3.49.51 \(\int \frac {e^4 (-42-60 x)-486 x^2-1620 x^3-2160 x^4-1440 x^5-480 x^6-64 x^7+(-6 e^4-810 x^2-2160 x^3-2160 x^4-960 x^5-160 x^6) \log (x)+(-540 x^2-1080 x^3-720 x^4-160 x^5) \log ^2(x)+(-180 x^2-240 x^3-80 x^4) \log ^3(x)+(-30 x^2-20 x^3) \log ^4(x)-2 x^2 \log ^5(x)}{243 x^2+810 x^3+1080 x^4+720 x^5+240 x^6+32 x^7+(405 x^2+1080 x^3+1080 x^4+480 x^5+80 x^6) \log (x)+(270 x^2+540 x^3+360 x^4+80 x^5) \log ^2(x)+(90 x^2+120 x^3+40 x^4) \log ^3(x)+(15 x^2+10 x^3) \log ^4(x)+x^2 \log ^5(x)} \, dx\)

Optimal. Leaf size=23 \[ 2 \left (-x+\frac {3 e^4}{x (3+2 x+\log (x))^4}\right ) \]

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Rubi [F]  time = 1.63, 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^4 (-42-60 x)-486 x^2-1620 x^3-2160 x^4-1440 x^5-480 x^6-64 x^7+\left (-6 e^4-810 x^2-2160 x^3-2160 x^4-960 x^5-160 x^6\right ) \log (x)+\left (-540 x^2-1080 x^3-720 x^4-160 x^5\right ) \log ^2(x)+\left (-180 x^2-240 x^3-80 x^4\right ) \log ^3(x)+\left (-30 x^2-20 x^3\right ) \log ^4(x)-2 x^2 \log ^5(x)}{243 x^2+810 x^3+1080 x^4+720 x^5+240 x^6+32 x^7+\left (405 x^2+1080 x^3+1080 x^4+480 x^5+80 x^6\right ) \log (x)+\left (270 x^2+540 x^3+360 x^4+80 x^5\right ) \log ^2(x)+\left (90 x^2+120 x^3+40 x^4\right ) \log ^3(x)+\left (15 x^2+10 x^3\right ) \log ^4(x)+x^2 \log ^5(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^4*(-42 - 60*x) - 486*x^2 - 1620*x^3 - 2160*x^4 - 1440*x^5 - 480*x^6 - 64*x^7 + (-6*E^4 - 810*x^2 - 2160
*x^3 - 2160*x^4 - 960*x^5 - 160*x^6)*Log[x] + (-540*x^2 - 1080*x^3 - 720*x^4 - 160*x^5)*Log[x]^2 + (-180*x^2 -
240*x^3 - 80*x^4)*Log[x]^3 + (-30*x^2 - 20*x^3)*Log[x]^4 - 2*x^2*Log[x]^5)/(243*x^2 + 810*x^3 + 1080*x^4 + 72
0*x^5 + 240*x^6 + 32*x^7 + (405*x^2 + 1080*x^3 + 1080*x^4 + 480*x^5 + 80*x^6)*Log[x] + (270*x^2 + 540*x^3 + 36
0*x^4 + 80*x^5)*Log[x]^2 + (90*x^2 + 120*x^3 + 40*x^4)*Log[x]^3 + (15*x^2 + 10*x^3)*Log[x]^4 + x^2*Log[x]^5),x
]

[Out]

