3.99.15 \(\int \frac {-6480-4320 x-924 x^2-70 x^3-x^4+(1296+864 x+216 x^2+24 x^3+x^4) \log (x)}{20736-24480 x-3143 x^2+5256 x^3+1806 x^4+216 x^5+9 x^6+(-10368+2664 x+4344 x^2+1250 x^3+144 x^4+6 x^5) \log (x)+(1296+864 x+216 x^2+24 x^3+x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=28 \[ \frac {x}{-4-x+\frac {\left (2 x+\frac {x}{6+x}\right )^2}{x}+\log (x)} \]

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Rubi [F]  time = 1.52, 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 {-6480-4320 x-924 x^2-70 x^3-x^4+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log (x)}{20736-24480 x-3143 x^2+5256 x^3+1806 x^4+216 x^5+9 x^6+\left (-10368+2664 x+4344 x^2+1250 x^3+144 x^4+6 x^5\right ) \log (x)+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6480 - 4320*x - 924*x^2 - 70*x^3 - x^4 + (1296 + 864*x + 216*x^2 + 24*x^3 + x^4)*Log[x])/(20736 - 24480*
x - 3143*x^2 + 5256*x^3 + 1806*x^4 + 216*x^5 + 9*x^6 + (-10368 + 2664*x + 4344*x^2 + 1250*x^3 + 144*x^4 + 6*x^
5)*Log[x] + (1296 + 864*x + 216*x^2 + 24*x^3 + x^4)*Log[x]^2),x]

[Out]

-1296*Defer[Int][(-144 + 85*x + 36*x^2 + 3*x^3 + 36*Log[x] + 12*x*Log[x] + x^2*Log[x])^(-2), x] - 5652*Defer[I
nt][x/(-144 + 85*x + 36*x^2 + 3*x^3 + 36*Log[x] + 12*x*Log[x] + x^2*Log[x])^2, x] - 3096*Defer[Int][x^2/(-144
+ 85*x + 36*x^2 + 3*x^3 + 36*Log[x] + 12*x*Log[x] + x^2*Log[x])^2, x] - 695*Defer[Int][x^3/(-144 + 85*x + 36*x
^2 + 3*x^3 + 36*Log[x] + 12*x*Log[x] + x^2*Log[x])^2, x] - 73*Defer[Int][x^4/(-144 + 85*x + 36*x^2 + 3*x^3 + 3
6*Log[x] + 12*x*Log[x] + x^2*Log[x])^2, x] - 3*Defer[Int][x^5/(-144 + 85*x + 36*x^2 + 3*x^3 + 36*Log[x] + 12*x
*Log[x] + x^2*Log[x])^2, x] + 36*Defer[Int][(-144 + 85*x + 36*x^2 + 3*x^3 + 36*Log[x] + 12*x*Log[x] + x^2*Log[
x])^(-1), x] + 12*Defer[Int][x/(-144 + 85*x + 36*x^2 + 3*x^3 + 36*Log[x] + 12*x*Log[x] + x^2*Log[x]), x] + Def
er[Int][x^2/(-144 + 85*x + 36*x^2 + 3*x^3 + 36*Log[x] + 12*x*Log[x] + x^2*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(6+x) \left (-1080-540 x-64 x^2-x^3+(6+x)^3 \log (x)\right )}{\left (144-85 x-36 x^2-3 x^3-(6+x)^2 \log (x)\right )^2} \, dx\\ &=\int \left (\frac {-1296-5652 x-3096 x^2-695 x^3-73 x^4-3 x^5}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2}+\frac {(6+x)^2}{-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)}\right ) \, dx\\ &=\int \frac {-1296-5652 x-3096 x^2-695 x^3-73 x^4-3 x^5}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2} \, dx+\int \frac {(6+x)^2}{-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)} \, dx\\ &=\int \left (-\frac {1296}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2}-\frac {5652 x}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2}-\frac {3096 x^2}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2}-\frac {695 x^3}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2}-\frac {73 x^4}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2}-\frac {3 x^5}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2}\right ) \, dx+\int \left (\frac {36}{-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)}+\frac {12 x}{-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)}+\frac {x^2}{-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)}\right ) \, dx\\ &=-\left (3 \int \frac {x^5}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2} \, dx\right )+12 \int \frac {x}{-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)} \, dx+36 \int \frac {1}{-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)} \, dx-73 \int \frac {x^4}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2} \, dx-695 \int \frac {x^3}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2} \, dx-1296 \int \frac {1}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2} \, dx-3096 \int \frac {x^2}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2} \, dx-5652 \int \frac {x}{\left (-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)\right )^2} \, dx+\int \frac {x^2}{-144+85 x+36 x^2+3 x^3+36 \log (x)+12 x \log (x)+x^2 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.85, size = 32, normalized size = 1.14 \begin {gather*} \frac {x (6+x)^2}{-144+85 x+36 x^2+3 x^3+(6+x)^2 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6480 - 4320*x - 924*x^2 - 70*x^3 - x^4 + (1296 + 864*x + 216*x^2 + 24*x^3 + x^4)*Log[x])/(20736 -
24480*x - 3143*x^2 + 5256*x^3 + 1806*x^4 + 216*x^5 + 9*x^6 + (-10368 + 2664*x + 4344*x^2 + 1250*x^3 + 144*x^4
+ 6*x^5)*Log[x] + (1296 + 864*x + 216*x^2 + 24*x^3 + x^4)*Log[x]^2),x]

