3.3.7 \(\int \frac {-663552 x-921600 x^2-467456 x^3-102400 x^4-8192 x^5+(664704 x+1383200 x^2+935040 x^3+256000 x^4+24576 x^5+(1152+800 x+128 x^2) \log (5)) \log (x)+(-2304 x-2400 x^2-512 x^3+(-1152-1600 x-384 x^2) \log (5)) \log ^2(x)+(2 x+2 \log (5)) \log ^3(x)}{\log ^3(x)} \, dx\)

Optimal. Leaf size=22 \[ \left (x+\log (5)-\frac {16 x \left (x+(6+2 x)^2\right )}{\log (x)}\right )^2 \]

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Rubi [C]  time = 1.00, antiderivative size = 180, normalized size of antiderivative = 8.18, number of steps used = 54, number of rules used = 9, integrand size = 114, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6688, 12, 6742, 2356, 2306, 2309, 2178, 2297, 2298} \begin {gather*} -1327104 \text {Ei}(2 \log (x))-4147200 \text {Ei}(3 \log (x))+96 (43225+\log (625)) \text {Ei}(3 \log (x))-96 (25+\log (625)) \text {Ei}(3 \log (x))+64 (20772+25 \log (5)) \text {Ei}(2 \log (x))-64 (36+25 \log (5)) \text {Ei}(2 \log (x))+\frac {4096 x^6}{\log ^2(x)}+\frac {51200 x^5}{\log ^2(x)}+\frac {233728 x^4}{\log ^2(x)}-\frac {128 x^4}{\log (x)}+\frac {460800 x^3}{\log ^2(x)}-\frac {32 x^3 (43225+\log (625))}{\log (x)}+\frac {1382400 x^3}{\log (x)}+x^2+\frac {331776 x^2}{\log ^2(x)}-\frac {32 x^2 (20772+25 \log (5))}{\log (x)}+\frac {663552 x^2}{\log (x)}-\frac {1152 x \log (5)}{\log (x)}+2 x \log (5) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-663552*x - 921600*x^2 - 467456*x^3 - 102400*x^4 - 8192*x^5 + (664704*x + 1383200*x^2 + 935040*x^3 + 2560
00*x^4 + 24576*x^5 + (1152 + 800*x + 128*x^2)*Log[5])*Log[x] + (-2304*x - 2400*x^2 - 512*x^3 + (-1152 - 1600*x
- 384*x^2)*Log[5])*Log[x]^2 + (2*x + 2*Log[5])*Log[x]^3)/Log[x]^3,x]

[Out]

