3.33.96 \(\int \frac {2 e^{3 x} x-144 e^{2 x} x^2+(e^{2 x} (-432 x^2+216 x^3)+e^x (10368 x^3-1728 x^4)) \log (x)+e^x (20736 x^3-17280 x^4+2592 x^5) \log ^2(x)+(-20736 x^6+3456 x^7) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx\)

Optimal. Leaf size=26 \[ \left (-x+\frac {24}{4+\frac {e^x}{3 x^2 \log (x)}}\right )^2 \]

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Rubi [F]  time = 8.65, 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 {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*E^(3*x)*x - 144*E^(2*x)*x^2 + (E^(2*x)*(-432*x^2 + 216*x^3) + E^x*(10368*x^3 - 1728*x^4))*Log[x] + E^x*
(20736*x^3 - 17280*x^4 + 2592*x^5)*Log[x]^2 + (-20736*x^6 + 3456*x^7)*Log[x]^3)/(E^(3*x) + 36*E^(2*x)*x^2*Log[
x] + 432*E^x*x^4*Log[x]^2 + 1728*x^6*Log[x]^3),x]

[Out]

x^2 - 124416*Defer[Int][(x^5*Log[x]^2)/(E^x + 12*x^2*Log[x])^3, x] - 248832*Defer[Int][(x^5*Log[x]^3)/(E^x + 1
2*x^2*Log[x])^3, x] + 124416*Defer[Int][(x^6*Log[x]^3)/(E^x + 12*x^2*Log[x])^3, x] + 10368*Defer[Int][(x^3*Log
[x])/(E^x + 12*x^2*Log[x])^2, x] + 1728*Defer[Int][(x^4*Log[x])/(E^x + 12*x^2*Log[x])^2, x] + 20736*Defer[Int]
[(x^3*Log[x]^2)/(E^x + 12*x^2*Log[x])^2, x] - 6912*Defer[Int][(x^4*Log[x]^2)/(E^x + 12*x^2*Log[x])^2, x] - 172
8*Defer[Int][(x^5*Log[x]^2)/(E^x + 12*x^2*Log[x])^2, x] - 144*Defer[Int][x^2/(E^x + 12*x^2*Log[x]), x] - 432*D
efer[Int][(x^2*Log[x])/(E^x + 12*x^2*Log[x]), x] + 144*Defer[Int][(x^3*Log[x])/(E^x + 12*x^2*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \left (e^{2 x} \left (e^x-72 x\right )+108 e^x x \left (e^x (-2+x)-8 (-6+x) x\right ) \log (x)+432 e^x x^2 \left (24-20 x+3 x^2\right ) \log ^2(x)+1728 (-6+x) x^5 \log ^3(x)\right )}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx\\ &=2 \int \frac {x \left (e^{2 x} \left (e^x-72 x\right )+108 e^x x \left (e^x (-2+x)-8 (-6+x) x\right ) \log (x)+432 e^x x^2 \left (24-20 x+3 x^2\right ) \log ^2(x)+1728 (-6+x) x^5 \log ^3(x)\right )}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx\\ &=2 \int \left (x+\frac {62208 x^5 \log ^2(x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^3}-\frac {864 x^3 (6+x) \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2}+\frac {72 x^2 (-1-3 \log (x)+x \log (x))}{e^x+12 x^2 \log (x)}\right ) \, dx\\ &=x^2+144 \int \frac {x^2 (-1-3 \log (x)+x \log (x))}{e^x+12 x^2 \log (x)} \, dx-1728 \int \frac {x^3 (6+x) \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx+124416 \int \frac {x^5 \log ^2(x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx\\ &=x^2+144 \int \left (-\frac {x^2}{e^x+12 x^2 \log (x)}-\frac {3 x^2 \log (x)}{e^x+12 x^2 \log (x)}+\frac {x^3 \log (x)}{e^x+12 x^2 \log (x)}\right ) \, dx-1728 \int \left (\frac {6 x^3 \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2}+\frac {x^4 \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2}\right ) \, dx+124416 \int \left (-\frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^3}-\frac {2 x^5 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3}+\frac {x^6 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3}\right ) \, dx\\ &=x^2-144 \int \frac {x^2}{e^x+12 x^2 \log (x)} \, dx+144 \int \frac {x^3 \log (x)}{e^x+12 x^2 \log (x)} \, dx-432 \int \frac {x^2 \log (x)}{e^x+12 x^2 \log (x)} \, dx-1728 \int \frac {x^4 \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx-10368 \int \frac {x^3 \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx-124416 \int \frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx+124416 \int \frac {x^6 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx-248832 \int \frac {x^5 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx\\ &=x^2-144 \int \frac {x^2}{e^x+12 x^2 \log (x)} \, dx+144 \int \frac {x^3 \log (x)}{e^x+12 x^2 \log (x)} \, dx-432 \int \frac {x^2 \log (x)}{e^x+12 x^2 \log (x)} \, dx-1728 \int \left (-\frac {x^4 \log (x)}{\left (e^x+12 x^2 \log (x)\right )^2}-\frac {2 x^4 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2}+\frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2}\right ) \, dx-10368 \int \left (-\frac {x^3 \log (x)}{\left (e^x+12 x^2 \log (x)\right )^2}-\frac {2 x^3 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2}+\frac {x^4 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2}\right ) \, dx-124416 \int \frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx+124416 \int \frac {x^6 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx-248832 \int \frac {x^5 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx\\ &=x^2-144 \int \frac {x^2}{e^x+12 x^2 \log (x)} \, dx+144 \int \frac {x^3 \log (x)}{e^x+12 x^2 \log (x)} \, dx-432 \int \frac {x^2 \log (x)}{e^x+12 x^2 \log (x)} \, dx+1728 \int \frac {x^4 \log (x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx-1728 \int \frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx+3456 \int \frac {x^4 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx+10368 \int \frac {x^3 \log (x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx-10368 \int \frac {x^4 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx+20736 \int \frac {x^3 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx-124416 \int \frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx+124416 \int \frac {x^6 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx-248832 \int \frac {x^5 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.10, size = 54, normalized size = 2.08 \begin {gather*} 2 \left (-6 x+\frac {x^2}{2}+\frac {18 e^{2 x}}{\left (e^x+12 x^2 \log (x)\right )^2}+\frac {6 e^x (-6+x)}{e^x+12 x^2 \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*E^(3*x)*x - 144*E^(2*x)*x^2 + (E^(2*x)*(-432*x^2 + 216*x^3) + E^x*(10368*x^3 - 1728*x^4))*Log[x]
+ E^x*(20736*x^3 - 17280*x^4 + 2592*x^5)*Log[x]^2 + (-20736*x^6 + 3456*x^7)*Log[x]^3)/(E^(3*x) + 36*E^(2*x)*x^
2*Log[x] + 432*E^x*x^4*Log[x]^2 + 1728*x^6*Log[x]^3),x]

