3.18.16 \(\int \frac {495-648 x+6905 x^2-6540 x^3+16420 x^4-11712 x^5+2304 x^6+(-45+26 x-244 x^2+96 x^3) \log (x)+\log ^2(x)}{625-750 x+6325 x^2-6060 x^3+16324 x^4-11712 x^5+2304 x^6+(-50+30 x-244 x^2+96 x^3) \log (x)+\log ^2(x)} \, dx\)

Optimal. Leaf size=24 \[ x+\frac {x}{-5+x-24 x^2+\frac {\log (x)}{5-2 x}} \]

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Rubi [F]  time = 0.96, 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 {495-648 x+6905 x^2-6540 x^3+16420 x^4-11712 x^5+2304 x^6+\left (-45+26 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)}{625-750 x+6325 x^2-6060 x^3+16324 x^4-11712 x^5+2304 x^6+\left (-50+30 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(495 - 648*x + 6905*x^2 - 6540*x^3 + 16420*x^4 - 11712*x^5 + 2304*x^6 + (-45 + 26*x - 244*x^2 + 96*x^3)*Lo
g[x] + Log[x]^2)/(625 - 750*x + 6325*x^2 - 6060*x^3 + 16324*x^4 - 11712*x^5 + 2304*x^6 + (-50 + 30*x - 244*x^2
+ 96*x^3)*Log[x] + Log[x]^2),x]

[Out]

x - 5*Defer[Int][(-25 + 15*x - 122*x^2 + 48*x^3 + Log[x])^(-2), x] - 73*Defer[Int][x/(-25 + 15*x - 122*x^2 + 4
8*x^3 + Log[x])^2, x] + 1250*Defer[Int][x^2/(-25 + 15*x - 122*x^2 + 48*x^3 + Log[x])^2, x] - 1208*Defer[Int][x
^3/(-25 + 15*x - 122*x^2 + 48*x^3 + Log[x])^2, x] + 288*Defer[Int][x^4/(-25 + 15*x - 122*x^2 + 48*x^3 + Log[x]
)^2, x] + 5*Defer[Int][(-25 + 15*x - 122*x^2 + 48*x^3 + Log[x])^(-1), x] - 4*Defer[Int][x/(-25 + 15*x - 122*x^
2 + 48*x^3 + Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {495-648 x+6905 x^2-6540 x^3+16420 x^4-11712 x^5+2304 x^6+\left (-45+26 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)}{\left (25-15 x+122 x^2-48 x^3-\log (x)\right )^2} \, dx\\ &=\int \left (1+\frac {-5-73 x+1250 x^2-1208 x^3+288 x^4}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2}+\frac {5-4 x}{-25+15 x-122 x^2+48 x^3+\log (x)}\right ) \, dx\\ &=x+\int \frac {-5-73 x+1250 x^2-1208 x^3+288 x^4}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2} \, dx+\int \frac {5-4 x}{-25+15 x-122 x^2+48 x^3+\log (x)} \, dx\\ &=x+\int \left (-\frac {5}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2}-\frac {73 x}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2}+\frac {1250 x^2}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2}-\frac {1208 x^3}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2}+\frac {288 x^4}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2}\right ) \, dx+\int \left (\frac {5}{-25+15 x-122 x^2+48 x^3+\log (x)}-\frac {4 x}{-25+15 x-122 x^2+48 x^3+\log (x)}\right ) \, dx\\ &=x-4 \int \frac {x}{-25+15 x-122 x^2+48 x^3+\log (x)} \, dx-5 \int \frac {1}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2} \, dx+5 \int \frac {1}{-25+15 x-122 x^2+48 x^3+\log (x)} \, dx-73 \int \frac {x}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2} \, dx+288 \int \frac {x^4}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2} \, dx-1208 \int \frac {x^3}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2} \, dx+1250 \int \frac {x^2}{\left (-25+15 x-122 x^2+48 x^3+\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 29, normalized size = 1.21 \begin {gather*} x-\frac {x (-5+2 x)}{-25+15 x-122 x^2+48 x^3+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(495 - 648*x + 6905*x^2 - 6540*x^3 + 16420*x^4 - 11712*x^5 + 2304*x^6 + (-45 + 26*x - 244*x^2 + 96*x
^3)*Log[x] + Log[x]^2)/(625 - 750*x + 6325*x^2 - 6060*x^3 + 16324*x^4 - 11712*x^5 + 2304*x^6 + (-50 + 30*x - 2
44*x^2 + 96*x^3)*Log[x] + Log[x]^2),x]

[Out]

x - (x*(-5 + 2*x))/(-25 + 15*x - 122*x^2 + 48*x^3 + Log[x])