x^2 - 1327104*ExpIntegralEi[2*Log[x]] - 4147200*ExpIntegralEi[3*Log[x]] + 2*x*Log[5] - 64*ExpIntegralEi[2*Log[
x]]*(36 + 25*Log[5]) + 64*ExpIntegralEi[2*Log[x]]*(20772 + 25*Log[5]) - 96*ExpIntegralEi[3*Log[x]]*(25 + Log[6
25]) + 96*ExpIntegralEi[3*Log[x]]*(43225 + Log[625]) + (331776*x^2)/Log[x]^2 + (460800*x^3)/Log[x]^2 + (233728
*x^4)/Log[x]^2 + (51200*x^5)/Log[x]^2 + (4096*x^6)/Log[x]^2 + (663552*x^2)/Log[x] + (1382400*x^3)/Log[x] - (12
8*x^4)/Log[x] - (1152*x*Log[5])/Log[x] - (32*x^2*(20772 + 25*Log[5]))/Log[x] - (32*x^3*(43225 + Log[625]))/Log
[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (16 x \left (36+25 x+4 x^2\right )-(x+\log (5)) \log (x)\right ) \left (-16 \left (36+25 x+4 x^2\right )+32 \left (18+25 x+6 x^2\right ) \log (x)-\log ^2(x)\right )}{\log ^3(x)} \, dx\\ &=2 \int \frac {\left (16 x \left (36+25 x+4 x^2\right )-(x+\log (5)) \log (x)\right ) \left (-16 \left (36+25 x+4 x^2\right )+32 \left (18+25 x+6 x^2\right ) \log (x)-\log ^2(x)\right )}{\log ^3(x)} \, dx\\ &=2 \int \left (x+\log (5)-\frac {256 x (4+x)^2 (9+4 x)^2}{\log ^3(x)}+\frac {16 \left (36+25 x+4 x^2\right ) \left (577 x+800 x^2+192 x^3+\log (5)\right )}{\log ^2(x)}+\frac {16 \left (-16 x^3-36 \log (5)-2 x (36+25 \log (5))-3 x^2 (25+\log (625))\right )}{\log (x)}\right ) \, dx\\ &=x^2+2 x \log (5)+32 \int \frac {\left (36+25 x+4 x^2\right ) \left (577 x+800 x^2+192 x^3+\log (5)\right )}{\log ^2(x)} \, dx+32 \int \frac {-16 x^3-36 \log (5)-2 x (36+25 \log (5))-3 x^2 (25+\log (625))}{\log (x)} \, dx-512 \int \frac {x (4+x)^2 (9+4 x)^2}{\log ^3(x)} \, dx\\ &=x^2+2 x \log (5)+32 \int \left (\frac {29220 x^3}{\log ^2(x)}+\frac {8000 x^4}{\log ^2(x)}+\frac {768 x^5}{\log ^2(x)}+\frac {36 \log (5)}{\log ^2(x)}+\frac {x (20772+25 \log (5))}{\log ^2(x)}+\frac {x^2 (43225+\log (625))}{\log ^2(x)}\right ) \, dx+32 \int \left (-\frac {16 x^3}{\log (x)}-\frac {36 \log (5)}{\log (x)}-\frac {2 x (36+25 \log (5))}{\log (x)}-\frac {3 x^2 (25+\log (625))}{\log (x)}\right ) \, dx-512 \int \left (\frac {1296 x}{\log ^3(x)}+\frac {1800 x^2}{\log ^3(x)}+\frac {913 x^3}{\log ^3(x)}+\frac {200 x^4}{\log ^3(x)}+\frac {16 x^5}{\log ^3(x)}\right ) \, dx\\ &=x^2+2 x \log (5)-512 \int \frac {x^3}{\log (x)} \, dx-8192 \int \frac {x^5}{\log ^3(x)} \, dx+24576 \int \frac {x^5}{\log ^2(x)} \, dx-102400 \int \frac {x^4}{\log ^3(x)} \, dx+256000 \int \frac {x^4}{\log ^2(x)} \, dx-467456 \int \frac {x^3}{\log ^3(x)} \, dx-663552 \int \frac {x}{\log ^3(x)} \, dx-921600 \int \frac {x^2}{\log ^3(x)} \, dx+935040 \int \frac {x^3}{\log ^2(x)} \, dx+(1152 \log (5)) \int \frac {1}{\log ^2(x)} \, dx-(1152 \log (5)) \int \frac {1}{\log (x)} \, dx-(64 (36+25 \log (5))) \int \frac {x}{\log (x)} \, dx+(32 (20772+25 \log (5))) \int \frac {x}{\log ^2(x)} \, dx-(96 (25+\log (625))) \int \frac {x^2}{\log (x)} \, dx+(32 (43225+\log (625))) \int \frac {x^2}{\log ^2(x)} \, dx\\ &=x^2+2 x \log (5)+\frac {331776 x^2}{\log ^2(x)}+\frac {460800 x^3}{\log ^2(x)}+\frac {233728 x^4}{\log ^2(x)}+\frac {51200 x^5}{\log ^2(x)}+\frac {4096 x^6}{\log ^2(x)}-\frac {935040 x^4}{\log (x)}-\frac {256000 x^5}{\log (x)}-\frac {24576 x^6}{\log (x)}-\frac {1152 x \log (5)}{\log (x)}-\frac {32 x^2 (20772+25 \log (5))}{\log (x)}-\frac {32 x^3 (43225+\log (625))}{\log (x)}-1152 \log (5) \text {li}(x)-512 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )-24576 \int \frac {x^5}{\log ^2(x)} \, dx+147456 \int \frac {x^5}{\log (x)} \, dx-256000 \int \frac {x^4}{\log ^2(x)} \, dx-663552 \int \frac {x}{\log ^2(x)} \, dx-934912 \int \frac {x^3}{\log ^2(x)} \, dx+1280000 \int \frac {x^4}{\log (x)} \, dx-1382400 \int \frac {x^2}{\log ^2(x)} \, dx+3740160 \int \frac {x^3}{\log (x)} \, dx+(1152 \log (5)) \int \frac {1}{\log (x)} \, dx-(64 (36+25 \log (5))) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )+(64 (20772+25 \log (5))) \int \frac {x}{\log (x)} \, dx-(96 (25+\log (625))) \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+(96 (43225+\log (625))) \int \frac {x^2}{\log (x)} \, dx\\ &=x^2-512 \text {Ei}(4 \log (x))+2 x \log (5)-64 \text {Ei}(2 \log (x)) (36+25 \log (5))-96 \text {Ei}(3 \log (x)) (25+\log (625))+\frac {331776 x^2}{\log ^2(x)}+\frac {460800 x^3}{\log ^2(x)}+\frac {233728 x^4}{\log ^2(x)}+\frac {51200 x^5}{\log ^2(x)}+\frac {4096 x^6}{\log ^2(x)}+\frac {663552 x^2}{\log (x)}+\frac {1382400 x^3}{\log (x)}-\frac {128 x^4}{\log (x)}-\frac {1152 x \log (5)}{\log (x)}-\frac {32 x^2 (20772+25 \log (5))}{\log (x)}-\frac {32 x^3 (43225+\log (625))}{\log (x)}-147456 \int \frac {x^5}{\log (x)} \, dx+147456 \operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )-1280000 \int \frac {x^4}{\log (x)} \, dx+1280000 \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )-1327104 \int \frac {x}{\log (x)} \, dx-3739648 \int \frac {x^3}{\log (x)} \, dx+3740160 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )-4147200 \int \frac {x^2}{\log (x)} \, dx+(64 (20772+25 \log (5))) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )+(96 (43225+\log (625))) \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )\\ &=x^2+3739648 \text {Ei}(4 \log (x))+1280000 \text {Ei}(5 \log (x))+147456 \text {Ei}(6 \log (x))+2 x \log (5)-64 \text {Ei}(2 \log (x)) (36+25 \log (5))+64 \text {Ei}(2 \log (x)) (20772+25 \log (5))-96 \text {Ei}(3 \log (x)) (25+\log (625))+96 \text {Ei}(3 \log (x)) (43225+\log (625))+\frac {331776 x^2}{\log ^2(x)}+\frac {460800 x^3}{\log ^2(x)}+\frac {233728 x^4}{\log ^2(x)}+\frac {51200 x^5}{\log ^2(x)}+\frac {4096 x^6}{\log ^2(x)}+\frac {663552 x^2}{\log (x)}+\frac {1382400 x^3}{\log (x)}-\frac {128 x^4}{\log (x)}-\frac {1152 x \log (5)}{\log (x)}-\frac {32 x^2 (20772+25 \log (5))}{\log (x)}-\frac {32 x^3 (43225+\log (625))}{\log (x)}-147456 \operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )-1280000 \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )-1327104 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-3739648 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )-4147200 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )\\ &=x^2-1327104 \text {Ei}(2 \log (x))-4147200 \text {Ei}(3 \log (x))+2 x \log (5)-64 \text {Ei}(2 \log (x)) (36+25 \log (5))+64 \text {Ei}(2 \log (x)) (20772+25 \log (5))-96 \text {Ei}(3 \log (x)) (25+\log (625))+96 \text {Ei}(3 \log (x)) (43225+\log (625))+\frac {331776 x^2}{\log ^2(x)}+\frac {460800 x^3}{\log ^2(x)}+\frac {233728 x^4}{\log ^2(x)}+\frac {51200 x^5}{\log ^2(x)}+\frac {4096 x^6}{\log ^2(x)}+\frac {663552 x^2}{\log (x)}+\frac {1382400 x^3}{\log (x)}-\frac {128 x^4}{\log (x)}-\frac {1152 x \log (5)}{\log (x)}-\frac {32 x^2 (20772+25 \log (5))}{\log (x)}-\frac {32 x^3 (43225+\log (625))}{\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 42, normalized size = 1.91 \begin {gather*} \frac {x \left (576+400 x+64 x^2-\log (x)\right ) \left (16 x \left (36+25 x+4 x^2\right )-(x+\log (25)) \log (x)\right )}{\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-663552*x - 921600*x^2 - 467456*x^3 - 102400*x^4 - 8192*x^5 + (664704*x + 1383200*x^2 + 935040*x^3
+ 256000*x^4 + 24576*x^5 + (1152 + 800*x + 128*x^2)*Log[5])*Log[x] + (-2304*x - 2400*x^2 - 512*x^3 + (-1152 -
1600*x - 384*x^2)*Log[5])*Log[x]^2 + (2*x + 2*Log[5])*Log[x]^3)/Log[x]^3,x]

