3.19.14 \(\int \frac {-1216-64 x+(-1600-768 x) \log (x)+320 x^2 \log ^2(x)}{(-30+6 x+(25 x-11 x^2) \log (x)+5 x^3 \log ^2(x)) \log ^5(\frac {24-20 x \log (x)}{5-x+x^2 \log (x)})} \, dx\)

Optimal. Leaf size=29 \[ \frac {16}{\log ^4\left (\frac {4}{5-\frac {25+x}{5+\frac {1}{1-x \log (x)}}}\right )} \]

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Rubi [F]  time = 2.47, 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 {-1216-64 x+(-1600-768 x) \log (x)+320 x^2 \log ^2(x)}{\left (-30+6 x+\left (25 x-11 x^2\right ) \log (x)+5 x^3 \log ^2(x)\right ) \log ^5\left (\frac {24-20 x \log (x)}{5-x+x^2 \log (x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1216 - 64*x + (-1600 - 768*x)*Log[x] + 320*x^2*Log[x]^2)/((-30 + 6*x + (25*x - 11*x^2)*Log[x] + 5*x^3*Lo
g[x]^2)*Log[(24 - 20*x*Log[x])/(5 - x + x^2*Log[x])]^5),x]

[Out]

1216*Defer[Int][1/((30 - 6*x - 25*x*Log[x] + 11*x^2*Log[x] - 5*x^3*Log[x]^2)*Log[(-4*(-6 + 5*x*Log[x]))/(5 - x
+ x^2*Log[x])]^5), x] + 64*Defer[Int][x/((30 - 6*x - 25*x*Log[x] + 11*x^2*Log[x] - 5*x^3*Log[x]^2)*Log[(-4*(-
6 + 5*x*Log[x]))/(5 - x + x^2*Log[x])]^5), x] + 1600*Defer[Int][Log[x]/((30 - 6*x - 25*x*Log[x] + 11*x^2*Log[x
] - 5*x^3*Log[x]^2)*Log[(-4*(-6 + 5*x*Log[x]))/(5 - x + x^2*Log[x])]^5), x] + 768*Defer[Int][(x*Log[x])/((30 -
6*x - 25*x*Log[x] + 11*x^2*Log[x] - 5*x^3*Log[x]^2)*Log[(-4*(-6 + 5*x*Log[x]))/(5 - x + x^2*Log[x])]^5), x] +
320*Defer[Int][(x^2*Log[x]^2)/((-30 + 6*x + 25*x*Log[x] - 11*x^2*Log[x] + 5*x^3*Log[x]^2)*Log[(-4*(-6 + 5*x*L
og[x]))/(5 - x + x^2*Log[x])]^5), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1216+64 x-(-1600-768 x) \log (x)-320 x^2 \log ^2(x)}{\left (30-6 x-\left (25 x-11 x^2\right ) \log (x)-5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )} \, dx\\ &=\int \frac {64 \left (19+x+25 \log (x)+12 x \log (x)-5 x^2 \log ^2(x)\right )}{\left (30-6 x-\left (25 x-11 x^2\right ) \log (x)-5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )} \, dx\\ &=64 \int \frac {19+x+25 \log (x)+12 x \log (x)-5 x^2 \log ^2(x)}{\left (30-6 x-\left (25 x-11 x^2\right ) \log (x)-5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )} \, dx\\ &=64 \int \left (\frac {19}{\left (30-6 x-25 x \log (x)+11 x^2 \log (x)-5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )}+\frac {x}{\left (30-6 x-25 x \log (x)+11 x^2 \log (x)-5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )}+\frac {25 \log (x)}{\left (30-6 x-25 x \log (x)+11 x^2 \log (x)-5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )}+\frac {12 x \log (x)}{\left (30-6 x-25 x \log (x)+11 x^2 \log (x)-5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )}+\frac {5 x^2 \log ^2(x)}{\left (-30+6 x+25 x \log (x)-11 x^2 \log (x)+5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )}\right ) \, dx\\ &=64 \int \frac {x}{\left (30-6 x-25 x \log (x)+11 x^2 \log (x)-5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )} \, dx+320 \int \frac {x^2 \log ^2(x)}{\left (-30+6 x+25 x \log (x)-11 x^2 \log (x)+5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )} \, dx+768 \int \frac {x \log (x)}{\left (30-6 x-25 x \log (x)+11 x^2 \log (x)-5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )} \, dx+1216 \int \frac {1}{\left (30-6 x-25 x \log (x)+11 x^2 \log (x)-5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )} \, dx+1600 \int \frac {\log (x)}{\left (30-6 x-25 x \log (x)+11 x^2 \log (x)-5 x^3 \log ^2(x)\right ) \log ^5\left (-\frac {4 (-6+5 x \log (x))}{5-x+x^2 \log (x)}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 26, normalized size = 0.90 \begin {gather*} \frac {16}{\log ^4\left (\frac {24-20 x \log (x)}{5-x+x^2 \log (x)}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1216 - 64*x + (-1600 - 768*x)*Log[x] + 320*x^2*Log[x]^2)/((-30 + 6*x + (25*x - 11*x^2)*Log[x] + 5*
x^3*Log[x]^2)*Log[(24 - 20*x*Log[x])/(5 - x + x^2*Log[x])]^5),x]

