3.52.26 \(\int \frac {-54 e^6 x^3+(-864 e^2+e^4 (-1296 x^2-540 x^3)) \log (x)+e^2 (864-10368 x-8640 x^2-1800 x^3) \log ^2(x)+(-24768-34560 x-14400 x^2-2000 x^3) \log ^3(x)}{27 e^6 x^3+e^4 (648 x^2+270 x^3) \log (x)+e^2 (5184 x+4320 x^2+900 x^3) \log ^2(x)+(13824+17280 x+7200 x^2+1000 x^3) \log ^3(x)} \, dx\)

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

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Rubi [F]  time = 1.81, 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 {-54 e^6 x^3+\left (-864 e^2+e^4 \left (-1296 x^2-540 x^3\right )\right ) \log (x)+e^2 \left (864-10368 x-8640 x^2-1800 x^3\right ) \log ^2(x)+\left (-24768-34560 x-14400 x^2-2000 x^3\right ) \log ^3(x)}{27 e^6 x^3+e^4 \left (648 x^2+270 x^3\right ) \log (x)+e^2 \left (5184 x+4320 x^2+900 x^3\right ) \log ^2(x)+\left (13824+17280 x+7200 x^2+1000 x^3\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-54*E^6*x^3 + (-864*E^2 + E^4*(-1296*x^2 - 540*x^3))*Log[x] + E^2*(864 - 10368*x - 8640*x^2 - 1800*x^3)*L
og[x]^2 + (-24768 - 34560*x - 14400*x^2 - 2000*x^3)*Log[x]^3)/(27*E^6*x^3 + E^4*(648*x^2 + 270*x^3)*Log[x] + E
^2*(5184*x + 4320*x^2 + 900*x^3)*Log[x]^2 + (13824 + 17280*x + 7200*x^2 + 1000*x^3)*Log[x]^3),x]

[Out]