[Out]

(x*(576 + 400*x + 64*x^2 - Log[x])*(16*x*(36 + 25*x + 4*x^2) - (x + Log[25])*Log[x]))/Log[x]^2

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fricas [B]  time = 0.54, size = 82, normalized size = 3.73 \begin {gather*} \frac {4096 \, x^{6} + 51200 \, x^{5} + 233728 \, x^{4} + 460800 \, x^{3} + {\left (x^{2} + 2 \, x \log \relax (5)\right )} \log \relax (x)^{2} + 331776 \, x^{2} - 32 \, {\left (4 \, x^{4} + 25 \, x^{3} + 36 \, x^{2} + {\left (4 \, x^{3} + 25 \, x^{2} + 36 \, x\right )} \log \relax (5)\right )} \log \relax (x)}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(5)+2*x)*log(x)^3+((-384*x^2-1600*x-1152)*log(5)-512*x^3-2400*x^2-2304*x)*log(x)^2+((128*x^2+
800*x+1152)*log(5)+24576*x^5+256000*x^4+935040*x^3+1383200*x^2+664704*x)*log(x)-8192*x^5-102400*x^4-467456*x^3
-921600*x^2-663552*x)/log(x)^3,x, algorithm="fricas")

[Out]

(4096*x^6 + 51200*x^5 + 233728*x^4 + 460800*x^3 + (x^2 + 2*x*log(5))*log(x)^2 + 331776*x^2 - 32*(4*x^4 + 25*x^
3 + 36*x^2 + (4*x^3 + 25*x^2 + 36*x)*log(5))*log(x))/log(x)^2

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giac [B]  time = 0.40, size = 112, normalized size = 5.09 \begin {gather*} \frac {4096 \, x^{6}}{\log \relax (x)^{2}} + \frac {51200 \, x^{5}}{\log \relax (x)^{2}} - \frac {128 \, x^{4}}{\log \relax (x)} - \frac {128 \, x^{3} \log \relax (5)}{\log \relax (x)} + x^{2} + 2 \, x \log \relax (5) + \frac {233728 \, x^{4}}{\log \relax (x)^{2}} - \frac {800 \, x^{3}}{\log \relax (x)} - \frac {800 \, x^{2} \log \relax (5)}{\log \relax (x)} + \frac {460800 \, x^{3}}{\log \relax (x)^{2}} - \frac {1152 \, x^{2}}{\log \relax (x)} - \frac {1152 \, x \log \relax (5)}{\log \relax (x)} + \frac {331776 \, x^{2}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(5)+2*x)*log(x)^3+((-384*x^2-1600*x-1152)*log(5)-512*x^3-2400*x^2-2304*x)*log(x)^2+((128*x^2+
800*x+1152)*log(5)+24576*x^5+256000*x^4+935040*x^3+1383200*x^2+664704*x)*log(x)-8192*x^5-102400*x^4-467456*x^3
-921600*x^2-663552*x)/log(x)^3,x, algorithm="giac")

[Out]

4096*x^6/log(x)^2 + 51200*x^5/log(x)^2 - 128*x^4/log(x) - 128*x^3*log(5)/log(x) + x^2 + 2*x*log(5) + 233728*x^
4/log(x)^2 - 800*x^3/log(x) - 800*x^2*log(5)/log(x) + 460800*x^3/log(x)^2 - 1152*x^2/log(x) - 1152*x*log(5)/lo
g(x) + 331776*x^2/log(x)^2

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maple [B]  time = 0.06, size = 82, normalized size = 3.73

