3.97.90 \(\int \frac {248+398 x+216 x^2+24 x^3+(2120+830 x+180 x^2) \log (x)+(1350+450 x) \log ^2(x)+375 \log ^3(x)}{216+216 x+72 x^2+8 x^3+(540+360 x+60 x^2) \log (x)+(450+150 x) \log ^2(x)+125 \log ^3(x)} \, dx\)

Optimal. Leaf size=30 \[ \frac {x+x^2 \left (3+\frac {4-x}{\left (\frac {2 (3+x)}{5}+\log (x)\right )^2}\right )}{x} \]

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Rubi [F]  time = 0.47, 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 {248+398 x+216 x^2+24 x^3+\left (2120+830 x+180 x^2\right ) \log (x)+(1350+450 x) \log ^2(x)+375 \log ^3(x)}{216+216 x+72 x^2+8 x^3+\left (540+360 x+60 x^2\right ) \log (x)+(450+150 x) \log ^2(x)+125 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(248 + 398*x + 216*x^2 + 24*x^3 + (2120 + 830*x + 180*x^2)*Log[x] + (1350 + 450*x)*Log[x]^2 + 375*Log[x]^3
)/(216 + 216*x + 72*x^2 + 8*x^3 + (540 + 360*x + 60*x^2)*Log[x] + (450 + 150*x)*Log[x]^2 + 125*Log[x]^3),x]

[Out]

3*x - 1000*Defer[Int][(6 + 2*x + 5*Log[x])^(-3), x] - 150*Defer[Int][x/(6 + 2*x + 5*Log[x])^3, x] + 100*Defer[
Int][x^2/(6 + 2*x + 5*Log[x])^3, x] + 100*Defer[Int][(6 + 2*x + 5*Log[x])^(-2), x] - 50*Defer[Int][x/(6 + 2*x
+ 5*Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {248+398 x+216 x^2+24 x^3+10 \left (212+83 x+18 x^2\right ) \log (x)+450 (3+x) \log ^2(x)+375 \log ^3(x)}{(6+2 x+5 \log (x))^3} \, dx\\ &=\int \left (3+\frac {50 \left (-20-3 x+2 x^2\right )}{(6+2 x+5 \log (x))^3}-\frac {50 (-2+x)}{(6+2 x+5 \log (x))^2}\right ) \, dx\\ &=3 x+50 \int \frac {-20-3 x+2 x^2}{(6+2 x+5 \log (x))^3} \, dx-50 \int \frac {-2+x}{(6+2 x+5 \log (x))^2} \, dx\\ &=3 x+50 \int \left (-\frac {20}{(6+2 x+5 \log (x))^3}-\frac {3 x}{(6+2 x+5 \log (x))^3}+\frac {2 x^2}{(6+2 x+5 \log (x))^3}\right ) \, dx-50 \int \left (-\frac {2}{(6+2 x+5 \log (x))^2}+\frac {x}{(6+2 x+5 \log (x))^2}\right ) \, dx\\ &=3 x-50 \int \frac {x}{(6+2 x+5 \log (x))^2} \, dx+100 \int \frac {x^2}{(6+2 x+5 \log (x))^3} \, dx+100 \int \frac {1}{(6+2 x+5 \log (x))^2} \, dx-150 \int \frac {x}{(6+2 x+5 \log (x))^3} \, dx-1000 \int \frac {1}{(6+2 x+5 \log (x))^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 21, normalized size = 0.70 \begin {gather*} 3 x-\frac {25 (-4+x) x}{(6+2 x+5 \log (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(248 + 398*x + 216*x^2 + 24*x^3 + (2120 + 830*x + 180*x^2)*Log[x] + (1350 + 450*x)*Log[x]^2 + 375*Lo
g[x]^3)/(216 + 216*x + 72*x^2 + 8*x^3 + (540 + 360*x + 60*x^2)*Log[x] + (450 + 150*x)*Log[x]^2 + 125*Log[x]^3)
,x]

[Out]

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

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fricas [A]  time = 0.53, size = 58, normalized size = 1.93 \begin {gather*} \frac {12 \, x^{3} + 75 \, x \log \relax (x)^{2} + 47 \, x^{2} + 60 \, {\left (x^{2} + 3 \, x\right )} \log \relax (x) + 208 \, x}{4 \, x^{2} + 20 \, {\left (x + 3\right )} \log \relax (x) + 25 \, \log \relax (x)^{2} + 24 \, x + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((375*log(x)^3+(450*x+1350)*log(x)^2+(180*x^2+830*x+2120)*log(x)+24*x^3+216*x^2+398*x+248)/(125*log(x
)^3+(150*x+450)*log(x)^2+(60*x^2+360*x+540)*log(x)+8*x^3+72*x^2+216*x+216),x, algorithm="fricas")

[Out]

(12*x^3 + 75*x*log(x)^2 + 47*x^2 + 60*(x^2 + 3*x)*log(x) + 208*x)/(4*x^2 + 20*(x + 3)*log(x) + 25*log(x)^2 + 2
4*x + 36)

