3.38.30 \(\int \frac {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+(-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7) \log (x)+(256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7) \log ^2(x)+(-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx\)

Optimal. Leaf size=31 \[ \frac {16}{25} \left (5-\frac {2}{x}-x-x^2\right )^2 \left (x-\frac {2}{\log (x)}\right )^2 \]

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Rubi [F]  time = 0.78, 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 {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-512 + 2560*x - 3712*x^2 + 768*x^3 + 1152*x^4 - 256*x^5 - 128*x^6 + (-512 + 1536*x - 1280*x^2 + 1472*x^3
- 1536*x^4 - 192*x^5 + 384*x^6 + 64*x^7)*Log[x] + (256*x - 1856*x^3 + 768*x^4 + 1728*x^5 - 512*x^6 - 320*x^7)*
Log[x]^2 + (-320*x^3 + 928*x^4 - 288*x^5 - 576*x^6 + 160*x^7 + 96*x^8)*Log[x]^3)/(25*x^3*Log[x]^3),x]

[Out]

(16*(2 - 5*x + x^2 + x^3)^2)/25 - (128*Defer[Int][(2 - 5*x + x^2 + x^3)^2/(x^3*Log[x]^3), x])/25 + (64*Defer[I
nt][((2 - 5*x + x^2 + x^3)*(-4 + 2*x - 3*x^2 + 5*x^3 + x^4))/(x^3*Log[x]^2), x])/25 - (64*Defer[Int][((2 - 5*x
+ x^2 + x^3)*(-2 - 5*x + 3*x^2 + 5*x^3))/(x^2*Log[x]), x])/25

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int \frac {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{x^3 \log ^3(x)} \, dx\\ &=\frac {1}{25} \int \frac {32 \left (2-5 x+x^2+x^3\right ) \left (-4 \left (2-5 x+x^2+x^3\right )+2 \left (-4+2 x-3 x^2+5 x^3+x^4\right ) \log (x)-2 x \left (-2-5 x+3 x^2+5 x^3\right ) \log ^2(x)+x^3 \left (-5+2 x+3 x^2\right ) \log ^3(x)\right )}{x^3 \log ^3(x)} \, dx\\ &=\frac {32}{25} \int \frac {\left (2-5 x+x^2+x^3\right ) \left (-4 \left (2-5 x+x^2+x^3\right )+2 \left (-4+2 x-3 x^2+5 x^3+x^4\right ) \log (x)-2 x \left (-2-5 x+3 x^2+5 x^3\right ) \log ^2(x)+x^3 \left (-5+2 x+3 x^2\right ) \log ^3(x)\right )}{x^3 \log ^3(x)} \, dx\\ &=\frac {32}{25} \int \left ((-1+x) (5+3 x) \left (2-5 x+x^2+x^3\right )-\frac {4 \left (2-5 x+x^2+x^3\right )^2}{x^3 \log ^3(x)}+\frac {2 \left (2-5 x+x^2+x^3\right ) \left (-4+2 x-3 x^2+5 x^3+x^4\right )}{x^3 \log ^2(x)}-\frac {2 \left (2-5 x+x^2+x^3\right ) \left (-2-5 x+3 x^2+5 x^3\right )}{x^2 \log (x)}\right ) \, dx\\ &=\frac {32}{25} \int (-1+x) (5+3 x) \left (2-5 x+x^2+x^3\right ) \, dx+\frac {64}{25} \int \frac {\left (2-5 x+x^2+x^3\right ) \left (-4+2 x-3 x^2+5 x^3+x^4\right )}{x^3 \log ^2(x)} \, dx-\frac {64}{25} \int \frac {\left (2-5 x+x^2+x^3\right ) \left (-2-5 x+3 x^2+5 x^3\right )}{x^2 \log (x)} \, dx-\frac {128}{25} \int \frac {\left (2-5 x+x^2+x^3\right )^2}{x^3 \log ^3(x)} \, dx\\ &=\frac {16}{25} \left (2-5 x+x^2+x^3\right )^2+\frac {64}{25} \int \frac {\left (2-5 x+x^2+x^3\right ) \left (-4+2 x-3 x^2+5 x^3+x^4\right )}{x^3 \log ^2(x)} \, dx-\frac {64}{25} \int \frac {\left (2-5 x+x^2+x^3\right ) \left (-2-5 x+3 x^2+5 x^3\right )}{x^2 \log (x)} \, dx-\frac {128}{25} \int \frac {\left (2-5 x+x^2+x^3\right )^2}{x^3 \log ^3(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 77, normalized size = 2.48 \begin {gather*} \frac {32}{25} \left (\frac {1}{2} x \left (-20+29 x-6 x^2-9 x^3+2 x^4+x^5\right )+\frac {2 \left (2-5 x+x^2+x^3\right )^2}{x^2 \log ^2(x)}-\frac {2 \left (2-5 x+x^2+x^3\right )^2}{x \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-512 + 2560*x - 3712*x^2 + 768*x^3 + 1152*x^4 - 256*x^5 - 128*x^6 + (-512 + 1536*x - 1280*x^2 + 147
2*x^3 - 1536*x^4 - 192*x^5 + 384*x^6 + 64*x^7)*Log[x] + (256*x - 1856*x^3 + 768*x^4 + 1728*x^5 - 512*x^6 - 320
*x^7)*Log[x]^2 + (-320*x^3 + 928*x^4 - 288*x^5 - 576*x^6 + 160*x^7 + 96*x^8)*Log[x]^3)/(25*x^3*Log[x]^3),x]