[Out]

16/Log[(24 - 20*x*Log[x])/(5 - x + x^2*Log[x])]^4

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fricas [A]  time = 0.73, size = 27, normalized size = 0.93 \begin {gather*} \frac {16}{\log \left (-\frac {4 \, {\left (5 \, x \log \relax (x) - 6\right )}}{x^{2} \log \relax (x) - x + 5}\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((320*x^2*log(x)^2+(-768*x-1600)*log(x)-64*x-1216)/(5*x^3*log(x)^2+(-11*x^2+25*x)*log(x)+6*x-30)/log(
(-20*x*log(x)+24)/(x^2*log(x)+5-x))^5,x, algorithm="fricas")

[Out]

16/log(-4*(5*x*log(x) - 6)/(x^2*log(x) - x + 5))^4

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giac [B]  time = 2.60, size = 603, normalized size = 20.79 \begin {gather*} \frac {16 \, {\left (5 \, x^{2} \log \relax (x)^{2} - 12 \, x \log \relax (x) - x - 25 \, \log \relax (x) - 19\right )}}{5 \, x^{2} \log \left (x^{2} \log \relax (x) - x + 5\right )^{4} \log \relax (x)^{2} - 20 \, x^{2} \log \left (x^{2} \log \relax (x) - x + 5\right )^{3} \log \left (-20 \, x \log \relax (x) + 24\right ) \log \relax (x)^{2} + 30 \, x^{2} \log \left (x^{2} \log \relax (x) - x + 5\right )^{2} \log \left (-20 \, x \log \relax (x) + 24\right )^{2} \log \relax (x)^{2} - 20 \, x^{2} \log \left (x^{2} \log \relax (x) - x + 5\right ) \log \left (-20 \, x \log \relax (x) + 24\right )^{3} \log \relax (x)^{2} + 5 \, x^{2} \log \left (-20 \, x \log \relax (x) + 24\right )^{4} \log \relax (x)^{2} - 12 \, x \log \left (x^{2} \log \relax (x) - x + 5\right )^{4} \log \relax (x) + 48 \, x \log \left (x^{2} \log \relax (x) - x + 5\right )^{3} \log \left (-20 \, x \log \relax (x) + 24\right ) \log \relax (x) - 72 \, x \log \left (x^{2} \log \relax (x) - x + 5\right )^{2} \log \left (-20 \, x \log \relax (x) + 24\right )^{2} \log \relax (x) + 48 \, x \log \left (x^{2} \log \relax (x) - x + 5\right ) \log \left (-20 \, x \log \relax (x) + 24\right )^{3} \log \relax (x) - 12 \, x \log \left (-20 \, x \log \relax (x) + 24\right )^{4} \log \relax (x) - x \log \left (x^{2} \log \relax (x) - x + 5\right )^{4} + 4 \, x \log \left (x^{2} \log \relax (x) - x + 5\right )^{3} \log \left (-20 \, x \log \relax (x) + 24\right ) - 6 \, x \log \left (x^{2} \log \relax (x) - x + 5\right )^{2} \log \left (-20 \, x \log \relax (x) + 24\right )^{2} + 4 \, x \log \left (x^{2} \log \relax (x) - x + 5\right ) \log \left (-20 \, x \log \relax (x) + 24\right )^{3} - x \log \left (-20 \, x \log \relax (x) + 24\right )^{4} - 25 \, \log \left (x^{2} \log \relax (x) - x + 5\right )^{4} \log \relax (x) + 100 \, \log \left (x^{2} \log \relax (x) - x + 5\right )^{3} \log \left (-20 \, x \log \relax (x) + 24\right ) \log \relax (x) - 150 \, \log \left (x^{2} \log \relax (x) - x + 5\right )^{2} \log \left (-20 \, x \log \relax (x) + 24\right )^{2} \log \relax (x) + 100 \, \log \left (x^{2} \log \relax (x) - x + 5\right ) \log \left (-20 \, x \log \relax (x) + 24\right )^{3} \log \relax (x) - 25 \, \log \left (-20 \, x \log \relax (x) + 24\right )^{4} \log \relax (x) - 19 \, \log \left (x^{2} \log \relax (x) - x + 5\right )^{4} + 76 \, \log \left (x^{2} \log \relax (x) - x + 5\right )^{3} \log \left (-20 \, x \log \relax (x) + 24\right ) - 114 \, \log \left (x^{2} \log \relax (x) - x + 5\right )^{2} \log \left (-20 \, x \log \relax (x) + 24\right )^{2} + 76 \, \log \left (x^{2} \log \relax (x) - x + 5\right ) \log \left (-20 \, x \log \relax (x) + 24\right )^{3} - 19 \, \log \left (-20 \, x \log \relax (x) + 24\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((320*x^2*log(x)^2+(-768*x-1600)*log(x)-64*x-1216)/(5*x^3*log(x)^2+(-11*x^2+25*x)*log(x)+6*x-30)/log(
(-20*x*log(x)+24)/(x^2*log(x)+5-x))^5,x, algorithm="giac")

