3.23.3 \(\int \frac {7454200-2564100 x+1823674 x^2-532140 x^3+66501 x^4-3780 x^5+81 x^6+(-12220-4798 x^2+840 x^3-36 x^4) \log (x)+4 x^2 \log ^2(x)}{1493284 x^2-513240 x^3+66096 x^4-3780 x^5+81 x^6+(-4888 x^2+840 x^3-36 x^4) \log (x)+4 x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=30 \[ x+\frac {5}{x \left (-1+\frac {2 \log (x)}{-3+(5+3 (10-x))^2}\right )} \]

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Rubi [F]  time = 1.45, 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 {7454200-2564100 x+1823674 x^2-532140 x^3+66501 x^4-3780 x^5+81 x^6+\left (-12220-4798 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)}{1493284 x^2-513240 x^3+66096 x^4-3780 x^5+81 x^6+\left (-4888 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(7454200 - 2564100*x + 1823674*x^2 - 532140*x^3 + 66501*x^4 - 3780*x^5 + 81*x^6 + (-12220 - 4798*x^2 + 840
*x^3 - 36*x^4)*Log[x] + 4*x^2*Log[x]^2)/(1493284*x^2 - 513240*x^3 + 66096*x^4 - 3780*x^5 + 81*x^6 + (-4888*x^2
+ 840*x^3 - 36*x^4)*Log[x] + 4*x^2*Log[x]^2),x]

[Out]

x + 330390*Defer[Int][(1222 - 210*x + 9*x^2 - 2*Log[x])^(-2), x] - 12220*Defer[Int][1/(x^2*(1222 - 210*x + 9*x
^2 - 2*Log[x])^2), x] - 1281000*Defer[Int][1/(x*(1222 - 210*x + 9*x^2 - 2*Log[x])^2), x] - 28350*Defer[Int][x/
(1222 - 210*x + 9*x^2 - 2*Log[x])^2, x] + 810*Defer[Int][x^2/(1222 - 210*x + 9*x^2 - 2*Log[x])^2, x] - 45*Defe
r[Int][(1222 - 210*x + 9*x^2 - 2*Log[x])^(-1), x] + 6110*Defer[Int][1/(x^2*(1222 - 210*x + 9*x^2 - 2*Log[x])),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {7454200-2564100 x+1823674 x^2-532140 x^3+66501 x^4-3780 x^5+81 x^6-2 \left (6110+2399 x^2-420 x^3+18 x^4\right ) \log (x)+4 x^2 \log ^2(x)}{x^2 \left (1222-210 x+9 x^2-2 \log (x)\right )^2} \, dx\\ &=\int \left (1+\frac {10 \left (-1222-128100 x+33039 x^2-2835 x^3+81 x^4\right )}{x^2 \left (1222-210 x+9 x^2-2 \log (x)\right )^2}-\frac {5 \left (-1222+9 x^2\right )}{x^2 \left (1222-210 x+9 x^2-2 \log (x)\right )}\right ) \, dx\\ &=x-5 \int \frac {-1222+9 x^2}{x^2 \left (1222-210 x+9 x^2-2 \log (x)\right )} \, dx+10 \int \frac {-1222-128100 x+33039 x^2-2835 x^3+81 x^4}{x^2 \left (1222-210 x+9 x^2-2 \log (x)\right )^2} \, dx\\ &=x-5 \int \left (\frac {9}{1222-210 x+9 x^2-2 \log (x)}-\frac {1222}{x^2 \left (1222-210 x+9 x^2-2 \log (x)\right )}\right ) \, dx+10 \int \left (\frac {33039}{\left (1222-210 x+9 x^2-2 \log (x)\right )^2}-\frac {1222}{x^2 \left (1222-210 x+9 x^2-2 \log (x)\right )^2}-\frac {128100}{x \left (1222-210 x+9 x^2-2 \log (x)\right )^2}-\frac {2835 x}{\left (1222-210 x+9 x^2-2 \log (x)\right )^2}+\frac {81 x^2}{\left (1222-210 x+9 x^2-2 \log (x)\right )^2}\right ) \, dx\\ &=x-45 \int \frac {1}{1222-210 x+9 x^2-2 \log (x)} \, dx+810 \int \frac {x^2}{\left (1222-210 x+9 x^2-2 \log (x)\right )^2} \, dx+6110 \int \frac {1}{x^2 \left (1222-210 x+9 x^2-2 \log (x)\right )} \, dx-12220 \int \frac {1}{x^2 \left (1222-210 x+9 x^2-2 \log (x)\right )^2} \, dx-28350 \int \frac {x}{\left (1222-210 x+9 x^2-2 \log (x)\right )^2} \, dx+330390 \int \frac {1}{\left (1222-210 x+9 x^2-2 \log (x)\right )^2} \, dx-1281000 \int \frac {1}{x \left (1222-210 x+9 x^2-2 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 33, normalized size = 1.10 \begin {gather*} x+\frac {5 \left (1222-210 x+9 x^2\right )}{x \left (-1222+210 x-9 x^2+2 \log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7454200 - 2564100*x + 1823674*x^2 - 532140*x^3 + 66501*x^4 - 3780*x^5 + 81*x^6 + (-12220 - 4798*x^2
+ 840*x^3 - 36*x^4)*Log[x] + 4*x^2*Log[x]^2)/(1493284*x^2 - 513240*x^3 + 66096*x^4 - 3780*x^5 + 81*x^6 + (-48
88*x^2 + 840*x^3 - 36*x^4)*Log[x] + 4*x^2*Log[x]^2),x]

[Out]

x + (5*(1222 - 210*x + 9*x^2))/(x*(-1222 + 210*x - 9*x^2 + 2*Log[x]))

