3.88.10 \(\int \frac {144-2244 x-1728 x^2-1425 x^3-360 x^4-144 x^5+e^5 (48-60 x-96 x^2)+(1104 x+360 x^2+288 x^3) \log (x)-144 x \log ^2(x)}{256 x+160 x^2+153 x^3+40 x^4+16 x^5+(-128 x-40 x^2-32 x^3) \log (x)+16 x \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ \left (3+e^5+x\right ) \left (-9+\frac {3}{4+\frac {5 x}{4}+x^2-\log (x)}\right ) \]

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Rubi [F]  time = 1.08, 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 {144-2244 x-1728 x^2-1425 x^3-360 x^4-144 x^5+e^5 \left (48-60 x-96 x^2\right )+\left (1104 x+360 x^2+288 x^3\right ) \log (x)-144 x \log ^2(x)}{256 x+160 x^2+153 x^3+40 x^4+16 x^5+\left (-128 x-40 x^2-32 x^3\right ) \log (x)+16 x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(144 - 2244*x - 1728*x^2 - 1425*x^3 - 360*x^4 - 144*x^5 + E^5*(48 - 60*x - 96*x^2) + (1104*x + 360*x^2 + 2
88*x^3)*Log[x] - 144*x*Log[x]^2)/(256*x + 160*x^2 + 153*x^3 + 40*x^4 + 16*x^5 + (-128*x - 40*x^2 - 32*x^3)*Log
[x] + 16*x*Log[x]^2),x]

[Out]

-9*x - 12*(11 + 5*E^5)*Defer[Int][(16 + 5*x + 4*x^2 - 4*Log[x])^(-2), x] + 48*(3 + E^5)*Defer[Int][1/(x*(16 +
5*x + 4*x^2 - 4*Log[x])^2), x] - 12*(29 + 8*E^5)*Defer[Int][x/(16 + 5*x + 4*x^2 - 4*Log[x])^2, x] - 96*Defer[I
nt][x^2/(16 + 5*x + 4*x^2 - 4*Log[x])^2, x] + 12*Defer[Int][(16 + 5*x + 4*x^2 - 4*Log[x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (48-748 x-576 x^2-475 x^3-120 x^4-48 x^5-4 e^5 \left (-4+5 x+8 x^2\right )+8 x \left (46+15 x+12 x^2\right ) \log (x)-48 x \log ^2(x)\right )}{x \left (16+5 x+4 x^2-4 \log (x)\right )^2} \, dx\\ &=3 \int \frac {48-748 x-576 x^2-475 x^3-120 x^4-48 x^5-4 e^5 \left (-4+5 x+8 x^2\right )+8 x \left (46+15 x+12 x^2\right ) \log (x)-48 x \log ^2(x)}{x \left (16+5 x+4 x^2-4 \log (x)\right )^2} \, dx\\ &=3 \int \left (-3+\frac {4 \left (4 \left (3+e^5\right )-\left (11+5 e^5\right ) x-\left (29+8 e^5\right ) x^2-8 x^3\right )}{x \left (16+5 x+4 x^2-4 \log (x)\right )^2}+\frac {4}{16+5 x+4 x^2-4 \log (x)}\right ) \, dx\\ &=-9 x+12 \int \frac {4 \left (3+e^5\right )-\left (11+5 e^5\right ) x-\left (29+8 e^5\right ) x^2-8 x^3}{x \left (16+5 x+4 x^2-4 \log (x)\right )^2} \, dx+12 \int \frac {1}{16+5 x+4 x^2-4 \log (x)} \, dx\\ &=-9 x+12 \int \left (-\frac {11 \left (1+\frac {5 e^5}{11}\right )}{\left (16+5 x+4 x^2-4 \log (x)\right )^2}+\frac {4 \left (3+e^5\right )}{x \left (16+5 x+4 x^2-4 \log (x)\right )^2}-\frac {\left (29+8 e^5\right ) x}{\left (16+5 x+4 x^2-4 \log (x)\right )^2}-\frac {8 x^2}{\left (16+5 x+4 x^2-4 \log (x)\right )^2}\right ) \, dx+12 \int \frac {1}{16+5 x+4 x^2-4 \log (x)} \, dx\\ &=-9 x+12 \int \frac {1}{16+5 x+4 x^2-4 \log (x)} \, dx-96 \int \frac {x^2}{\left (16+5 x+4 x^2-4 \log (x)\right )^2} \, dx+\left (48 \left (3+e^5\right )\right ) \int \frac {1}{x \left (16+5 x+4 x^2-4 \log (x)\right )^2} \, dx-\left (12 \left (11+5 e^5\right )\right ) \int \frac {1}{\left (16+5 x+4 x^2-4 \log (x)\right )^2} \, dx-\left (12 \left (29+8 e^5\right )\right ) \int \frac {x}{\left (16+5 x+4 x^2-4 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 30, normalized size = 1.11 \begin {gather*} -3 \left (3 x+\frac {4 \left (3+e^5+x\right )}{-16-5 x-4 x^2+4 \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(144 - 2244*x - 1728*x^2 - 1425*x^3 - 360*x^4 - 144*x^5 + E^5*(48 - 60*x - 96*x^2) + (1104*x + 360*x
^2 + 288*x^3)*Log[x] - 144*x*Log[x]^2)/(256*x + 160*x^2 + 153*x^3 + 40*x^4 + 16*x^5 + (-128*x - 40*x^2 - 32*x^
3)*Log[x] + 16*x*Log[x]^2),x]

[Out]

-3*(3*x + (4*(3 + E^5 + x))/(-16 - 5*x - 4*x^2 + 4*Log[x]))