[Out]

16*(5*x^2*log(x)^2 - 12*x*log(x) - x - 25*log(x) - 19)/(5*x^2*log(x^2*log(x) - x + 5)^4*log(x)^2 - 20*x^2*log(
x^2*log(x) - x + 5)^3*log(-20*x*log(x) + 24)*log(x)^2 + 30*x^2*log(x^2*log(x) - x + 5)^2*log(-20*x*log(x) + 24
)^2*log(x)^2 - 20*x^2*log(x^2*log(x) - x + 5)*log(-20*x*log(x) + 24)^3*log(x)^2 + 5*x^2*log(-20*x*log(x) + 24)
^4*log(x)^2 - 12*x*log(x^2*log(x) - x + 5)^4*log(x) + 48*x*log(x^2*log(x) - x + 5)^3*log(-20*x*log(x) + 24)*lo
g(x) - 72*x*log(x^2*log(x) - x + 5)^2*log(-20*x*log(x) + 24)^2*log(x) + 48*x*log(x^2*log(x) - x + 5)*log(-20*x
*log(x) + 24)^3*log(x) - 12*x*log(-20*x*log(x) + 24)^4*log(x) - x*log(x^2*log(x) - x + 5)^4 + 4*x*log(x^2*log(
x) - x + 5)^3*log(-20*x*log(x) + 24) - 6*x*log(x^2*log(x) - x + 5)^2*log(-20*x*log(x) + 24)^2 + 4*x*log(x^2*lo
g(x) - x + 5)*log(-20*x*log(x) + 24)^3 - x*log(-20*x*log(x) + 24)^4 - 25*log(x^2*log(x) - x + 5)^4*log(x) + 10
0*log(x^2*log(x) - x + 5)^3*log(-20*x*log(x) + 24)*log(x) - 150*log(x^2*log(x) - x + 5)^2*log(-20*x*log(x) + 2
4)^2*log(x) + 100*log(x^2*log(x) - x + 5)*log(-20*x*log(x) + 24)^3*log(x) - 25*log(-20*x*log(x) + 24)^4*log(x)
- 19*log(x^2*log(x) - x + 5)^4 + 76*log(x^2*log(x) - x + 5)^3*log(-20*x*log(x) + 24) - 114*log(x^2*log(x) - x
+ 5)^2*log(-20*x*log(x) + 24)^2 + 76*log(x^2*log(x) - x + 5)*log(-20*x*log(x) + 24)^3 - 19*log(-20*x*log(x) +
24)^4)