[Out]

(x*(6 + x)^2)/(-144 + 85*x + 36*x^2 + 3*x^3 + (6 + x)^2*Log[x])

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fricas [A]  time = 0.75, size = 41, normalized size = 1.46 \begin {gather*} \frac {x^{3} + 12 \, x^{2} + 36 \, x}{3 \, x^{3} + 36 \, x^{2} + {\left (x^{2} + 12 \, x + 36\right )} \log \relax (x) + 85 \, x - 144} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+24*x^3+216*x^2+864*x+1296)*log(x)-x^4-70*x^3-924*x^2-4320*x-6480)/((x^4+24*x^3+216*x^2+864*x+1
296)*log(x)^2+(6*x^5+144*x^4+1250*x^3+4344*x^2+2664*x-10368)*log(x)+9*x^6+216*x^5+1806*x^4+5256*x^3-3143*x^2-2
4480*x+20736),x, algorithm="fricas")

[Out]

(x^3 + 12*x^2 + 36*x)/(3*x^3 + 36*x^2 + (x^2 + 12*x + 36)*log(x) + 85*x - 144)

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giac [A]  time = 0.17, size = 45, normalized size = 1.61 \begin {gather*} \frac {x^{3} + 12 \, x^{2} + 36 \, x}{3 \, x^{3} + x^{2} \log \relax (x) + 36 \, x^{2} + 12 \, x \log \relax (x) + 85 \, x + 36 \, \log \relax (x) - 144} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+24*x^3+216*x^2+864*x+1296)*log(x)-x^4-70*x^3-924*x^2-4320*x-6480)/((x^4+24*x^3+216*x^2+864*x+1
296)*log(x)^2+(6*x^5+144*x^4+1250*x^3+4344*x^2+2664*x-10368)*log(x)+9*x^6+216*x^5+1806*x^4+5256*x^3-3143*x^2-2
4480*x+20736),x, algorithm="giac")

[Out]

(x^3 + 12*x^2 + 36*x)/(3*x^3 + x^2*log(x) + 36*x^2 + 12*x*log(x) + 85*x + 36*log(x) - 144)

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maple [A]  time = 0.08, size = 40, normalized size = 1.43

method	result	size
risch	\(\frac {\left (x +6\right )^{2} x}{x^{2} \ln \relax (x )+3 x^{3}+12 x \ln \relax (x )+36 x^{2}+36 \ln \relax (x )+85 x -144}\)	\(40\)
norman	\(\frac {x^{3}+12 x^{2}+36 x}{x^{2} \ln \relax (x )+3 x^{3}+12 x \ln \relax (x )+36 x^{2}+36 \ln \relax (x )+85 x -144}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4+24*x^3+216*x^2+864*x+1296)*ln(x)-x^4-70*x^3-924*x^2-4320*x-6480)/((x^4+24*x^3+216*x^2+864*x+1296)*ln
(x)^2+(6*x^5+144*x^4+1250*x^3+4344*x^2+2664*x-10368)*ln(x)+9*x^6+216*x^5+1806*x^4+5256*x^3-3143*x^2-24480*x+20
736),x,method=_RETURNVERBOSE)

