3.46.9 \(\int \frac {e^x (162-5832 x+1458 x^2+729 x^3)+e^x (-81 x+2916 x^2-729 x^3) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+(3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7) \log (x)+(-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7) \log ^2(x)+(x+27 x^3+243 x^5+729 x^7) \log ^3(x)} \, dx\)

Optimal. Leaf size=25 \[ -\frac {e^x}{\left (\frac {\log (x)}{9}+x (-4-x+x \log (x))\right )^2} \]

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Rubi [F]  time = 5.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(162 - 5832*x + 1458*x^2 + 729*x^3) + E^x*(-81*x + 2916*x^2 - 729*x^3)*Log[x])/(-46656*x^4 - 34992*x^
5 - 8748*x^6 - 729*x^7 + (3888*x^3 + 1944*x^4 + 35235*x^5 + 17496*x^6 + 2187*x^7)*Log[x] + (-108*x^2 - 27*x^3
- 1944*x^4 - 486*x^5 - 8748*x^6 - 2187*x^7)*Log[x]^2 + (x + 27*x^3 + 243*x^5 + 729*x^7)*Log[x]^3),x]

[Out]

5832*Defer[Int][E^x/(-36*x - 9*x^2 + Log[x] + 9*x^2*Log[x])^3, x] + (486 - 5832*I)*Defer[Int][E^x/((I - 3*x)*(
-36*x - 9*x^2 + Log[x] + 9*x^2*Log[x])^3), x] + 162*Defer[Int][E^x/(x*(-36*x - 9*x^2 + Log[x] + 9*x^2*Log[x])^
3), x] + 1458*Defer[Int][(E^x*x)/(-36*x - 9*x^2 + Log[x] + 9*x^2*Log[x])^3, x] - (486 + 5832*I)*Defer[Int][E^x
/((I + 3*x)*(-36*x - 9*x^2 + Log[x] + 9*x^2*Log[x])^3), x] - 81*Defer[Int][E^x/(-36*x - 9*x^2 + Log[x] + 9*x^2
*Log[x])^2, x] - 486*Defer[Int][E^x/((I - 3*x)*(-36*x - 9*x^2 + Log[x] + 9*x^2*Log[x])^2), x] + 486*Defer[Int]
[E^x/((I + 3*x)*(-36*x - 9*x^2 + Log[x] + 9*x^2*Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {81 e^x \left (-2+72 x-18 x^2-9 x^3+x \left (1-36 x+9 x^2\right ) \log (x)\right )}{x \left (9 x (4+x)-\left (1+9 x^2\right ) \log (x)\right )^3} \, dx\\ &=81 \int \frac {e^x \left (-2+72 x-18 x^2-9 x^3+x \left (1-36 x+9 x^2\right ) \log (x)\right )}{x \left (9 x (4+x)-\left (1+9 x^2\right ) \log (x)\right )^3} \, dx\\ &=81 \int \left (\frac {2 e^x \left (1-36 x+324 x^3+81 x^4\right )}{x \left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {e^x \left (-1+36 x-9 x^2\right )}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2}\right ) \, dx\\ &=81 \int \frac {e^x \left (-1+36 x-9 x^2\right )}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+162 \int \frac {e^x \left (1-36 x+324 x^3+81 x^4\right )}{x \left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx\\ &=81 \int \left (-\frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2}+\frac {36 e^x x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2}\right ) \, dx+162 \int \left (\frac {36 e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {9 e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}-\frac {18 e^x (4+x)}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}\right ) \, dx\\ &=-\left (81 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx\right )+162 \int \frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+1458 \int \frac {e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-2916 \int \frac {e^x (4+x)}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+2916 \int \frac {e^x x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+5832 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx\\ &=-\left (81 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx\right )+162 \int \frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+1458 \int \frac {e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-2916 \int \left (\frac {4 e^x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {e^x x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}\right ) \, dx+2916 \int \left (-\frac {e^x}{6 (i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2}+\frac {e^x}{6 (i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2}\right ) \, dx+5832 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx\\ &=-\left (81 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx\right )+162 \int \frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-486 \int \frac {e^x}{(i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+486 \int \frac {e^x}{(i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+1458 \int \frac {e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-2916 \int \frac {e^x x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+5832 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-11664 \int \frac {e^x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx\\ &=-\left (81 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx\right )+162 \int \frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-486 \int \frac {e^x}{(i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+486 \int \frac {e^x}{(i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+1458 \int \frac {e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-2916 \int \left (-\frac {e^x}{6 (i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {e^x}{6 (i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}\right ) \, dx+5832 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-11664 \int \left (\frac {i e^x}{2 (i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {i e^x}{2 (i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}\right ) \, dx\\ &=-\left (5832 i \int \frac {e^x}{(i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx\right )-5832 i \int \frac {e^x}{(i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-81 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+162 \int \frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+486 \int \frac {e^x}{(i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-486 \int \frac {e^x}{(i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-486 \int \frac {e^x}{(i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+486 \int \frac {e^x}{(i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+1458 \int \frac {e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+5832 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.00, size = 25, normalized size = 1.00 \begin {gather*} -\frac {81 e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(162 - 5832*x + 1458*x^2 + 729*x^3) + E^x*(-81*x + 2916*x^2 - 729*x^3)*Log[x])/(-46656*x^4 - 34
992*x^5 - 8748*x^6 - 729*x^7 + (3888*x^3 + 1944*x^4 + 35235*x^5 + 17496*x^6 + 2187*x^7)*Log[x] + (-108*x^2 - 2
7*x^3 - 1944*x^4 - 486*x^5 - 8748*x^6 - 2187*x^7)*Log[x]^2 + (x + 27*x^3 + 243*x^5 + 729*x^7)*Log[x]^3),x]