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fricas [A]  time = 0.47, size = 42, normalized size = 1.56 \begin {gather*} -\frac {3 \, {\left (12 \, x^{3} + 15 \, x^{2} - 12 \, x \log \relax (x) + 44 \, x - 4 \, e^{5} - 12\right )}}{4 \, x^{2} + 5 \, x - 4 \, \log \relax (x) + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-144*x*log(x)^2+(288*x^3+360*x^2+1104*x)*log(x)+(-96*x^2-60*x+48)*exp(5)-144*x^5-360*x^4-1425*x^3-1
728*x^2-2244*x+144)/(16*x*log(x)^2+(-32*x^3-40*x^2-128*x)*log(x)+16*x^5+40*x^4+153*x^3+160*x^2+256*x),x, algor
ithm="fricas")

[Out]

-3*(12*x^3 + 15*x^2 - 12*x*log(x) + 44*x - 4*e^5 - 12)/(4*x^2 + 5*x - 4*log(x) + 16)

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giac [A]  time = 0.18, size = 42, normalized size = 1.56 \begin {gather*} -\frac {3 \, {\left (12 \, x^{3} + 15 \, x^{2} - 12 \, x \log \relax (x) + 44 \, x - 4 \, e^{5} - 12\right )}}{4 \, x^{2} + 5 \, x - 4 \, \log \relax (x) + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-144*x*log(x)^2+(288*x^3+360*x^2+1104*x)*log(x)+(-96*x^2-60*x+48)*exp(5)-144*x^5-360*x^4-1425*x^3-1
728*x^2-2244*x+144)/(16*x*log(x)^2+(-32*x^3-40*x^2-128*x)*log(x)+16*x^5+40*x^4+153*x^3+160*x^2+256*x),x, algor
ithm="giac")

[Out]

-3*(12*x^3 + 15*x^2 - 12*x*log(x) + 44*x - 4*e^5 - 12)/(4*x^2 + 5*x - 4*log(x) + 16)

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maple [A]  time = 0.11, size = 28, normalized size = 1.04

method	result	size
risch	\(-9 x +\frac {12 \,{\mathrm e}^{5}+36+12 x}{4 x^{2}+5 x -4 \ln \relax (x )+16}\)	\(28\)
norman	\(\frac {-\frac {303 x}{4}-45 \ln \relax (x )+36 x \ln \relax (x )-36 x^{3}+216+12 \,{\mathrm e}^{5}}{4 x^{2}+5 x -4 \ln \relax (x )+16}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-144*x*ln(x)^2+(288*x^3+360*x^2+1104*x)*ln(x)+(-96*x^2-60*x+48)*exp(5)-144*x^5-360*x^4-1425*x^3-1728*x^2-
2244*x+144)/(16*x*ln(x)^2+(-32*x^3-40*x^2-128*x)*ln(x)+16*x^5+40*x^4+153*x^3+160*x^2+256*x),x,method=_RETURNVE
RBOSE)

[Out]

-9*x+12*(exp(5)+3+x)/(4*x^2+5*x-4*ln(x)+16)

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maxima [A]  time = 0.39, size = 42, normalized size = 1.56 \begin {gather*} -\frac {3 \, {\left (12 \, x^{3} + 15 \, x^{2} - 12 \, x \log \relax (x) + 44 \, x - 4 \, e^{5} - 12\right )}}{4 \, x^{2} + 5 \, x - 4 \, \log \relax (x) + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-144*x*log(x)^2+(288*x^3+360*x^2+1104*x)*log(x)+(-96*x^2-60*x+48)*exp(5)-144*x^5-360*x^4-1425*x^3-1
728*x^2-2244*x+144)/(16*x*log(x)^2+(-32*x^3-40*x^2-128*x)*log(x)+16*x^5+40*x^4+153*x^3+160*x^2+256*x),x, algor
ithm="maxima")

[Out]

-3*(12*x^3 + 15*x^2 - 12*x*log(x) + 44*x - 4*e^5 - 12)/(4*x^2 + 5*x - 4*log(x) + 16)

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mupad [B]  time = 5.59, size = 42, normalized size = 1.56 \begin {gather*} -\frac {3\,\left (44\,x-4\,{\mathrm {e}}^5-12\,x\,\ln \relax (x)+15\,x^2+12\,x^3-12\right )}{5\,x-4\,\ln \relax (x)+4\,x^2+16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2244*x + 144*x*log(x)^2 + exp(5)*(60*x + 96*x^2 - 48) + 1728*x^2 + 1425*x^3 + 360*x^4 + 144*x^5 - log(x)
*(1104*x + 360*x^2 + 288*x^3) - 144)/(256*x + 16*x*log(x)^2 + 160*x^2 + 153*x^3 + 40*x^4 + 16*x^5 - log(x)*(12
8*x + 40*x^2 + 32*x^3)),x)

[Out]

-(3*(44*x - 4*exp(5) - 12*x*log(x) + 15*x^2 + 12*x^3 - 12))/(5*x - 4*log(x) + 4*x^2 + 16)

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sympy [A]  time = 0.15, size = 29, normalized size = 1.07 \begin {gather*} - 9 x + \frac {- 12 x - 12 e^{5} - 36}{- 4 x^{2} - 5 x + 4 \log {\relax (x )} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-144*x*ln(x)**2+(288*x**3+360*x**2+1104*x)*ln(x)+(-96*x**2-60*x+48)*exp(5)-144*x**5-360*x**4-1425*x
**3-1728*x**2-2244*x+144)/(16*x*ln(x)**2+(-32*x**3-40*x**2-128*x)*ln(x)+16*x**5+40*x**4+153*x**3+160*x**2+256*
x),x)

[Out]

-9*x + (-12*x - 12*exp(5) - 36)/(-4*x**2 - 5*x + 4*log(x) - 16)

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