-2*x - 36/(12 + 5*x)^2 + (1296*E^4*Defer[Int][(3*E^2*x + 24*Log[x] + 10*x*Log[x])^(-3), x])/5 + (3359232*E^6*D
efer[Int][1/((12 + 5*x)^3*(3*E^2*x + 24*Log[x] + 10*x*Log[x])^3), x])/25 - (559872*E^6*Defer[Int][1/((12 + 5*x
)^2*(3*E^2*x + 24*Log[x] + 10*x*Log[x])^3), x])/25 - (7776*E^4*(10 - 3*E^2)*Defer[Int][1/((12 + 5*x)*(3*E^2*x
+ 24*Log[x] + 10*x*Log[x])^3), x])/25 + (279936*E^4*Defer[Int][1/((12 + 5*x)^3*(3*E^2*x + 24*Log[x] + 10*x*Log
[x])^2), x])/5 - (31104*E^4*Defer[Int][1/((12 + 5*x)^2*(3*E^2*x + 24*Log[x] + 10*x*Log[x])^2), x])/5 - (216*E^
2*(10 - 3*E^2)*Defer[Int][1/((12 + 5*x)*(3*E^2*x + 24*Log[x] + 10*x*Log[x])^2), x])/5 + 7776*E^2*Defer[Int][1/
((12 + 5*x)^3*(3*E^2*x + 24*Log[x] + 10*x*Log[x])), x] - 432*E^2*Defer[Int][1/((12 + 5*x)^2*(3*E^2*x + 24*Log[
x] + 10*x*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-27 e^6 x^3-54 e^2 \left (8+e^2 x^2 (12+5 x)\right ) \log (x)-36 e^2 \left (-12+144 x+120 x^2+25 x^3\right ) \log ^2(x)-8 \left (1548+2160 x+900 x^2+125 x^3\right ) \log ^3(x)\right )}{\left (3 e^2 x+2 (12+5 x) \log (x)\right )^3} \, dx\\ &=2 \int \frac {-27 e^6 x^3-54 e^2 \left (8+e^2 x^2 (12+5 x)\right ) \log (x)-36 e^2 \left (-12+144 x+120 x^2+25 x^3\right ) \log ^2(x)-8 \left (1548+2160 x+900 x^2+125 x^3\right ) \log ^3(x)}{\left (3 e^2 x+2 (12+5 x) \log (x)\right )^3} \, dx\\ &=2 \int \left (\frac {-1548-2160 x-900 x^2-125 x^3}{(12+5 x)^3}+\frac {648 e^4 x \left (144+6 \left (20+3 e^2\right ) x+25 x^2\right )}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3}+\frac {108 e^2 \left (-288-24 \left (10+3 e^2\right ) x-5 \left (10-3 e^2\right ) x^2\right )}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2}-\frac {216 e^2 (-6+5 x)}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )}\right ) \, dx\\ &=2 \int \frac {-1548-2160 x-900 x^2-125 x^3}{(12+5 x)^3} \, dx+\left (216 e^2\right ) \int \frac {-288-24 \left (10+3 e^2\right ) x-5 \left (10-3 e^2\right ) x^2}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2} \, dx-\left (432 e^2\right ) \int \frac {-6+5 x}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )} \, dx+\left (1296 e^4\right ) \int \frac {x \left (144+6 \left (20+3 e^2\right ) x+25 x^2\right )}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3} \, dx\\ &=2 \int \left (-1+\frac {180}{(12+5 x)^3}\right ) \, dx+\left (216 e^2\right ) \int \left (\frac {1296 e^2}{5 (12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2}-\frac {144 e^2}{5 (12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2}+\frac {-10+3 e^2}{5 (12+5 x) \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2}\right ) \, dx-\left (432 e^2\right ) \int \left (-\frac {18}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )}+\frac {1}{(12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )}\right ) \, dx+\left (1296 e^4\right ) \int \left (\frac {1}{5 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3}+\frac {2592 e^2}{25 (12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3}-\frac {432 e^2}{25 (12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3}+\frac {6 \left (-10+3 e^2\right )}{25 (12+5 x) \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3}\right ) \, dx\\ &=-2 x-\frac {36}{(12+5 x)^2}-\left (432 e^2\right ) \int \frac {1}{(12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )} \, dx+\left (7776 e^2\right ) \int \frac {1}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )} \, dx+\frac {1}{5} \left (1296 e^4\right ) \int \frac {1}{\left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3} \, dx-\frac {1}{5} \left (31104 e^4\right ) \int \frac {1}{(12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2} \, dx+\frac {1}{5} \left (279936 e^4\right ) \int \frac {1}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2} \, dx-\frac {1}{25} \left (559872 e^6\right ) \int \frac {1}{(12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3} \, dx+\frac {1}{25} \left (3359232 e^6\right ) \int \frac {1}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3} \, dx-\frac {1}{5} \left (216 e^2 \left (10-3 e^2\right )\right ) \int \frac {1}{(12+5 x) \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2} \, dx-\frac {1}{25} \left (7776 e^4 \left (10-3 e^2\right )\right ) \int \frac {1}{(12+5 x) \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 65, normalized size = 2.32 \begin {gather*} -\frac {2 \left (9 e^4 x^3+12 e^2 x^2 (12+5 x) \log (x)+4 \left (18+144 x+120 x^2+25 x^3\right ) \log ^2(x)\right )}{\left (3 e^2 x+2 (12+5 x) \log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-54*E^6*x^3 + (-864*E^2 + E^4*(-1296*x^2 - 540*x^3))*Log[x] + E^2*(864 - 10368*x - 8640*x^2 - 1800*
x^3)*Log[x]^2 + (-24768 - 34560*x - 14400*x^2 - 2000*x^3)*Log[x]^3)/(27*E^6*x^3 + E^4*(648*x^2 + 270*x^3)*Log[
x] + E^2*(5184*x + 4320*x^2 + 900*x^3)*Log[x]^2 + (13824 + 17280*x + 7200*x^2 + 1000*x^3)*Log[x]^3),x]

[Out]

(-2*(9*E^4*x^3 + 12*E^2*x^2*(12 + 5*x)*Log[x] + 4*(18 + 144*x + 120*x^2 + 25*x^3)*Log[x]^2))/(3*E^2*x + 2*(12
+ 5*x)*Log[x])^2

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fricas [B]  time = 0.53, size = 89, normalized size = 3.18 \begin {gather*} -\frac {2 \, {\left (9 \, x^{3} e^{4} + 12 \, {\left (5 \, x^{3} + 12 \, x^{2}\right )} e^{2} \log \relax (x) + 4 \, {\left (25 \, x^{3} + 120 \, x^{2} + 144 \, x + 18\right )} \log \relax (x)^{2}\right )}}{9 \, x^{2} e^{4} + 12 \, {\left (5 \, x^{2} + 12 \, x\right )} e^{2} \log \relax (x) + 4 \, {\left (25 \, x^{2} + 120 \, x + 144\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2000*x^3-14400*x^2-34560*x-24768)*log(x)^3+(-1800*x^3-8640*x^2-10368*x+864)*exp(2)*log(x)^2+((-54
0*x^3-1296*x^2)*exp(2)^2-864*exp(2))*log(x)-54*x^3*exp(2)^3)/((1000*x^3+7200*x^2+17280*x+13824)*log(x)^3+(900*
x^3+4320*x^2+5184*x)*exp(2)*log(x)^2+(270*x^3+648*x^2)*exp(2)^2*log(x)+27*x^3*exp(2)^3),x, algorithm="fricas")