[Out]

(-81*E^x)/(-36*x - 9*x^2 + Log[x] + 9*x^2*Log[x])^2

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fricas [B]  time = 0.62, size = 60, normalized size = 2.40 \begin {gather*} -\frac {81 \, e^{x}}{81 \, x^{4} + 648 \, x^{3} + {\left (81 \, x^{4} + 18 \, x^{2} + 1\right )} \log \relax (x)^{2} + 1296 \, x^{2} - 18 \, {\left (9 \, x^{4} + 36 \, x^{3} + x^{2} + 4 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-729*x^3+2916*x^2-81*x)*exp(x)*log(x)+(729*x^3+1458*x^2-5832*x+162)*exp(x))/((729*x^7+243*x^5+27*x
^3+x)*log(x)^3+(-2187*x^7-8748*x^6-486*x^5-1944*x^4-27*x^3-108*x^2)*log(x)^2+(2187*x^7+17496*x^6+35235*x^5+194
4*x^4+3888*x^3)*log(x)-729*x^7-8748*x^6-34992*x^5-46656*x^4),x, algorithm="fricas")

[Out]

-81*e^x/(81*x^4 + 648*x^3 + (81*x^4 + 18*x^2 + 1)*log(x)^2 + 1296*x^2 - 18*(9*x^4 + 36*x^3 + x^2 + 4*x)*log(x)
)

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giac [B]  time = 0.19, size = 70, normalized size = 2.80 \begin {gather*} -\frac {81 \, e^{x}}{81 \, x^{4} \log \relax (x)^{2} - 162 \, x^{4} \log \relax (x) + 81 \, x^{4} - 648 \, x^{3} \log \relax (x) + 18 \, x^{2} \log \relax (x)^{2} + 648 \, x^{3} - 18 \, x^{2} \log \relax (x) + 1296 \, x^{2} - 72 \, x \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-729*x^3+2916*x^2-81*x)*exp(x)*log(x)+(729*x^3+1458*x^2-5832*x+162)*exp(x))/((729*x^7+243*x^5+27*x
^3+x)*log(x)^3+(-2187*x^7-8748*x^6-486*x^5-1944*x^4-27*x^3-108*x^2)*log(x)^2+(2187*x^7+17496*x^6+35235*x^5+194
4*x^4+3888*x^3)*log(x)-729*x^7-8748*x^6-34992*x^5-46656*x^4),x, algorithm="giac")

[Out]

-81*e^x/(81*x^4*log(x)^2 - 162*x^4*log(x) + 81*x^4 - 648*x^3*log(x) + 18*x^2*log(x)^2 + 648*x^3 - 18*x^2*log(x
) + 1296*x^2 - 72*x*log(x) + log(x)^2)

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maple [A]  time = 0.03, size = 25, normalized size = 1.00

method	result	size
risch	\(-\frac {81 \,{\mathrm e}^{x}}{\left (9 x^{2} \ln \relax (x )-9 x^{2}+\ln \relax (x )-36 x \right )^{2}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-729*x^3+2916*x^2-81*x)*exp(x)*ln(x)+(729*x^3+1458*x^2-5832*x+162)*exp(x))/((729*x^7+243*x^5+27*x^3+x)*l
n(x)^3+(-2187*x^7-8748*x^6-486*x^5-1944*x^4-27*x^3-108*x^2)*ln(x)^2+(2187*x^7+17496*x^6+35235*x^5+1944*x^4+388
8*x^3)*ln(x)-729*x^7-8748*x^6-34992*x^5-46656*x^4),x,method=_RETURNVERBOSE)