method	result	size
risch	\(2 x \ln \relax (5)+x^{2}-\frac {32 x \left (-128 x^{5}+4 x^{2} \ln \relax (5) \ln \relax (x )-1600 x^{4}+4 x^{3} \ln \relax (x )+25 x \ln \relax (5) \ln \relax (x )-7304 x^{3}+25 x^{2} \ln \relax (x )+36 \ln \relax (5) \ln \relax (x )-14400 x^{2}+36 x \ln \relax (x )-10368 x \right )}{\ln \relax (x )^{2}}\)	\(82\)
norman	\(\frac {x^{2} \ln \relax (x )^{2}+\left (-1152-800 \ln \relax (5)\right ) x^{2} \ln \relax (x )+\left (-800-128 \ln \relax (5)\right ) x^{3} \ln \relax (x )+331776 x^{2}+460800 x^{3}+233728 x^{4}+51200 x^{5}+4096 x^{6}-128 x^{4} \ln \relax (x )-1152 x \ln \relax (5) \ln \relax (x )+2 \ln \relax (x )^{2} \ln \relax (5) x}{\ln \relax (x )^{2}}\)	\(87\)
default	\(-\frac {1152 x^{2}}{\ln \relax (x )}+x^{2}+\frac {331776 x^{2}}{\ln \relax (x )^{2}}-\frac {128 x^{4}}{\ln \relax (x )}+2 x \ln \relax (5)-\frac {800 x^{3}}{\ln \relax (x )}+1152 \ln \relax (5) \left (-\frac {x}{\ln \relax (x )}-\expIntegralEi \left (1, -\ln \relax (x )\right )\right )+800 \ln \relax (5) \left (-\frac {x^{2}}{\ln \relax (x )}-2 \expIntegralEi \left (1, -2 \ln \relax (x )\right )\right )+1152 \ln \relax (5) \expIntegralEi \left (1, -\ln \relax (x )\right )+128 \ln \relax (5) \left (-\frac {x^{3}}{\ln \relax (x )}-3 \expIntegralEi \left (1, -3 \ln \relax (x )\right )\right )+1600 \ln \relax (5) \expIntegralEi \left (1, -2 \ln \relax (x )\right )+\frac {4096 x^{6}}{\ln \relax (x )^{2}}+384 \ln \relax (5) \expIntegralEi \left (1, -3 \ln \relax (x )\right )+\frac {51200 x^{5}}{\ln \relax (x )^{2}}+\frac {233728 x^{4}}{\ln \relax (x )^{2}}+\frac {460800 x^{3}}{\ln \relax (x )^{2}}\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*ln(5)+2*x)*ln(x)^3+((-384*x^2-1600*x-1152)*ln(5)-512*x^3-2400*x^2-2304*x)*ln(x)^2+((128*x^2+800*x+1152
)*ln(5)+24576*x^5+256000*x^4+935040*x^3+1383200*x^2+664704*x)*ln(x)-8192*x^5-102400*x^4-467456*x^3-921600*x^2-
663552*x)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

2*x*ln(5)+x^2-32*x*(-128*x^5+4*x^2*ln(5)*ln(x)-1600*x^4+4*x^3*ln(x)+25*x*ln(5)*ln(x)-7304*x^3+25*x^2*ln(x)+36*
ln(5)*ln(x)-14400*x^2+36*x*ln(x)-10368*x)/ln(x)^2

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maxima [C]  time = 0.55, size = 165, normalized size = 7.50 \begin {gather*} x^{2} + 2 \, x \log \relax (5) - 384 \, {\rm Ei}\left (3 \, \log \relax (x)\right ) \log \relax (5) - 1600 \, {\rm Ei}\left (2 \, \log \relax (x)\right ) \log \relax (5) - 1152 \, {\rm Ei}\left (\log \relax (x)\right ) \log \relax (5) + 1152 \, \Gamma \left (-1, -\log \relax (x)\right ) \log \relax (5) + 1600 \, \Gamma \left (-1, -2 \, \log \relax (x)\right ) \log \relax (5) + 384 \, \Gamma \left (-1, -3 \, \log \relax (x)\right ) \log \relax (5) - 512 \, {\rm Ei}\left (4 \, \log \relax (x)\right ) - 2400 \, {\rm Ei}\left (3 \, \log \relax (x)\right ) - 2304 \, {\rm Ei}\left (2 \, \log \relax (x)\right ) + 1329408 \, \Gamma \left (-1, -2 \, \log \relax (x)\right ) + 4149600 \, \Gamma \left (-1, -3 \, \log \relax (x)\right ) + 3740160 \, \Gamma \left (-1, -4 \, \log \relax (x)\right ) + 1280000 \, \Gamma \left (-1, -5 \, \log \relax (x)\right ) + 147456 \, \Gamma \left (-1, -6 \, \log \relax (x)\right ) + 2654208 \, \Gamma \left (-2, -2 \, \log \relax (x)\right ) + 8294400 \, \Gamma \left (-2, -3 \, \log \relax (x)\right ) + 7479296 \, \Gamma \left (-2, -4 \, \log \relax (x)\right ) + 2560000 \, \Gamma \left (-2, -5 \, \log \relax (x)\right ) + 294912 \, \Gamma \left (-2, -6 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(5)+2*x)*log(x)^3+((-384*x^2-1600*x-1152)*log(5)-512*x^3-2400*x^2-2304*x)*log(x)^2+((128*x^2+
800*x+1152)*log(5)+24576*x^5+256000*x^4+935040*x^3+1383200*x^2+664704*x)*log(x)-8192*x^5-102400*x^4-467456*x^3
-921600*x^2-663552*x)/log(x)^3,x, algorithm="maxima")