[Out]

2*(-6*x + x^2/2 + (18*E^(2*x))/(E^x + 12*x^2*Log[x])^2 + (6*E^x*(-6 + x))/(E^x + 12*x^2*Log[x]))

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fricas [B]  time = 0.51, size = 72, normalized size = 2.77 \begin {gather*} \frac {24 \, {\left (x^{4} - 6 \, x^{3} - 36 \, x^{2}\right )} e^{x} \log \relax (x) + 144 \, {\left (x^{6} - 12 \, x^{5}\right )} \log \relax (x)^{2} + {\left (x^{2} - 36\right )} e^{\left (2 \, x\right )}}{144 \, x^{4} \log \relax (x)^{2} + 24 \, x^{2} e^{x} \log \relax (x) + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3456*x^7-20736*x^6)*log(x)^3+(2592*x^5-17280*x^4+20736*x^3)*exp(x)*log(x)^2+((216*x^3-432*x^2)*exp
(x)^2+(-1728*x^4+10368*x^3)*exp(x))*log(x)+2*x*exp(x)^3-144*exp(x)^2*x^2)/(1728*x^6*log(x)^3+432*x^4*exp(x)*lo
g(x)^2+36*x^2*exp(x)^2*log(x)+exp(x)^3),x, algorithm="fricas")