[Out]

(32*((x*(-20 + 29*x - 6*x^2 - 9*x^3 + 2*x^4 + x^5))/2 + (2*(2 - 5*x + x^2 + x^3)^2)/(x^2*Log[x]^2) - (2*(2 - 5
*x + x^2 + x^3)^2)/(x*Log[x])))/25

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fricas [B]  time = 0.71, size = 109, normalized size = 3.52 \begin {gather*} \frac {16 \, {\left (4 \, x^{6} + 8 \, x^{5} - 36 \, x^{4} - 24 \, x^{3} + {\left (x^{8} + 2 \, x^{7} - 9 \, x^{6} - 6 \, x^{5} + 29 \, x^{4} - 20 \, x^{3}\right )} \log \relax (x)^{2} + 116 \, x^{2} - 4 \, {\left (x^{7} + 2 \, x^{6} - 9 \, x^{5} - 6 \, x^{4} + 29 \, x^{3} - 20 \, x^{2} + 4 \, x\right )} \log \relax (x) - 80 \, x + 16\right )}}{25 \, x^{2} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((96*x^8+160*x^7-576*x^6-288*x^5+928*x^4-320*x^3)*log(x)^3+(-320*x^7-512*x^6+1728*x^5+768*x^4-1
856*x^3+256*x)*log(x)^2+(64*x^7+384*x^6-192*x^5-1536*x^4+1472*x^3-1280*x^2+1536*x-512)*log(x)-128*x^6-256*x^5+
1152*x^4+768*x^3-3712*x^2+2560*x-512)/x^3/log(x)^3,x, algorithm="fricas")

[Out]

16/25*(4*x^6 + 8*x^5 - 36*x^4 - 24*x^3 + (x^8 + 2*x^7 - 9*x^6 - 6*x^5 + 29*x^4 - 20*x^3)*log(x)^2 + 116*x^2 -
4*(x^7 + 2*x^6 - 9*x^5 - 6*x^4 + 29*x^3 - 20*x^2 + 4*x)*log(x) - 80*x + 16)/(x^2*log(x)^2)