[Out]

-2*(9*x^3*e^4 + 12*(5*x^3 + 12*x^2)*e^2*log(x) + 4*(25*x^3 + 120*x^2 + 144*x + 18)*log(x)^2)/(9*x^2*e^4 + 12*(
5*x^2 + 12*x)*e^2*log(x) + 4*(25*x^2 + 120*x + 144)*log(x)^2)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2000*x^3-14400*x^2-34560*x-24768)*log(x)^3+(-1800*x^3-8640*x^2-10368*x+864)*exp(2)*log(x)^2+((-54
0*x^3-1296*x^2)*exp(2)^2-864*exp(2))*log(x)-54*x^3*exp(2)^3)/((1000*x^3+7200*x^2+17280*x+13824)*log(x)^3+(900*
x^3+4320*x^2+5184*x)*exp(2)*log(x)^2+(270*x^3+648*x^2)*exp(2)^2*log(x)+27*x^3*exp(2)^3),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.25, size = 80, normalized size = 2.86

method	result	size
risch	\(-\frac {2 \left (25 x^{3}+120 x^{2}+144 x +18\right )}{25 x^{2}+120 x +144}+\frac {108 \left (3 \,{\mathrm e}^{2} x +20 x \ln \relax (x )+48 \ln \relax (x )\right ) {\mathrm e}^{2} x}{\left (25 x^{2}+120 x +144\right ) \left (3 \,{\mathrm e}^{2} x +10 x \ln \relax (x )+24 \ln \relax (x )\right )^{2}}\)	\(80\)
norman	\(\frac {\frac {26928 \ln \relax (x )^{2}}{5}+\frac {432 x^{2} {\mathrm e}^{4}}{5}+3456 x \ln \relax (x )^{2}+288 x^{2} {\mathrm e}^{2} \ln \relax (x )+\frac {6912 x \,{\mathrm e}^{2} \ln \relax (x )}{5}-18 x^{3} {\mathrm e}^{4}-200 x^{3} \ln \relax (x )^{2}-120 x^{3} {\mathrm e}^{2} \ln \relax (x )}{\left (3 \,{\mathrm e}^{2} x +10 x \ln \relax (x )+24 \ln \relax (x )\right )^{2}}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2000*x^3-14400*x^2-34560*x-24768)*ln(x)^3+(-1800*x^3-8640*x^2-10368*x+864)*exp(2)*ln(x)^2+((-540*x^3-12
96*x^2)*exp(2)^2-864*exp(2))*ln(x)-54*x^3*exp(2)^3)/((1000*x^3+7200*x^2+17280*x+13824)*ln(x)^3+(900*x^3+4320*x
^2+5184*x)*exp(2)*ln(x)^2+(270*x^3+648*x^2)*exp(2)^2*ln(x)+27*x^3*exp(2)^3),x,method=_RETURNVERBOSE)

[Out]

-2*(25*x^3+120*x^2+144*x+18)/(25*x^2+120*x+144)+108*(3*exp(2)*x+20*x*ln(x)+48*ln(x))*exp(2)*x/(25*x^2+120*x+14
4)/(3*exp(2)*x+10*x*ln(x)+24*ln(x))^2

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maxima [B]  time = 0.41, size = 93, normalized size = 3.32 \begin {gather*} -\frac {2 \, {\left (9 \, x^{3} e^{4} + 4 \, {\left (25 \, x^{3} + 120 \, x^{2} + 144 \, x + 18\right )} \log \relax (x)^{2} + 12 \, {\left (5 \, x^{3} e^{2} + 12 \, x^{2} e^{2}\right )} \log \relax (x)\right )}}{9 \, x^{2} e^{4} + 4 \, {\left (25 \, x^{2} + 120 \, x + 144\right )} \log \relax (x)^{2} + 12 \, {\left (5 \, x^{2} e^{2} + 12 \, x e^{2}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2000*x^3-14400*x^2-34560*x-24768)*log(x)^3+(-1800*x^3-8640*x^2-10368*x+864)*exp(2)*log(x)^2+((-54
0*x^3-1296*x^2)*exp(2)^2-864*exp(2))*log(x)-54*x^3*exp(2)^3)/((1000*x^3+7200*x^2+17280*x+13824)*log(x)^3+(900*
x^3+4320*x^2+5184*x)*exp(2)*log(x)^2+(270*x^3+648*x^2)*exp(2)^2*log(x)+27*x^3*exp(2)^3),x, algorithm="maxima")