[Out]

(x+6)^2*x/(x^2*ln(x)+3*x^3+12*x*ln(x)+36*x^2+36*ln(x)+85*x-144)

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maxima [A]  time = 0.39, size = 41, normalized size = 1.46 \begin {gather*} \frac {x^{3} + 12 \, x^{2} + 36 \, x}{3 \, x^{3} + 36 \, x^{2} + {\left (x^{2} + 12 \, x + 36\right )} \log \relax (x) + 85 \, x - 144} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+24*x^3+216*x^2+864*x+1296)*log(x)-x^4-70*x^3-924*x^2-4320*x-6480)/((x^4+24*x^3+216*x^2+864*x+1
296)*log(x)^2+(6*x^5+144*x^4+1250*x^3+4344*x^2+2664*x-10368)*log(x)+9*x^6+216*x^5+1806*x^4+5256*x^3-3143*x^2-2
4480*x+20736),x, algorithm="maxima")

[Out]

(x^3 + 12*x^2 + 36*x)/(3*x^3 + 36*x^2 + (x^2 + 12*x + 36)*log(x) + 85*x - 144)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {4320\,x-\ln \relax (x)\,\left (x^4+24\,x^3+216\,x^2+864\,x+1296\right )+924\,x^2+70\,x^3+x^4+6480}{{\ln \relax (x)}^2\,\left (x^4+24\,x^3+216\,x^2+864\,x+1296\right )-24480\,x+\ln \relax (x)\,\left (6\,x^5+144\,x^4+1250\,x^3+4344\,x^2+2664\,x-10368\right )-3143\,x^2+5256\,x^3+1806\,x^4+216\,x^5+9\,x^6+20736} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4320*x - log(x)*(864*x + 216*x^2 + 24*x^3 + x^4 + 1296) + 924*x^2 + 70*x^3 + x^4 + 6480)/(log(x)^2*(864*
x + 216*x^2 + 24*x^3 + x^4 + 1296) - 24480*x + log(x)*(2664*x + 4344*x^2 + 1250*x^3 + 144*x^4 + 6*x^5 - 10368)
- 3143*x^2 + 5256*x^3 + 1806*x^4 + 216*x^5 + 9*x^6 + 20736),x)

[Out]

int(-(4320*x - log(x)*(864*x + 216*x^2 + 24*x^3 + x^4 + 1296) + 924*x^2 + 70*x^3 + x^4 + 6480)/(log(x)^2*(864*
x + 216*x^2 + 24*x^3 + x^4 + 1296) - 24480*x + log(x)*(2664*x + 4344*x^2 + 1250*x^3 + 144*x^4 + 6*x^5 - 10368)
- 3143*x^2 + 5256*x^3 + 1806*x^4 + 216*x^5 + 9*x^6 + 20736), x)

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sympy [A]  time = 0.32, size = 37, normalized size = 1.32 \begin {gather*} \frac {x^{3} + 12 x^{2} + 36 x}{3 x^{3} + 36 x^{2} + 85 x + \left (x^{2} + 12 x + 36\right ) \log {\relax (x )} - 144} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**4+24*x**3+216*x**2+864*x+1296)*ln(x)-x**4-70*x**3-924*x**2-4320*x-6480)/((x**4+24*x**3+216*x**2
+864*x+1296)*ln(x)**2+(6*x**5+144*x**4+1250*x**3+4344*x**2+2664*x-10368)*ln(x)+9*x**6+216*x**5+1806*x**4+5256*
x**3-3143*x**2-24480*x+20736),x)

[Out]

(x**3 + 12*x**2 + 36*x)/(3*x**3 + 36*x**2 + 85*x + (x**2 + 12*x + 36)*log(x) - 144)

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