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giac [B]  time = 0.20, size = 114, normalized size = 3.68 \begin {gather*} \frac {16}{25} \, x^{6} + \frac {32}{25} \, x^{5} - \frac {144}{25} \, x^{4} - \frac {96}{25} \, x^{3} + \frac {464}{25} \, x^{2} - \frac {64}{5} \, x - \frac {64 \, {\left (x^{7} \log \relax (x) + 2 \, x^{6} \log \relax (x) - x^{6} - 9 \, x^{5} \log \relax (x) - 2 \, x^{5} - 6 \, x^{4} \log \relax (x) + 9 \, x^{4} + 29 \, x^{3} \log \relax (x) + 6 \, x^{3} - 20 \, x^{2} \log \relax (x) - 29 \, x^{2} + 4 \, x \log \relax (x) + 20 \, x - 4\right )}}{25 \, x^{2} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((96*x^8+160*x^7-576*x^6-288*x^5+928*x^4-320*x^3)*log(x)^3+(-320*x^7-512*x^6+1728*x^5+768*x^4-1
856*x^3+256*x)*log(x)^2+(64*x^7+384*x^6-192*x^5-1536*x^4+1472*x^3-1280*x^2+1536*x-512)*log(x)-128*x^6-256*x^5+
1152*x^4+768*x^3-3712*x^2+2560*x-512)/x^3/log(x)^3,x, algorithm="giac")

[Out]

16/25*x^6 + 32/25*x^5 - 144/25*x^4 - 96/25*x^3 + 464/25*x^2 - 64/5*x - 64/25*(x^7*log(x) + 2*x^6*log(x) - x^6
- 9*x^5*log(x) - 2*x^5 - 6*x^4*log(x) + 9*x^4 + 29*x^3*log(x) + 6*x^3 - 20*x^2*log(x) - 29*x^2 + 4*x*log(x) +
20*x - 4)/(x^2*log(x)^2)

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maple [B]  time = 0.03, size = 115, normalized size = 3.71

method	result	size
risch	\(\frac {16 x^{6}}{25}+\frac {32 x^{5}}{25}-\frac {144 x^{4}}{25}-\frac {96 x^{3}}{25}+\frac {464 x^{2}}{25}-\frac {64 x}{5}-\frac {64 \left (x^{7} \ln \relax (x )+2 x^{6} \ln \relax (x )-x^{6}-9 x^{5} \ln \relax (x )-2 x^{5}-6 x^{4} \ln \relax (x )+9 x^{4}+29 x^{3} \ln \relax (x )+6 x^{3}-20 x^{2} \ln \relax (x )-29 x^{2}+4 x \ln \relax (x )+20 x -4\right )}{25 x^{2} \ln \relax (x )^{2}}\)	\(115\)
default	\(-\frac {64 x}{5}+\frac {384 x^{2}}{25 \ln \relax (x )}-\frac {1856 x}{25 \ln \relax (x )}+\frac {16 x^{6}}{25}+\frac {32 x^{5}}{25}-\frac {144 x^{4}}{25}-\frac {96 x^{3}}{25}+\frac {464 x^{2}}{25}+\frac {256}{25 x^{2} \ln \relax (x )^{2}}-\frac {256}{5 x \ln \relax (x )^{2}}-\frac {576 x^{2}}{25 \ln \relax (x )^{2}}+\frac {1856}{25 \ln \relax (x )^{2}}-\frac {128 x^{4}}{25 \ln \relax (x )}+\frac {256}{5 \ln \relax (x )}+\frac {576 x^{3}}{25 \ln \relax (x )}+\frac {64 x^{4}}{25 \ln \relax (x )^{2}}-\frac {64 x^{5}}{25 \ln \relax (x )}-\frac {384 x}{25 \ln \relax (x )^{2}}-\frac {256}{25 x \ln \relax (x )}+\frac {128 x^{3}}{25 \ln \relax (x )^{2}}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*((96*x^8+160*x^7-576*x^6-288*x^5+928*x^4-320*x^3)*ln(x)^3+(-320*x^7-512*x^6+1728*x^5+768*x^4-1856*x^3
+256*x)*ln(x)^2+(64*x^7+384*x^6-192*x^5-1536*x^4+1472*x^3-1280*x^2+1536*x-512)*ln(x)-128*x^6-256*x^5+1152*x^4+
768*x^3-3712*x^2+2560*x-512)/x^3/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