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maple [C]  time = 0.10, size = 238, normalized size = 8.21

method	result	size
risch	\(\frac {256}{\left (4 \ln \relax (2)+2 \ln \relax (5)+2 i \pi +2 \ln \left (x \ln \relax (x )-\frac {6}{5}\right )-2 \ln \left (x^{2} \ln \relax (x )+5-x \right )-i \pi \,\mathrm {csgn}\left (i \left (x \ln \relax (x )-\frac {6}{5}\right )\right ) \mathrm {csgn}\left (\frac {i}{x^{2} \ln \relax (x )+5-x}\right ) \mathrm {csgn}\left (\frac {i \left (x \ln \relax (x )-\frac {6}{5}\right )}{x^{2} \ln \relax (x )+5-x}\right )+i \pi \,\mathrm {csgn}\left (i \left (x \ln \relax (x )-\frac {6}{5}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \ln \relax (x )-\frac {6}{5}\right )}{x^{2} \ln \relax (x )+5-x}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{x^{2} \ln \relax (x )+5-x}\right ) \mathrm {csgn}\left (\frac {i \left (x \ln \relax (x )-\frac {6}{5}\right )}{x^{2} \ln \relax (x )+5-x}\right )^{2}-2 i \pi \mathrm {csgn}\left (\frac {i \left (x \ln \relax (x )-\frac {6}{5}\right )}{x^{2} \ln \relax (x )+5-x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (x \ln \relax (x )-\frac {6}{5}\right )}{x^{2} \ln \relax (x )+5-x}\right )^{3}\right )^{4}}\)	\(238\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((320*x^2*ln(x)^2+(-768*x-1600)*ln(x)-64*x-1216)/(5*x^3*ln(x)^2+(-11*x^2+25*x)*ln(x)+6*x-30)/ln((-20*x*ln(x
)+24)/(x^2*ln(x)+5-x))^5,x,method=_RETURNVERBOSE)

[Out]

256/(4*ln(2)+2*ln(5)+2*I*Pi+2*ln(x*ln(x)-6/5)-2*ln(x^2*ln(x)+5-x)-I*Pi*csgn(I*(x*ln(x)-6/5))*csgn(I/(x^2*ln(x)
+5-x))*csgn(I*(x*ln(x)-6/5)/(x^2*ln(x)+5-x))+I*Pi*csgn(I*(x*ln(x)-6/5))*csgn(I*(x*ln(x)-6/5)/(x^2*ln(x)+5-x))^
2+I*Pi*csgn(I/(x^2*ln(x)+5-x))*csgn(I*(x*ln(x)-6/5)/(x^2*ln(x)+5-x))^2-2*I*Pi*csgn(I*(x*ln(x)-6/5)/(x^2*ln(x)+
5-x))^2+I*Pi*csgn(I*(x*ln(x)-6/5)/(x^2*ln(x)+5-x))^3)^4