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fricas [A]  time = 0.63, size = 43, normalized size = 1.79 \begin {gather*} \frac {48 \, x^{4} - 122 \, x^{3} + 13 \, x^{2} + x \log \relax (x) - 20 \, x}{48 \, x^{3} - 122 \, x^{2} + 15 \, x + \log \relax (x) - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^2+(96*x^3-244*x^2+26*x-45)*log(x)+2304*x^6-11712*x^5+16420*x^4-6540*x^3+6905*x^2-648*x+495)/
(log(x)^2+(96*x^3-244*x^2+30*x-50)*log(x)+2304*x^6-11712*x^5+16324*x^4-6060*x^3+6325*x^2-750*x+625),x, algorit
hm="fricas")

[Out]

(48*x^4 - 122*x^3 + 13*x^2 + x*log(x) - 20*x)/(48*x^3 - 122*x^2 + 15*x + log(x) - 25)

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giac [A]  time = 0.27, size = 32, normalized size = 1.33 \begin {gather*} x - \frac {2 \, x^{2} - 5 \, x}{48 \, x^{3} - 122 \, x^{2} + 15 \, x + \log \relax (x) - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^2+(96*x^3-244*x^2+26*x-45)*log(x)+2304*x^6-11712*x^5+16420*x^4-6540*x^3+6905*x^2-648*x+495)/
(log(x)^2+(96*x^3-244*x^2+30*x-50)*log(x)+2304*x^6-11712*x^5+16324*x^4-6060*x^3+6325*x^2-750*x+625),x, algorit
hm="giac")

[Out]

x - (2*x^2 - 5*x)/(48*x^3 - 122*x^2 + 15*x + log(x) - 25)

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maple [A]  time = 0.05, size = 30, normalized size = 1.25

method	result	size
risch	\(x -\frac {x \left (2 x -5\right )}{48 x^{3}-122 x^{2}+\ln \relax (x )+15 x -25}\)	\(30\)
norman	\(\frac {x \ln \relax (x )-122 x^{3}-20 x +13 x^{2}+48 x^{4}}{48 x^{3}-122 x^{2}+\ln \relax (x )+15 x -25}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)^2+(96*x^3-244*x^2+26*x-45)*ln(x)+2304*x^6-11712*x^5+16420*x^4-6540*x^3+6905*x^2-648*x+495)/(ln(x)^2
+(96*x^3-244*x^2+30*x-50)*ln(x)+2304*x^6-11712*x^5+16324*x^4-6060*x^3+6325*x^2-750*x+625),x,method=_RETURNVERB
OSE)

[Out]

x-x*(2*x-5)/(48*x^3-122*x^2+ln(x)+15*x-25)

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maxima [A]  time = 0.53, size = 43, normalized size = 1.79 \begin {gather*} \frac {48 \, x^{4} - 122 \, x^{3} + 13 \, x^{2} + x \log \relax (x) - 20 \, x}{48 \, x^{3} - 122 \, x^{2} + 15 \, x + \log \relax (x) - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^2+(96*x^3-244*x^2+26*x-45)*log(x)+2304*x^6-11712*x^5+16420*x^4-6540*x^3+6905*x^2-648*x+495)/
(log(x)^2+(96*x^3-244*x^2+30*x-50)*log(x)+2304*x^6-11712*x^5+16324*x^4-6060*x^3+6325*x^2-750*x+625),x, algorit
hm="maxima")

[Out]

(48*x^4 - 122*x^3 + 13*x^2 + x*log(x) - 20*x)/(48*x^3 - 122*x^2 + 15*x + log(x) - 25)

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mupad [B]  time = 1.33, size = 31, normalized size = 1.29 \begin {gather*} x+\frac {5\,x-2\,x^2}{15\,x+\ln \relax (x)-122\,x^2+48\,x^3-25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2 - 648*x + 6905*x^2 - 6540*x^3 + 16420*x^4 - 11712*x^5 + 2304*x^6 + log(x)*(26*x - 244*x^2 + 96*x
^3 - 45) + 495)/(log(x)^2 - 750*x + 6325*x^2 - 6060*x^3 + 16324*x^4 - 11712*x^5 + 2304*x^6 + log(x)*(30*x - 24
4*x^2 + 96*x^3 - 50) + 625),x)

[Out]

x + (5*x - 2*x^2)/(15*x + log(x) - 122*x^2 + 48*x^3 - 25)

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sympy [A]  time = 0.16, size = 27, normalized size = 1.12 \begin {gather*} x + \frac {- 2 x^{2} + 5 x}{48 x^{3} - 122 x^{2} + 15 x + \log {\relax (x )} - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x)**2+(96*x**3-244*x**2+26*x-45)*ln(x)+2304*x**6-11712*x**5+16420*x**4-6540*x**3+6905*x**2-648*x
+495)/(ln(x)**2+(96*x**3-244*x**2+30*x-50)*ln(x)+2304*x**6-11712*x**5+16324*x**4-6060*x**3+6325*x**2-750*x+625
),x)

[Out]

x + (-2*x**2 + 5*x)/(48*x**3 - 122*x**2 + 15*x + log(x) - 25)

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