^{8\,x^4}\,{\mathrm {e}}^{16\,x^7}\,{\mathrm {e}}^{24\,x^5}\,{\mathrm {e}}^{32\,x^6}}{3^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^3 - x*log(3) + 8*x^4 + 24*x^5 + 32*x^6 + 16*x^7 - 4/5)*(12*x^2 - 4*log(3) + 128*x^3 + 480*x^4 + 768*
x^5 + 448*x^6),x)

[Out]

(4*exp(x^3)*exp(-4/5)*exp(8*x^4)*exp(16*x^7)*exp(24*x^5)*exp(32*x^6))/3^x

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sympy [A]  time = 0.18, size = 34, normalized size = 1.17 \begin {gather*} 4 e^{16 x^{7} + 32 x^{6} + 24 x^{5} + 8 x^{4} + x^{3} - x \log {\relax (3 )} - \frac {4}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*ln(3)+448*x**6+768*x**5+480*x**4+128*x**3+12*x**2)/exp(x*ln(3)-16*x**7-32*x**6-24*x**5-8*x**4-x*
*3+4/5),x)

[Out]

4*exp(16*x**7 + 32*x**6 + 24*x**5 + 8*x**4 + x**3 - x*log(3) - 4/5)

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