[Out]

(24*(x^4 - 6*x^3 - 36*x^2)*e^x*log(x) + 144*(x^6 - 12*x^5)*log(x)^2 + (x^2 - 36)*e^(2*x))/(144*x^4*log(x)^2 +
24*x^2*e^x*log(x) + e^(2*x))

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giac [B]  time = 0.70, size = 95, normalized size = 3.65 \begin {gather*} \frac {144 \, x^{6} \log \relax (x)^{2} - 1728 \, x^{5} \log \relax (x)^{2} + 24 \, x^{4} e^{x} \log \relax (x) - 1872 \, x^{4} \log \relax (x)^{2} - 144 \, x^{3} e^{x} \log \relax (x) - 1176 \, x^{2} e^{x} \log \relax (x) + x^{2} e^{\left (2 \, x\right )} - 49 \, e^{\left (2 \, x\right )}}{144 \, x^{4} \log \relax (x)^{2} + 24 \, x^{2} e^{x} \log \relax (x) + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3456*x^7-20736*x^6)*log(x)^3+(2592*x^5-17280*x^4+20736*x^3)*exp(x)*log(x)^2+((216*x^3-432*x^2)*exp
(x)^2+(-1728*x^4+10368*x^3)*exp(x))*log(x)+2*x*exp(x)^3-144*exp(x)^2*x^2)/(1728*x^6*log(x)^3+432*x^4*exp(x)*lo
g(x)^2+36*x^2*exp(x)^2*log(x)+exp(x)^3),x, algorithm="giac")

[Out]

(144*x^6*log(x)^2 - 1728*x^5*log(x)^2 + 24*x^4*e^x*log(x) - 1872*x^4*log(x)^2 - 144*x^3*e^x*log(x) - 1176*x^2*
e^x*log(x) + x^2*e^(2*x) - 49*e^(2*x))/(144*x^4*log(x)^2 + 24*x^2*e^x*log(x) + e^(2*x))

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maple [A]  time = 0.04, size = 47, normalized size = 1.81

method	result	size
risch	\(x^{2}-12 x +\frac {12 \left (12 x^{3} \ln \relax (x )-72 x^{2} \ln \relax (x )+{\mathrm e}^{x} x -3 \,{\mathrm e}^{x}\right ) {\mathrm e}^{x}}{\left (12 x^{2} \ln \relax (x )+{\mathrm e}^{x}\right )^{2}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3456*x^7-20736*x^6)*ln(x)^3+(2592*x^5-17280*x^4+20736*x^3)*exp(x)*ln(x)^2+((216*x^3-432*x^2)*exp(x)^2+(-
1728*x^4+10368*x^3)*exp(x))*ln(x)+2*x*exp(x)^3-144*exp(x)^2*x^2)/(1728*x^6*ln(x)^3+432*x^4*exp(x)*ln(x)^2+36*x
^2*exp(x)^2*ln(x)+exp(x)^3),x,method=_RETURNVERBOSE)

[Out]

x^2-12*x+12*(12*x^3*ln(x)-72*x^2*ln(x)+exp(x)*x-3*exp(x))*exp(x)/(12*x^2*ln(x)+exp(x))^2

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maxima [B]  time = 0.63, size = 70, normalized size = 2.69 \begin {gather*} \frac {x^{2} e^{\left (2 \, x\right )} + 24 \, {\left (x^{4} - 6 \, x^{3}\right )} e^{x} \log \relax (x) + 144 \, {\left (x^{6} - 12 \, x^{5} + 36 \, x^{4}\right )} \log \relax (x)^{2}}{144 \, x^{4} \log \relax (x)^{2} + 24 \, x^{2} e^{x} \log \relax (x) + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3456*x^7-20736*x^6)*log(x)^3+(2592*x^5-17280*x^4+20736*x^3)*exp(x)*log(x)^2+((216*x^3-432*x^2)*exp
(x)^2+(-1728*x^4+10368*x^3)*exp(x))*log(x)+2*x*exp(x)^3-144*exp(x)^2*x^2)/(1728*x^6*log(x)^3+432*x^4*exp(x)*lo
g(x)^2+36*x^2*exp(x)^2*log(x)+exp(x)^3),x, algorithm="maxima")