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fricas [A]  time = 0.86, size = 49, normalized size = 1.63 \begin {gather*} \frac {9 \, x^{4} - 210 \, x^{3} - 2 \, x^{2} \log \relax (x) + 1177 \, x^{2} + 1050 \, x - 6110}{9 \, x^{3} - 210 \, x^{2} - 2 \, x \log \relax (x) + 1222 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*log(x)^2+(-36*x^4+840*x^3-4798*x^2-12220)*log(x)+81*x^6-3780*x^5+66501*x^4-532140*x^3+1823674
*x^2-2564100*x+7454200)/(4*x^2*log(x)^2+(-36*x^4+840*x^3-4888*x^2)*log(x)+81*x^6-3780*x^5+66096*x^4-513240*x^3
+1493284*x^2),x, algorithm="fricas")

[Out]

(9*x^4 - 210*x^3 - 2*x^2*log(x) + 1177*x^2 + 1050*x - 6110)/(9*x^3 - 210*x^2 - 2*x*log(x) + 1222*x)

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giac [A]  time = 2.71, size = 35, normalized size = 1.17 \begin {gather*} x - \frac {5 \, {\left (9 \, x^{2} - 210 \, x + 1222\right )}}{9 \, x^{3} - 210 \, x^{2} - 2 \, x \log \relax (x) + 1222 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*log(x)^2+(-36*x^4+840*x^3-4798*x^2-12220)*log(x)+81*x^6-3780*x^5+66501*x^4-532140*x^3+1823674
*x^2-2564100*x+7454200)/(4*x^2*log(x)^2+(-36*x^4+840*x^3-4888*x^2)*log(x)+81*x^6-3780*x^5+66096*x^4-513240*x^3
+1493284*x^2),x, algorithm="giac")

[Out]

x - 5*(9*x^2 - 210*x + 1222)/(9*x^3 - 210*x^2 - 2*x*log(x) + 1222*x)

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maple [A]  time = 0.05, size = 34, normalized size = 1.13

method	result	size
risch	\(x -\frac {5 \left (9 x^{2}-210 x +1222\right )}{x \left (9 x^{2}-210 x -2 \ln \relax (x )+1222\right )}\)	\(34\)
norman	\(\frac {-6110-210 x^{3}+1050 x +1177 x^{2}-2 x^{2} \ln \relax (x )+9 x^{4}}{x \left (9 x^{2}-210 x -2 \ln \relax (x )+1222\right )}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2*ln(x)^2+(-36*x^4+840*x^3-4798*x^2-12220)*ln(x)+81*x^6-3780*x^5+66501*x^4-532140*x^3+1823674*x^2-256
4100*x+7454200)/(4*x^2*ln(x)^2+(-36*x^4+840*x^3-4888*x^2)*ln(x)+81*x^6-3780*x^5+66096*x^4-513240*x^3+1493284*x
^2),x,method=_RETURNVERBOSE)

[Out]

x-5*(9*x^2-210*x+1222)/x/(9*x^2-210*x-2*ln(x)+1222)

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maxima [A]  time = 0.57, size = 49, normalized size = 1.63 \begin {gather*} \frac {9 \, x^{4} - 210 \, x^{3} - 2 \, x^{2} \log \relax (x) + 1177 \, x^{2} + 1050 \, x - 6110}{9 \, x^{3} - 210 \, x^{2} - 2 \, x \log \relax (x) + 1222 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*log(x)^2+(-36*x^4+840*x^3-4798*x^2-12220)*log(x)+81*x^6-3780*x^5+66501*x^4-532140*x^3+1823674
*x^2-2564100*x+7454200)/(4*x^2*log(x)^2+(-36*x^4+840*x^3-4888*x^2)*log(x)+81*x^6-3780*x^5+66096*x^4-513240*x^3
+1493284*x^2),x, algorithm="maxima")

[Out]

(9*x^4 - 210*x^3 - 2*x^2*log(x) + 1177*x^2 + 1050*x - 6110)/(9*x^3 - 210*x^2 - 2*x*log(x) + 1222*x)

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mupad [B]  time = 1.45, size = 32, normalized size = 1.07 \begin {gather*} x+\frac {45\,x^2-1050\,x+6110}{x\,\left (210\,x+2\,\ln \relax (x)-9\,x^2-1222\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2*log(x)^2 - log(x)*(4798*x^2 - 840*x^3 + 36*x^4 + 12220) - 2564100*x + 1823674*x^2 - 532140*x^3 + 66
501*x^4 - 3780*x^5 + 81*x^6 + 7454200)/(4*x^2*log(x)^2 - log(x)*(4888*x^2 - 840*x^3 + 36*x^4) + 1493284*x^2 -
513240*x^3 + 66096*x^4 - 3780*x^5 + 81*x^6),x)

[Out]

x + (45*x^2 - 1050*x + 6110)/(x*(210*x + 2*log(x) - 9*x^2 - 1222))

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sympy [A]  time = 0.16, size = 31, normalized size = 1.03 \begin {gather*} x + \frac {45 x^{2} - 1050 x + 6110}{- 9 x^{3} + 210 x^{2} + 2 x \log {\relax (x )} - 1222 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2*ln(x)**2+(-36*x**4+840*x**3-4798*x**2-12220)*ln(x)+81*x**6-3780*x**5+66501*x**4-532140*x**3+
1823674*x**2-2564100*x+7454200)/(4*x**2*ln(x)**2+(-36*x**4+840*x**3-4888*x**2)*ln(x)+81*x**6-3780*x**5+66096*x
**4-513240*x**3+1493284*x**2),x)

[Out]

x + (45*x**2 - 1050*x + 6110)/(-9*x**3 + 210*x**2 + 2*x*log(x) - 1222*x)

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