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maxima [C]  time = 1.51, size = 332, normalized size = 11.45 \begin {gather*} \frac {16}{\pi ^{4} - 8 i \, \pi ^{3} \log \relax (2) - 24 \, \pi ^{2} \log \relax (2)^{2} + 32 i \, \pi \log \relax (2)^{3} + 16 \, \log \relax (2)^{4} - 4 \, {\left (i \, \pi + 2 \, \log \relax (2) + \log \left (5 \, x \log \relax (x) - 6\right )\right )} \log \left (x^{2} \log \relax (x) - x + 5\right )^{3} + \log \left (x^{2} \log \relax (x) - x + 5\right )^{4} - 4 \, {\left (-i \, \pi - 2 \, \log \relax (2)\right )} \log \left (5 \, x \log \relax (x) - 6\right )^{3} + \log \left (5 \, x \log \relax (x) - 6\right )^{4} - 6 \, {\left (\pi ^{2} - 4 i \, \pi \log \relax (2) - 4 \, \log \relax (2)^{2} + 2 \, {\left (-i \, \pi - 2 \, \log \relax (2)\right )} \log \left (5 \, x \log \relax (x) - 6\right ) - \log \left (5 \, x \log \relax (x) - 6\right )^{2}\right )} \log \left (x^{2} \log \relax (x) - x + 5\right )^{2} - 6 \, {\left (\pi ^{2} - 4 i \, \pi \log \relax (2) - 4 \, \log \relax (2)^{2}\right )} \log \left (5 \, x \log \relax (x) - 6\right )^{2} - 4 \, {\left (-i \, \pi ^{3} - 6 \, \pi ^{2} \log \relax (2) + 12 i \, \pi \log \relax (2)^{2} + 8 \, \log \relax (2)^{3} + 3 \, {\left (i \, \pi + 2 \, \log \relax (2)\right )} \log \left (5 \, x \log \relax (x) - 6\right )^{2} + \log \left (5 \, x \log \relax (x) - 6\right )^{3} - 3 \, {\left (\pi ^{2} - 4 i \, \pi \log \relax (2) - 4 \, \log \relax (2)^{2}\right )} \log \left (5 \, x \log \relax (x) - 6\right )\right )} \log \left (x^{2} \log \relax (x) - x + 5\right ) - 4 \, {\left (i \, \pi ^{3} + 6 \, \pi ^{2} \log \relax (2) - 12 i \, \pi \log \relax (2)^{2} - 8 \, \log \relax (2)^{3}\right )} \log \left (5 \, x \log \relax (x) - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((320*x^2*log(x)^2+(-768*x-1600)*log(x)-64*x-1216)/(5*x^3*log(x)^2+(-11*x^2+25*x)*log(x)+6*x-30)/log(
(-20*x*log(x)+24)/(x^2*log(x)+5-x))^5,x, algorithm="maxima")

[Out]

16/(pi^4 - 8*I*pi^3*log(2) - 24*pi^2*log(2)^2 + 32*I*pi*log(2)^3 + 16*log(2)^4 - 4*(I*pi + 2*log(2) + log(5*x*
log(x) - 6))*log(x^2*log(x) - x + 5)^3 + log(x^2*log(x) - x + 5)^4 - 4*(-I*pi - 2*log(2))*log(5*x*log(x) - 6)^
3 + log(5*x*log(x) - 6)^4 - 6*(pi^2 - 4*I*pi*log(2) - 4*log(2)^2 + 2*(-I*pi - 2*log(2))*log(5*x*log(x) - 6) -
log(5*x*log(x) - 6)^2)*log(x^2*log(x) - x + 5)^2 - 6*(pi^2 - 4*I*pi*log(2) - 4*log(2)^2)*log(5*x*log(x) - 6)^2
- 4*(-I*pi^3 - 6*pi^2*log(2) + 12*I*pi*log(2)^2 + 8*log(2)^3 + 3*(I*pi + 2*log(2))*log(5*x*log(x) - 6)^2 + lo
g(5*x*log(x) - 6)^3 - 3*(pi^2 - 4*I*pi*log(2) - 4*log(2)^2)*log(5*x*log(x) - 6))*log(x^2*log(x) - x + 5) - 4*(
I*pi^3 + 6*pi^2*log(2) - 12*I*pi*log(2)^2 - 8*log(2)^3)*log(5*x*log(x) - 6))

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mupad [B]  time = 1.81, size = 27, normalized size = 0.93 \begin {gather*} \frac {16}{{\ln \left (-\frac {20\,x\,\ln \relax (x)-24}{x^2\,\ln \relax (x)-x+5}\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(64*x + log(x)*(768*x + 1600) - 320*x^2*log(x)^2 + 1216)/(log(-(20*x*log(x) - 24)/(x^2*log(x) - x + 5))^5
*(6*x + 5*x^3*log(x)^2 + log(x)*(25*x - 11*x^2) - 30)),x)

[Out]

16/log(-(20*x*log(x) - 24)/(x^2*log(x) - x + 5))^4

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sympy [A]  time = 0.47, size = 22, normalized size = 0.76 \begin {gather*} \frac {16}{\log {\left (\frac {- 20 x \log {\relax (x )} + 24}{x^{2} \log {\relax (x )} - x + 5} \right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((320*x**2*ln(x)**2+(-768*x-1600)*ln(x)-64*x-1216)/(5*x**3*ln(x)**2+(-11*x**2+25*x)*ln(x)+6*x-30)/ln(
(-20*x*ln(x)+24)/(x**2*ln(x)+5-x))**5,x)

[Out]

16/log((-20*x*log(x) + 24)/(x**2*log(x) - x + 5))**4

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