[Out]

-81*exp(x)/(9*x^2*ln(x)-9*x^2+ln(x)-36*x)^2

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maxima [B]  time = 0.41, size = 60, normalized size = 2.40 \begin {gather*} -\frac {81 \, e^{x}}{81 \, x^{4} + 648 \, x^{3} + {\left (81 \, x^{4} + 18 \, x^{2} + 1\right )} \log \relax (x)^{2} + 1296 \, x^{2} - 18 \, {\left (9 \, x^{4} + 36 \, x^{3} + x^{2} + 4 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-729*x^3+2916*x^2-81*x)*exp(x)*log(x)+(729*x^3+1458*x^2-5832*x+162)*exp(x))/((729*x^7+243*x^5+27*x
^3+x)*log(x)^3+(-2187*x^7-8748*x^6-486*x^5-1944*x^4-27*x^3-108*x^2)*log(x)^2+(2187*x^7+17496*x^6+35235*x^5+194
4*x^4+3888*x^3)*log(x)-729*x^7-8748*x^6-34992*x^5-46656*x^4),x, algorithm="maxima")

[Out]

-81*e^x/(81*x^4 + 648*x^3 + (81*x^4 + 18*x^2 + 1)*log(x)^2 + 1296*x^2 - 18*(9*x^4 + 36*x^3 + x^2 + 4*x)*log(x)
)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^x\,\left (729\,x^3+1458\,x^2-5832\,x+162\right )-{\mathrm {e}}^x\,\ln \relax (x)\,\left (729\,x^3-2916\,x^2+81\,x\right )}{{\ln \relax (x)}^2\,\left (2187\,x^7+8748\,x^6+486\,x^5+1944\,x^4+27\,x^3+108\,x^2\right )-{\ln \relax (x)}^3\,\left (729\,x^7+243\,x^5+27\,x^3+x\right )-\ln \relax (x)\,\left (2187\,x^7+17496\,x^6+35235\,x^5+1944\,x^4+3888\,x^3\right )+46656\,x^4+34992\,x^5+8748\,x^6+729\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(1458*x^2 - 5832*x + 729*x^3 + 162) - exp(x)*log(x)*(81*x - 2916*x^2 + 729*x^3))/(log(x)^2*(108*x
^2 + 27*x^3 + 1944*x^4 + 486*x^5 + 8748*x^6 + 2187*x^7) - log(x)^3*(x + 27*x^3 + 243*x^5 + 729*x^7) - log(x)*(
3888*x^3 + 1944*x^4 + 35235*x^5 + 17496*x^6 + 2187*x^7) + 46656*x^4 + 34992*x^5 + 8748*x^6 + 729*x^7),x)

[Out]

int(-(exp(x)*(1458*x^2 - 5832*x + 729*x^3 + 162) - exp(x)*log(x)*(81*x - 2916*x^2 + 729*x^3))/(log(x)^2*(108*x
^2 + 27*x^3 + 1944*x^4 + 486*x^5 + 8748*x^6 + 2187*x^7) - log(x)^3*(x + 27*x^3 + 243*x^5 + 729*x^7) - log(x)*(
3888*x^3 + 1944*x^4 + 35235*x^5 + 17496*x^6 + 2187*x^7) + 46656*x^4 + 34992*x^5 + 8748*x^6 + 729*x^7), x)

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sympy [B]  time = 0.54, size = 78, normalized size = 3.12 \begin {gather*} - \frac {81 e^{x}}{81 x^{4} \log {\relax (x )}^{2} - 162 x^{4} \log {\relax (x )} + 81 x^{4} - 648 x^{3} \log {\relax (x )} + 648 x^{3} + 18 x^{2} \log {\relax (x )}^{2} - 18 x^{2} \log {\relax (x )} + 1296 x^{2} - 72 x \log {\relax (x )} + \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-729*x**3+2916*x**2-81*x)*exp(x)*ln(x)+(729*x**3+1458*x**2-5832*x+162)*exp(x))/((729*x**7+243*x**5
+27*x**3+x)*ln(x)**3+(-2187*x**7-8748*x**6-486*x**5-1944*x**4-27*x**3-108*x**2)*ln(x)**2+(2187*x**7+17496*x**6
+35235*x**5+1944*x**4+3888*x**3)*ln(x)-729*x**7-8748*x**6-34992*x**5-46656*x**4),x)

[Out]

-81*exp(x)/(81*x**4*log(x)**2 - 162*x**4*log(x) + 81*x**4 - 648*x**3*log(x) + 648*x**3 + 18*x**2*log(x)**2 - 1
8*x**2*log(x) + 1296*x**2 - 72*x*log(x) + log(x)**2)

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