[Out]

-2*(9*x^3*e^4 + 4*(25*x^3 + 120*x^2 + 144*x + 18)*log(x)^2 + 12*(5*x^3*e^2 + 12*x^2*e^2)*log(x))/(9*x^2*e^4 +
4*(25*x^2 + 120*x + 144)*log(x)^2 + 12*(5*x^2*e^2 + 12*x*e^2)*log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\ln \relax (x)}^3\,\left (2000\,x^3+14400\,x^2+34560\,x+24768\right )+54\,x^3\,{\mathrm {e}}^6+\ln \relax (x)\,\left (864\,{\mathrm {e}}^2+{\mathrm {e}}^4\,\left (540\,x^3+1296\,x^2\right )\right )+{\mathrm {e}}^2\,{\ln \relax (x)}^2\,\left (1800\,x^3+8640\,x^2+10368\,x-864\right )}{{\ln \relax (x)}^3\,\left (1000\,x^3+7200\,x^2+17280\,x+13824\right )+27\,x^3\,{\mathrm {e}}^6+{\mathrm {e}}^4\,\ln \relax (x)\,\left (270\,x^3+648\,x^2\right )+{\mathrm {e}}^2\,{\ln \relax (x)}^2\,\left (900\,x^3+4320\,x^2+5184\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^3*(34560*x + 14400*x^2 + 2000*x^3 + 24768) + 54*x^3*exp(6) + log(x)*(864*exp(2) + exp(4)*(1296*x^
2 + 540*x^3)) + exp(2)*log(x)^2*(10368*x + 8640*x^2 + 1800*x^3 - 864))/(log(x)^3*(17280*x + 7200*x^2 + 1000*x^
3 + 13824) + 27*x^3*exp(6) + exp(4)*log(x)*(648*x^2 + 270*x^3) + exp(2)*log(x)^2*(5184*x + 4320*x^2 + 900*x^3)
),x)

[Out]

int(-(log(x)^3*(34560*x + 14400*x^2 + 2000*x^3 + 24768) + 54*x^3*exp(6) + log(x)*(864*exp(2) + exp(4)*(1296*x^
2 + 540*x^3)) + exp(2)*log(x)^2*(10368*x + 8640*x^2 + 1800*x^3 - 864))/(log(x)^3*(17280*x + 7200*x^2 + 1000*x^
3 + 13824) + 27*x^3*exp(6) + exp(4)*log(x)*(648*x^2 + 270*x^3) + exp(2)*log(x)^2*(5184*x + 4320*x^2 + 900*x^3)
), x)

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sympy [B]  time = 0.48, size = 128, normalized size = 4.57 \begin {gather*} - 2 x + \frac {324 x^{2} e^{4} + \left (2160 x^{2} e^{2} + 5184 x e^{2}\right ) \log {\relax (x )}}{225 x^{4} e^{4} + 1080 x^{3} e^{4} + 1296 x^{2} e^{4} + \left (1500 x^{4} e^{2} + 10800 x^{3} e^{2} + 25920 x^{2} e^{2} + 20736 x e^{2}\right ) \log {\relax (x )} + \left (2500 x^{4} + 24000 x^{3} + 86400 x^{2} + 138240 x + 82944\right ) \log {\relax (x )}^{2}} - \frac {36}{25 x^{2} + 120 x + 144} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2000*x**3-14400*x**2-34560*x-24768)*ln(x)**3+(-1800*x**3-8640*x**2-10368*x+864)*exp(2)*ln(x)**2+(
(-540*x**3-1296*x**2)*exp(2)**2-864*exp(2))*ln(x)-54*x**3*exp(2)**3)/((1000*x**3+7200*x**2+17280*x+13824)*ln(x
)**3+(900*x**3+4320*x**2+5184*x)*exp(2)*ln(x)**2+(270*x**3+648*x**2)*exp(2)**2*ln(x)+27*x**3*exp(2)**3),x)

[Out]

-2*x + (324*x**2*exp(4) + (2160*x**2*exp(2) + 5184*x*exp(2))*log(x))/(225*x**4*exp(4) + 1080*x**3*exp(4) + 129
6*x**2*exp(4) + (1500*x**4*exp(2) + 10800*x**3*exp(2) + 25920*x**2*exp(2) + 20736*x*exp(2))*log(x) + (2500*x**
4 + 24000*x**3 + 86400*x**2 + 138240*x + 82944)*log(x)**2) - 36/(25*x**2 + 120*x + 144)

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