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giac [B]  time = 0.21, size = 66, normalized size = 2.20 \begin {gather*} 3 \, x - \frac {25 \, {\left (2 \, x^{3} - 3 \, x^{2} - 20 \, x\right )}}{8 \, x^{3} + 40 \, x^{2} \log \relax (x) + 50 \, x \log \relax (x)^{2} + 68 \, x^{2} + 220 \, x \log \relax (x) + 125 \, \log \relax (x)^{2} + 192 \, x + 300 \, \log \relax (x) + 180} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((375*log(x)^3+(450*x+1350)*log(x)^2+(180*x^2+830*x+2120)*log(x)+24*x^3+216*x^2+398*x+248)/(125*log(x
)^3+(150*x+450)*log(x)^2+(60*x^2+360*x+540)*log(x)+8*x^3+72*x^2+216*x+216),x, algorithm="giac")

[Out]

3*x - 25*(2*x^3 - 3*x^2 - 20*x)/(8*x^3 + 40*x^2*log(x) + 50*x*log(x)^2 + 68*x^2 + 220*x*log(x) + 125*log(x)^2
+ 192*x + 300*log(x) + 180)

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maple [A]  time = 0.05, size = 22, normalized size = 0.73

method	result	size
risch	\(3 x -\frac {25 \left (x -4\right ) x}{\left (5 \ln \relax (x )+2 x +6\right )^{2}}\)	\(22\)
norman	\(\frac {-540 \ln \relax (x )-225 \ln \relax (x )^{2}-8 x +11 x^{2}+12 x^{3}+75 x \ln \relax (x )^{2}+60 x^{2} \ln \relax (x )-324}{\left (5 \ln \relax (x )+2 x +6\right )^{2}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((375*ln(x)^3+(450*x+1350)*ln(x)^2+(180*x^2+830*x+2120)*ln(x)+24*x^3+216*x^2+398*x+248)/(125*ln(x)^3+(150*x
+450)*ln(x)^2+(60*x^2+360*x+540)*ln(x)+8*x^3+72*x^2+216*x+216),x,method=_RETURNVERBOSE)

[Out]

3*x-25*(x-4)*x/(5*ln(x)+2*x+6)^2

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maxima [A]  time = 0.40, size = 58, normalized size = 1.93 \begin {gather*} \frac {12 \, x^{3} + 75 \, x \log \relax (x)^{2} + 47 \, x^{2} + 60 \, {\left (x^{2} + 3 \, x\right )} \log \relax (x) + 208 \, x}{4 \, x^{2} + 20 \, {\left (x + 3\right )} \log \relax (x) + 25 \, \log \relax (x)^{2} + 24 \, x + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((375*log(x)^3+(450*x+1350)*log(x)^2+(180*x^2+830*x+2120)*log(x)+24*x^3+216*x^2+398*x+248)/(125*log(x
)^3+(150*x+450)*log(x)^2+(60*x^2+360*x+540)*log(x)+8*x^3+72*x^2+216*x+216),x, algorithm="maxima")

[Out]

(12*x^3 + 75*x*log(x)^2 + 47*x^2 + 60*(x^2 + 3*x)*log(x) + 208*x)/(4*x^2 + 20*(x + 3)*log(x) + 25*log(x)^2 + 2
4*x + 36)

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mupad [B]  time = 6.14, size = 38, normalized size = 1.27 \begin {gather*} \frac {x\,\left (12\,x^2+60\,x\,\ln \relax (x)+47\,x+75\,{\ln \relax (x)}^2+180\,\ln \relax (x)+208\right )}{{\left (2\,x+5\,\ln \relax (x)+6\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((398*x + 375*log(x)^3 + log(x)*(830*x + 180*x^2 + 2120) + 216*x^2 + 24*x^3 + log(x)^2*(450*x + 1350) + 248
)/(216*x + 125*log(x)^3 + log(x)*(360*x + 60*x^2 + 540) + 72*x^2 + 8*x^3 + log(x)^2*(150*x + 450) + 216),x)

[Out]

(x*(47*x + 180*log(x) + 75*log(x)^2 + 60*x*log(x) + 12*x^2 + 208))/(2*x + 5*log(x) + 6)^2

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sympy [A]  time = 0.17, size = 41, normalized size = 1.37 \begin {gather*} 3 x + \frac {- x^{2} + 4 x}{\frac {4 x^{2}}{25} + \frac {24 x}{25} + \left (\frac {4 x}{5} + \frac {12}{5}\right ) \log {\relax (x )} + \log {\relax (x )}^{2} + \frac {36}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((375*ln(x)**3+(450*x+1350)*ln(x)**2+(180*x**2+830*x+2120)*ln(x)+24*x**3+216*x**2+398*x+248)/(125*ln(
x)**3+(150*x+450)*ln(x)**2+(60*x**2+360*x+540)*ln(x)+8*x**3+72*x**2+216*x+216),x)

[Out]

3*x + (-x**2 + 4*x)/(4*x**2/25 + 24*x/25 + (4*x/5 + 12/5)*log(x) + log(x)**2 + 36/25)

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