-2*x - 24*E^4*Defer[Int][1/(x^2*(3 + 2*x + Log[x])^5), x] - 48*E^4*Defer[Int][1/(x*(3 + 2*x + Log[x])^5), x] -
6*E^4*Defer[Int][1/(x^2*(3 + 2*x + Log[x])^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-x^2 (3+2 x)^5-3 e^4 (7+10 x)-\left (3 e^4+5 x^2 (3+2 x)^4\right ) \log (x)-10 x^2 (3+2 x)^3 \log ^2(x)-10 x^2 (3+2 x)^2 \log ^3(x)-5 x^2 (3+2 x) \log ^4(x)-x^2 \log ^5(x)\right )}{x^2 (3+2 x+\log (x))^5} \, dx\\ &=2 \int \frac {-x^2 (3+2 x)^5-3 e^4 (7+10 x)-\left (3 e^4+5 x^2 (3+2 x)^4\right ) \log (x)-10 x^2 (3+2 x)^3 \log ^2(x)-10 x^2 (3+2 x)^2 \log ^3(x)-5 x^2 (3+2 x) \log ^4(x)-x^2 \log ^5(x)}{x^2 (3+2 x+\log (x))^5} \, dx\\ &=2 \int \left (-1-\frac {12 e^4 (1+2 x)}{x^2 (3+2 x+\log (x))^5}-\frac {3 e^4}{x^2 (3+2 x+\log (x))^4}\right ) \, dx\\ &=-2 x-\left (6 e^4\right ) \int \frac {1}{x^2 (3+2 x+\log (x))^4} \, dx-\left (24 e^4\right ) \int \frac {1+2 x}{x^2 (3+2 x+\log (x))^5} \, dx\\ &=-2 x-\left (6 e^4\right ) \int \frac {1}{x^2 (3+2 x+\log (x))^4} \, dx-\left (24 e^4\right ) \int \left (\frac {1}{x^2 (3+2 x+\log (x))^5}+\frac {2}{x (3+2 x+\log (x))^5}\right ) \, dx\\ &=-2 x-\left (6 e^4\right ) \int \frac {1}{x^2 (3+2 x+\log (x))^4} \, dx-\left (24 e^4\right ) \int \frac {1}{x^2 (3+2 x+\log (x))^5} \, dx-\left (48 e^4\right ) \int \frac {1}{x (3+2 x+\log (x))^5} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 21, normalized size = 0.91 \begin {gather*} -2 \left (x-\frac {3 e^4}{x (3+2 x+\log (x))^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(-42 - 60*x) - 486*x^2 - 1620*x^3 - 2160*x^4 - 1440*x^5 - 480*x^6 - 64*x^7 + (-6*E^4 - 810*x^2
- 2160*x^3 - 2160*x^4 - 960*x^5 - 160*x^6)*Log[x] + (-540*x^2 - 1080*x^3 - 720*x^4 - 160*x^5)*Log[x]^2 + (-180
*x^2 - 240*x^3 - 80*x^4)*Log[x]^3 + (-30*x^2 - 20*x^3)*Log[x]^4 - 2*x^2*Log[x]^5)/(243*x^2 + 810*x^3 + 1080*x^
4 + 720*x^5 + 240*x^6 + 32*x^7 + (405*x^2 + 1080*x^3 + 1080*x^4 + 480*x^5 + 80*x^6)*Log[x] + (270*x^2 + 540*x^
3 + 360*x^4 + 80*x^5)*Log[x]^2 + (90*x^2 + 120*x^3 + 40*x^4)*Log[x]^3 + (15*x^2 + 10*x^3)*Log[x]^4 + x^2*Log[x
]^5),x]

[Out]

-2*(x - (3*E^4)/(x*(3 + 2*x + Log[x])^4))

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fricas [B]  time = 0.65, size = 194, normalized size = 8.43 \begin {gather*} -\frac {2 \, {\left (16 \, x^{6} + x^{2} \log \relax (x)^{4} + 96 \, x^{5} + 216 \, x^{4} + 4 \, {\left (2 \, x^{3} + 3 \, x^{2}\right )} \log \relax (x)^{3} + 216 \, x^{3} + 6 \, {\left (4 \, x^{4} + 12 \, x^{3} + 9 \, x^{2}\right )} \log \relax (x)^{2} + 81 \, x^{2} + 4 \, {\left (8 \, x^{5} + 36 \, x^{4} + 54 \, x^{3} + 27 \, x^{2}\right )} \log \relax (x) - 3 \, e^{4}\right )}}{16 \, x^{5} + x \log \relax (x)^{4} + 96 \, x^{4} + 4 \, {\left (2 \, x^{2} + 3 \, x\right )} \log \relax (x)^{3} + 216 \, x^{3} + 6 \, {\left (4 \, x^{3} + 12 \, x^{2} + 9 \, x\right )} \log \relax (x)^{2} + 216 \, x^{2} + 4 \, {\left (8 \, x^{4} + 36 \, x^{3} + 54 \, x^{2} + 27 \, x\right )} \log \relax (x) + 81 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2*log(x)^5+(-20*x^3-30*x^2)*log(x)^4+(-80*x^4-240*x^3-180*x^2)*log(x)^3+(-160*x^5-720*x^4-1080
*x^3-540*x^2)*log(x)^2+(-6*exp(2)^2-160*x^6-960*x^5-2160*x^4-2160*x^3-810*x^2)*log(x)+(-60*x-42)*exp(2)^2-64*x
^7-480*x^6-1440*x^5-2160*x^4-1620*x^3-486*x^2)/(x^2*log(x)^5+(10*x^3+15*x^2)*log(x)^4+(40*x^4+120*x^3+90*x^2)*
log(x)^3+(80*x^5+360*x^4+540*x^3+270*x^2)*log(x)^2+(80*x^6+480*x^5+1080*x^4+1080*x^3+405*x^2)*log(x)+32*x^7+24
0*x^6+720*x^5+1080*x^4+810*x^3+243*x^2),x, algorithm="fricas")