[Out]

(x^2*e^(2*x) + 24*(x^4 - 6*x^3)*e^x*log(x) + 144*(x^6 - 12*x^5 + 36*x^4)*log(x)^2)/(144*x^4*log(x)^2 + 24*x^2*
e^x*log(x) + e^(2*x))

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mupad [B]  time = 2.23, size = 155, normalized size = 5.96 \begin {gather*} x^2-12\,x+\frac {12\,\left (12\,x\,{\mathrm {e}}^{2\,x}-72\,x^3\,{\mathrm {e}}^x+12\,x^4\,{\mathrm {e}}^x-8\,x^2\,{\mathrm {e}}^{2\,x}+x^3\,{\mathrm {e}}^{2\,x}\right )}{\left ({\mathrm {e}}^x+12\,x^2\,\ln \relax (x)\right )\,\left (x^2\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x+12\,x^3\right )}+\frac {36\,{\mathrm {e}}^x\,\left (12\,x^5\,{\mathrm {e}}^x-2\,x^3\,{\mathrm {e}}^{2\,x}+x^4\,{\mathrm {e}}^{2\,x}\right )}{x^2\,\left ({\mathrm {e}}^{2\,x}+144\,x^4\,{\ln \relax (x)}^2+24\,x^2\,{\mathrm {e}}^x\,\ln \relax (x)\right )\,\left (x^2\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x+12\,x^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(3*x) + log(x)*(exp(x)*(10368*x^3 - 1728*x^4) - exp(2*x)*(432*x^2 - 216*x^3)) - log(x)^3*(20736*x^
6 - 3456*x^7) - 144*x^2*exp(2*x) + exp(x)*log(x)^2*(20736*x^3 - 17280*x^4 + 2592*x^5))/(exp(3*x) + 1728*x^6*lo
g(x)^3 + 36*x^2*exp(2*x)*log(x) + 432*x^4*exp(x)*log(x)^2),x)

[Out]

x^2 - 12*x + (12*(12*x*exp(2*x) - 72*x^3*exp(x) + 12*x^4*exp(x) - 8*x^2*exp(2*x) + x^3*exp(2*x)))/((exp(x) + 1
2*x^2*log(x))*(x^2*exp(x) - 2*x*exp(x) + 12*x^3)) + (36*exp(x)*(12*x^5*exp(x) - 2*x^3*exp(2*x) + x^4*exp(2*x))
)/(x^2*(exp(2*x) + 144*x^4*log(x)^2 + 24*x^2*exp(x)*log(x))*(x^2*exp(x) - 2*x*exp(x) + 12*x^3))

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sympy [B]  time = 0.42, size = 61, normalized size = 2.35 \begin {gather*} x^{2} + \frac {- 1728 x^{5} \log {\relax (x )}^{2} + 5184 x^{4} \log {\relax (x )}^{2} - 144 x^{3} e^{x} \log {\relax (x )}}{144 x^{4} \log {\relax (x )}^{2} + 24 x^{2} e^{x} \log {\relax (x )} + e^{2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3456*x**7-20736*x**6)*ln(x)**3+(2592*x**5-17280*x**4+20736*x**3)*exp(x)*ln(x)**2+((216*x**3-432*x*
*2)*exp(x)**2+(-1728*x**4+10368*x**3)*exp(x))*ln(x)+2*x*exp(x)**3-144*exp(x)**2*x**2)/(1728*x**6*ln(x)**3+432*
x**4*exp(x)*ln(x)**2+36*x**2*exp(x)**2*ln(x)+exp(x)**3),x)

[Out]

x**2 + (-1728*x**5*log(x)**2 + 5184*x**4*log(x)**2 - 144*x**3*exp(x)*log(x))/(144*x**4*log(x)**2 + 24*x**2*exp
(x)*log(x) + exp(2*x))

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