16/25*x^6+32/25*x^5-144/25*x^4-96/25*x^3+464/25*x^2-64/5*x-64/25*(x^7*ln(x)+2*x^6*ln(x)-x^6-9*x^5*ln(x)-2*x^5-
6*x^4*ln(x)+9*x^4+29*x^3*ln(x)+6*x^3-20*x^2*ln(x)-29*x^2+4*x*ln(x)+20*x-4)/x^2/ln(x)^2

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maxima [C]  time = 0.46, size = 181, normalized size = 5.84 \begin {gather*} \frac {16}{25} \, x^{6} + \frac {32}{25} \, x^{5} - \frac {144}{25} \, x^{4} - \frac {96}{25} \, x^{3} + \frac {464}{25} \, x^{2} - \frac {64}{5} \, x + \frac {256}{5 \, \log \relax (x)} + \frac {1856}{25 \, \log \relax (x)^{2}} - \frac {64}{5} \, {\rm Ei}\left (5 \, \log \relax (x)\right ) - \frac {512}{25} \, {\rm Ei}\left (4 \, \log \relax (x)\right ) + \frac {1728}{25} \, {\rm Ei}\left (3 \, \log \relax (x)\right ) + \frac {768}{25} \, {\rm Ei}\left (2 \, \log \relax (x)\right ) + \frac {256}{25} \, {\rm Ei}\left (-\log \relax (x)\right ) - \frac {1856}{25} \, {\rm Ei}\left (\log \relax (x)\right ) + \frac {1024}{25} \, \Gamma \left (-1, 2 \, \log \relax (x)\right ) + \frac {1472}{25} \, \Gamma \left (-1, -\log \relax (x)\right ) - \frac {3072}{25} \, \Gamma \left (-1, -2 \, \log \relax (x)\right ) - \frac {576}{25} \, \Gamma \left (-1, -3 \, \log \relax (x)\right ) + \frac {1536}{25} \, \Gamma \left (-1, -4 \, \log \relax (x)\right ) + \frac {64}{5} \, \Gamma \left (-1, -5 \, \log \relax (x)\right ) - \frac {1536}{25} \, \Gamma \left (-1, \log \relax (x)\right ) + \frac {2048}{25} \, \Gamma \left (-2, 2 \, \log \relax (x)\right ) - \frac {768}{25} \, \Gamma \left (-2, -\log \relax (x)\right ) - \frac {4608}{25} \, \Gamma \left (-2, -2 \, \log \relax (x)\right ) + \frac {2304}{25} \, \Gamma \left (-2, -3 \, \log \relax (x)\right ) + \frac {2048}{25} \, \Gamma \left (-2, -4 \, \log \relax (x)\right ) - \frac {512}{5} \, \Gamma \left (-2, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((96*x^8+160*x^7-576*x^6-288*x^5+928*x^4-320*x^3)*log(x)^3+(-320*x^7-512*x^6+1728*x^5+768*x^4-1
856*x^3+256*x)*log(x)^2+(64*x^7+384*x^6-192*x^5-1536*x^4+1472*x^3-1280*x^2+1536*x-512)*log(x)-128*x^6-256*x^5+
1152*x^4+768*x^3-3712*x^2+2560*x-512)/x^3/log(x)^3,x, algorithm="maxima")

[Out]

16/25*x^6 + 32/25*x^5 - 144/25*x^4 - 96/25*x^3 + 464/25*x^2 - 64/5*x + 256/5/log(x) + 1856/25/log(x)^2 - 64/5*
Ei(5*log(x)) - 512/25*Ei(4*log(x)) + 1728/25*Ei(3*log(x)) + 768/25*Ei(2*log(x)) + 256/25*Ei(-log(x)) - 1856/25
*Ei(log(x)) + 1024/25*gamma(-1, 2*log(x)) + 1472/25*gamma(-1, -log(x)) - 3072/25*gamma(-1, -2*log(x)) - 576/25
*gamma(-1, -3*log(x)) + 1536/25*gamma(-1, -4*log(x)) + 64/5*gamma(-1, -5*log(x)) - 1536/25*gamma(-1, log(x)) +
2048/25*gamma(-2, 2*log(x)) - 768/25*gamma(-2, -log(x)) - 4608/25*gamma(-2, -2*log(x)) + 2304/25*gamma(-2, -3
*log(x)) + 2048/25*gamma(-2, -4*log(x)) - 512/5*gamma(-2, log(x))