[Out]

-2*(16*x^6 + x^2*log(x)^4 + 96*x^5 + 216*x^4 + 4*(2*x^3 + 3*x^2)*log(x)^3 + 216*x^3 + 6*(4*x^4 + 12*x^3 + 9*x^
2)*log(x)^2 + 81*x^2 + 4*(8*x^5 + 36*x^4 + 54*x^3 + 27*x^2)*log(x) - 3*e^4)/(16*x^5 + x*log(x)^4 + 96*x^4 + 4*
(2*x^2 + 3*x)*log(x)^3 + 216*x^3 + 6*(4*x^3 + 12*x^2 + 9*x)*log(x)^2 + 216*x^2 + 4*(8*x^4 + 36*x^3 + 54*x^2 +
27*x)*log(x) + 81*x)

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giac [B]  time = 0.32, size = 212, normalized size = 9.22 \begin {gather*} -\frac {2 \, {\left (16 \, x^{6} + 32 \, x^{5} \log \relax (x) + 24 \, x^{4} \log \relax (x)^{2} + 8 \, x^{3} \log \relax (x)^{3} + x^{2} \log \relax (x)^{4} + 96 \, x^{5} + 144 \, x^{4} \log \relax (x) + 72 \, x^{3} \log \relax (x)^{2} + 12 \, x^{2} \log \relax (x)^{3} + 216 \, x^{4} + 216 \, x^{3} \log \relax (x) + 54 \, x^{2} \log \relax (x)^{2} + 216 \, x^{3} + 108 \, x^{2} \log \relax (x) + 81 \, x^{2} - 3 \, e^{4}\right )}}{16 \, x^{5} + 32 \, x^{4} \log \relax (x) + 24 \, x^{3} \log \relax (x)^{2} + 8 \, x^{2} \log \relax (x)^{3} + x \log \relax (x)^{4} + 96 \, x^{4} + 144 \, x^{3} \log \relax (x) + 72 \, x^{2} \log \relax (x)^{2} + 12 \, x \log \relax (x)^{3} + 216 \, x^{3} + 216 \, x^{2} \log \relax (x) + 54 \, x \log \relax (x)^{2} + 216 \, x^{2} + 108 \, x \log \relax (x) + 81 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2*log(x)^5+(-20*x^3-30*x^2)*log(x)^4+(-80*x^4-240*x^3-180*x^2)*log(x)^3+(-160*x^5-720*x^4-1080
*x^3-540*x^2)*log(x)^2+(-6*exp(2)^2-160*x^6-960*x^5-2160*x^4-2160*x^3-810*x^2)*log(x)+(-60*x-42)*exp(2)^2-64*x
^7-480*x^6-1440*x^5-2160*x^4-1620*x^3-486*x^2)/(x^2*log(x)^5+(10*x^3+15*x^2)*log(x)^4+(40*x^4+120*x^3+90*x^2)*
log(x)^3+(80*x^5+360*x^4+540*x^3+270*x^2)*log(x)^2+(80*x^6+480*x^5+1080*x^4+1080*x^3+405*x^2)*log(x)+32*x^7+24
0*x^6+720*x^5+1080*x^4+810*x^3+243*x^2),x, algorithm="giac")

[Out]

-2*(16*x^6 + 32*x^5*log(x) + 24*x^4*log(x)^2 + 8*x^3*log(x)^3 + x^2*log(x)^4 + 96*x^5 + 144*x^4*log(x) + 72*x^
3*log(x)^2 + 12*x^2*log(x)^3 + 216*x^4 + 216*x^3*log(x) + 54*x^2*log(x)^2 + 216*x^3 + 108*x^2*log(x) + 81*x^2
- 3*e^4)/(16*x^5 + 32*x^4*log(x) + 24*x^3*log(x)^2 + 8*x^2*log(x)^3 + x*log(x)^4 + 96*x^4 + 144*x^3*log(x) + 7
2*x^2*log(x)^2 + 12*x*log(x)^3 + 216*x^3 + 216*x^2*log(x) + 54*x*log(x)^2 + 216*x^2 + 108*x*log(x) + 81*x)

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maple [A]  time = 0.05, size = 21, normalized size = 0.91