[Out]

x^2 + 2*x*log(5) - 384*Ei(3*log(x))*log(5) - 1600*Ei(2*log(x))*log(5) - 1152*Ei(log(x))*log(5) + 1152*gamma(-1
, -log(x))*log(5) + 1600*gamma(-1, -2*log(x))*log(5) + 384*gamma(-1, -3*log(x))*log(5) - 512*Ei(4*log(x)) - 24
00*Ei(3*log(x)) - 2304*Ei(2*log(x)) + 1329408*gamma(-1, -2*log(x)) + 4149600*gamma(-1, -3*log(x)) + 3740160*ga
mma(-1, -4*log(x)) + 1280000*gamma(-1, -5*log(x)) + 147456*gamma(-1, -6*log(x)) + 2654208*gamma(-2, -2*log(x))
+ 8294400*gamma(-2, -3*log(x)) + 7479296*gamma(-2, -4*log(x)) + 2560000*gamma(-2, -5*log(x)) + 294912*gamma(-
2, -6*log(x))

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mupad [B]  time = 0.49, size = 74, normalized size = 3.36 \begin {gather*} \frac {x\,\left (4096\,x^5+51200\,x^4+233728\,x^3+460800\,x^2+331776\,x\right )-x\,\ln \relax (x)\,\left (1152\,x+1152\,\ln \relax (5)+800\,x\,\ln \relax (5)+128\,x^2\,\ln \relax (5)+800\,x^2+128\,x^3\right )}{{\ln \relax (x)}^2}+x\,\left (x+\ln \left (25\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(663552*x + log(x)^2*(2304*x + log(5)*(1600*x + 384*x^2 + 1152) + 2400*x^2 + 512*x^3) - log(x)*(664704*x
+ log(5)*(800*x + 128*x^2 + 1152) + 1383200*x^2 + 935040*x^3 + 256000*x^4 + 24576*x^5) - log(x)^3*(2*x + 2*log
(5)) + 921600*x^2 + 467456*x^3 + 102400*x^4 + 8192*x^5)/log(x)^3,x)

[Out]

(x*(331776*x + 460800*x^2 + 233728*x^3 + 51200*x^4 + 4096*x^5) - x*log(x)*(1152*x + 1152*log(5) + 800*x*log(5)
+ 128*x^2*log(5) + 800*x^2 + 128*x^3))/log(x)^2 + x*(x + log(25))

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sympy [B]  time = 0.16, size = 83, normalized size = 3.77 \begin {gather*} x^{2} + 2 x \log {\relax (5 )} + \frac {4096 x^{6} + 51200 x^{5} + 233728 x^{4} + 460800 x^{3} + 331776 x^{2} + \left (- 128 x^{4} - 800 x^{3} - 128 x^{3} \log {\relax (5 )} - 800 x^{2} \log {\relax (5 )} - 1152 x^{2} - 1152 x \log {\relax (5 )}\right ) \log {\relax (x )}}{\log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*ln(5)+2*x)*ln(x)**3+((-384*x**2-1600*x-1152)*ln(5)-512*x**3-2400*x**2-2304*x)*ln(x)**2+((128*x**
2+800*x+1152)*ln(5)+24576*x**5+256000*x**4+935040*x**3+1383200*x**2+664704*x)*ln(x)-8192*x**5-102400*x**4-4674
56*x**3-921600*x**2-663552*x)/ln(x)**3,x)

[Out]

x**2 + 2*x*log(5) + (4096*x**6 + 51200*x**5 + 233728*x**4 + 460800*x**3 + 331776*x**2 + (-128*x**4 - 800*x**3
- 128*x**3*log(5) - 800*x**2*log(5) - 1152*x**2 - 1152*x*log(5))*log(x))/log(x)**2

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