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mupad [B]  time = 2.36, size = 127, normalized size = 4.10 \begin {gather*} \frac {\frac {16\,{\left (2\,x^3+2\,x^2-10\,x+4\right )}^2}{25}-\ln \relax (x)\,\left (\frac {16\,\left (x^4+x^3-5\,x^2+4\,x\right )\,\left (2\,x^3+2\,x^2-10\,x+4\right )}{25}+\frac {16\,\left (x^4+x^3-5\,x^2\right )\,\left (2\,x^3+2\,x^2-10\,x+4\right )}{25}\right )}{x^2\,{\ln \relax (x)}^2}+\frac {16\,\left (x^4+x^3-5\,x^2\right )\,\left (x^4+x^3-5\,x^2+4\,x\right )}{25\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((512*x)/5 - (log(x)^3*(320*x^3 - 928*x^4 + 288*x^5 + 576*x^6 - 160*x^7 - 96*x^8))/25 + (log(x)*(1536*x -
1280*x^2 + 1472*x^3 - 1536*x^4 - 192*x^5 + 384*x^6 + 64*x^7 - 512))/25 - (3712*x^2)/25 + (768*x^3)/25 + (1152*
x^4)/25 - (256*x^5)/25 - (128*x^6)/25 + (log(x)^2*(256*x - 1856*x^3 + 768*x^4 + 1728*x^5 - 512*x^6 - 320*x^7))
/25 - 512/25)/(x^3*log(x)^3),x)

[Out]

((16*(2*x^2 - 10*x + 2*x^3 + 4)^2)/25 - log(x)*((16*(4*x - 5*x^2 + x^3 + x^4)*(2*x^2 - 10*x + 2*x^3 + 4))/25 +
(16*(x^3 - 5*x^2 + x^4)*(2*x^2 - 10*x + 2*x^3 + 4))/25))/(x^2*log(x)^2) + (16*(x^3 - 5*x^2 + x^4)*(4*x - 5*x^
2 + x^3 + x^4))/(25*x^2)

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sympy [B]  time = 0.22, size = 116, normalized size = 3.74 \begin {gather*} \frac {16 x^{6}}{25} + \frac {32 x^{5}}{25} - \frac {144 x^{4}}{25} - \frac {96 x^{3}}{25} + \frac {464 x^{2}}{25} - \frac {64 x}{5} + \frac {64 x^{6} + 128 x^{5} - 576 x^{4} - 384 x^{3} + 1856 x^{2} - 1280 x + \left (- 64 x^{7} - 128 x^{6} + 576 x^{5} + 384 x^{4} - 1856 x^{3} + 1280 x^{2} - 256 x\right ) \log {\relax (x )} + 256}{25 x^{2} \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((96*x**8+160*x**7-576*x**6-288*x**5+928*x**4-320*x**3)*ln(x)**3+(-320*x**7-512*x**6+1728*x**5+
768*x**4-1856*x**3+256*x)*ln(x)**2+(64*x**7+384*x**6-192*x**5-1536*x**4+1472*x**3-1280*x**2+1536*x-512)*ln(x)-
128*x**6-256*x**5+1152*x**4+768*x**3-3712*x**2+2560*x-512)/x**3/ln(x)**3,x)

[Out]

16*x**6/25 + 32*x**5/25 - 144*x**4/25 - 96*x**3/25 + 464*x**2/25 - 64*x/5 + (64*x**6 + 128*x**5 - 576*x**4 - 3
84*x**3 + 1856*x**2 - 1280*x + (-64*x**7 - 128*x**6 + 576*x**5 + 384*x**4 - 1856*x**3 + 1280*x**2 - 256*x)*log
(x) + 256)/(25*x**2*log(x)**2)

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