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^2*ln(x)^5+(-20*x^3-30*x^2)*ln(x)^4+(-80*x^4-240*x^3-180*x^2)*ln(x)^3+(-160*x^5-720*x^4-1080*x^3-540*
x^2)*ln(x)^2+(-6*exp(2)^2-160*x^6-960*x^5-2160*x^4-2160*x^3-810*x^2)*ln(x)+(-60*x-42)*exp(2)^2-64*x^7-480*x^6-
1440*x^5-2160*x^4-1620*x^3-486*x^2)/(x^2*ln(x)^5+(10*x^3+15*x^2)*ln(x)^4+(40*x^4+120*x^3+90*x^2)*ln(x)^3+(80*x
^5+360*x^4+540*x^3+270*x^2)*ln(x)^2+(80*x^6+480*x^5+1080*x^4+1080*x^3+405*x^2)*ln(x)+32*x^7+240*x^6+720*x^5+10
80*x^4+810*x^3+243*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.44, size = 194, normalized size = 8.43 \begin {gather*} -\frac {2 \, {\left (16 \, x^{6} + x^{2} \log \relax (x)^{4} + 96 \, x^{5} + 216 \, x^{4} + 4 \, {\left (2 \, x^{3} + 3 \, x^{2}\right )} \log \relax (x)^{3} + 216 \, x^{3} + 6 \, {\left (4 \, x^{4} + 12 \, x^{3} + 9 \, x^{2}\right )} \log \relax (x)^{2} + 81 \, x^{2} + 4 \, {\left (8 \, x^{5} + 36 \, x^{4} + 54 \, x^{3} + 27 \, x^{2}\right )} \log \relax (x) - 3 \, e^{4}\right )}}{16 \, x^{5} + x \log \relax (x)^{4} + 96 \, x^{4} + 4 \, {\left (2 \, x^{2} + 3 \, x\right )} \log \relax (x)^{3} + 216 \, x^{3} + 6 \, {\left (4 \, x^{3} + 12 \, x^{2} + 9 \, x\right )} \log \relax (x)^{2} + 216 \, x^{2} + 4 \, {\left (8 \, x^{4} + 36 \, x^{3} + 54 \, x^{2} + 27 \, x\right )} \log \relax (x) + 81 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2*log(x)^5+(-20*x^3-30*x^2)*log(x)^4+(-80*x^4-240*x^3-180*x^2)*log(x)^3+(-160*x^5-720*x^4-1080
*x^3-540*x^2)*log(x)^2+(-6*exp(2)^2-160*x^6-960*x^5-2160*x^4-2160*x^3-810*x^2)*log(x)+(-60*x-42)*exp(2)^2-64*x
^7-480*x^6-1440*x^5-2160*x^4-1620*x^3-486*x^2)/(x^2*log(x)^5+(10*x^3+15*x^2)*log(x)^4+(40*x^4+120*x^3+90*x^2)*
log(x)^3+(80*x^5+360*x^4+540*x^3+270*x^2)*log(x)^2+(80*x^6+480*x^5+1080*x^4+1080*x^3+405*x^2)*log(x)+32*x^7+24
0*x^6+720*x^5+1080*x^4+810*x^3+243*x^2),x, algorithm="maxima")

[Out]

-2*(16*x^6 + x^2*log(x)^4 + 96*x^5 + 216*x^4 + 4*(2*x^3 + 3*x^2)*log(x)^3 + 216*x^3 + 6*(4*x^4 + 12*x^3 + 9*x^
2)*log(x)^2 + 81*x^2 + 4*(8*x^5 + 36*x^4 + 54*x^3 + 27*x^2)*log(x) - 3*e^4)/(16*x^5 + x*log(x)^4 + 96*x^4 + 4*
(2*x^2 + 3*x)*log(x)^3 + 216*x^3 + 6*(4*x^3 + 12*x^2 + 9*x)*log(x)^2 + 216*x^2 + 4*(8*x^4 + 36*x^3 + 54*x^2 +
27*x)*log(x) + 81*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\ln \relax (x)}^4\,\left (20\,x^3+30\,x^2\right )+\ln \relax (x)\,\left (160\,x^6+960\,x^5+2160\,x^4+2160\,x^3+810\,x^2+6\,{\mathrm {e}}^4\right )+2\,x^2\,{\ln \relax (x)}^5+{\ln \relax (x)}^3\,\left (80\,x^4+240\,x^3+180\,x^2\right )+486\,x^2+1620\,x^3+2160\,x^4+1440\,x^5+480\,x^6+64\,x^7+{\ln \relax (x)}^2\,\left (160\,x^5+720\,x^4+1080\,x^3+540\,x^2\right )+{\mathrm {e}}^4\,\left (60\,x+42\right )}{{\ln \relax (x)}^4\,\left (10\,x^3+15\,x^2\right )+x^2\,{\ln \relax (x)}^5+{\ln \relax (x)}^3\,\left (40\,x^4+120\,x^3+90\,x^2\right )+\ln \relax (x)\,\left (80\,x^6+480\,x^5+1080\,x^4+1080\,x^3+405\,x^2\right )+243\,x^2+810\,x^3+1080\,x^4+720\,x^5+240\,x^6+32\,x^7+{\ln \relax (x)}^2\,\left (80\,x^5+360\,x^4+540\,x^3+270\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^4*(30*x^2 + 20*x^3) + log(x)*(6*exp(4) + 810*x^2 + 2160*x^3 + 2160*x^4 + 960*x^5 + 160*x^6) + 2*x
^2*log(x)^5 + log(x)^3*(180*x^2 + 240*x^3 + 80*x^4) + 486*x^2 + 1620*x^3 + 2160*x^4 + 1440*x^5 + 480*x^6 + 64*
x^7 + log(x)^2*(540*x^2 + 1080*x^3 + 720*x^4 + 160*x^5) + exp(4)*(60*x + 42))/(log(x)^4*(15*x^2 + 10*x^3) + x^
2*log(x)^5 + log(x)^3*(90*x^2 + 120*x^3 + 40*x^4) + log(x)*(405*x^2 + 1080*x^3 + 1080*x^4 + 480*x^5 + 80*x^6)
+ 243*x^2 + 810*x^3 + 1080*x^4 + 720*x^5 + 240*x^6 + 32*x^7 + log(x)^2*(270*x^2 + 540*x^3 + 360*x^4 + 80*x^5))
,x)

[Out]

int(-(log(x)^4*(30*x^2 + 20*x^3) + log(x)*(6*exp(4) + 810*x^2 + 2160*x^3 + 2160*x^4 + 960*x^5 + 160*x^6) + 2*x
^2*log(x)^5 + log(x)^3*(180*x^2 + 240*x^3 + 80*x^4) + 486*x^2 + 1620*x^3 + 2160*x^4 + 1440*x^5 + 480*x^6 + 64*
x^7 + log(x)^2*(540*x^2 + 1080*x^3 + 720*x^4 + 160*x^5) + exp(4)*(60*x + 42))/(log(x)^4*(15*x^2 + 10*x^3) + x^
2*log(x)^5 + log(x)^3*(90*x^2 + 120*x^3 + 40*x^4) + log(x)*(405*x^2 + 1080*x^3 + 1080*x^4 + 480*x^5 + 80*x^6)
+ 243*x^2 + 810*x^3 + 1080*x^4 + 720*x^5 + 240*x^6 + 32*x^7 + log(x)^2*(270*x^2 + 540*x^3 + 360*x^4 + 80*x^5))
, x)

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sympy [B]  time = 0.37, size = 92, normalized size = 4.00 \begin {gather*} - 2 x + \frac {6 e^{4}}{16 x^{5} + 96 x^{4} + 216 x^{3} + 216 x^{2} + x \log {\relax (x )}^{4} + 81 x + \left (8 x^{2} + 12 x\right ) \log {\relax (x )}^{3} + \left (24 x^{3} + 72 x^{2} + 54 x\right ) \log {\relax (x )}^{2} + \left (32 x^{4} + 144 x^{3} + 216 x^{2} + 108 x\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2*ln(x)**5+(-20*x**3-30*x**2)*ln(x)**4+(-80*x**4-240*x**3-180*x**2)*ln(x)**3+(-160*x**5-720*x
**4-1080*x**3-540*x**2)*ln(x)**2+(-6*exp(2)**2-160*x**6-960*x**5-2160*x**4-2160*x**3-810*x**2)*ln(x)+(-60*x-42
)*exp(2)**2-64*x**7-480*x**6-1440*x**5-2160*x**4-1620*x**3-486*x**2)/(x**2*ln(x)**5+(10*x**3+15*x**2)*ln(x)**4
+(40*x**4+120*x**3+90*x**2)*ln(x)**3+(80*x**5+360*x**4+540*x**3+270*x**2)*ln(x)**2+(80*x**6+480*x**5+1080*x**4
+1080*x**3+405*x**2)*ln(x)+32*x**7+240*x**6+720*x**5+1080*x**4+810*x**3+243*x**2),x)

[Out]

-2*x + 6*exp(4)/(16*x**5 + 96*x**4 + 216*x**3 + 216*x**2 + x*log(x)**4 + 81*x + (8*x**2 + 12*x)*log(x)**3 + (2
4*x**3 + 72*x**2 + 54*x)*log(x)**2 + (32*x**4 + 144*x**3 + 216*x**2 + 108*x)*log(x))

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