3.66.100 \(\int \frac {-100 x^5-750 x^6+(300 x^3+6750 x^4+45500 x^5+7500 x^6+300 x^7) \log (\frac {2+30 x+3 x^2}{1+15 x+x^2})+(-200 x-6000 x^2-45500 x^3-7500 x^4-300 x^5) \log ^2(\frac {2+30 x+3 x^2}{1+15 x+x^2})}{(2+60 x+455 x^2+75 x^3+3 x^4) \log ^3(\frac {2+30 x+3 x^2}{1+15 x+x^2})} \, dx\)

Optimal. Leaf size=24 \[ \left (-5+\frac {5 x^2}{\log \left (2+\frac {x}{15+\frac {1}{x}+x}\right )}\right )^2 \]

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Rubi [F]  time = 4.95, 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 {-100 x^5-750 x^6+\left (300 x^3+6750 x^4+45500 x^5+7500 x^6+300 x^7\right ) \log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )+\left (-200 x-6000 x^2-45500 x^3-7500 x^4-300 x^5\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}{\left (2+60 x+455 x^2+75 x^3+3 x^4\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-100*x^5 - 750*x^6 + (300*x^3 + 6750*x^4 + 45500*x^5 + 7500*x^6 + 300*x^7)*Log[(2 + 30*x + 3*x^2)/(1 + 15
*x + x^2)] + (-200*x - 6000*x^2 - 45500*x^3 - 7500*x^4 - 300*x^5)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^2)/
((2 + 60*x + 455*x^2 + 75*x^3 + 3*x^4)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^3),x]

[Out]

-117500*Defer[Int][Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^(-3), x] + 98000*Sqrt[3/73]*Defer[Int][1/((-30 + 2
*Sqrt[219] - 6*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^3), x] - (333000*Defer[Int][1/((-15 + Sqrt[221] - 2
*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^3), x])/Sqrt[221] + (18650*Defer[Int][x/Log[(2 + 30*x + 3*x^2)/(1
+ 15*x + x^2)]^3, x])/3 - 250*Defer[Int][x^2/Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^3, x] + (2486350*(221 -
15*Sqrt[221])*Defer[Int][1/((15 - Sqrt[221] + 2*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^3), x])/221 - (33
3000*Defer[Int][1/((15 + Sqrt[221] + 2*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^3), x])/Sqrt[221] + (248635
0*(221 + 15*Sqrt[221])*Defer[Int][1/((15 + Sqrt[221] + 2*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^3), x])/2
21 - (4380400*(73 - 5*Sqrt[219])*Defer[Int][1/((30 - 2*Sqrt[219] + 6*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2
)]^3), x])/219 + 98000*Sqrt[3/73]*Defer[Int][1/((30 + 2*Sqrt[219] + 6*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^
2)]^3), x] - (4380400*(73 + 5*Sqrt[219])*Defer[Int][1/((30 + 2*Sqrt[219] + 6*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15
*x + x^2)]^3), x])/219 + 250*Defer[Int][Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^(-2), x] - 1000*Sqrt[3/73]*De
fer[Int][1/((-30 + 2*Sqrt[219] - 6*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^2), x] + (1500*Defer[Int][1/((-
15 + Sqrt[221] - 2*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^2), x])/Sqrt[221] + 100*Defer[Int][x^3/Log[(2 +
30*x + 3*x^2)/(1 + 15*x + x^2)]^2, x] - (11150*(221 - 15*Sqrt[221])*Defer[Int][1/((15 - Sqrt[221] + 2*x)*Log[
(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^2), x])/221 + (1500*Defer[Int][1/((15 + Sqrt[221] + 2*x)*Log[(2 + 30*x +
3*x^2)/(1 + 15*x + x^2)]^2), x])/Sqrt[221] - (11150*(221 + 15*Sqrt[221])*Defer[Int][1/((15 + Sqrt[221] + 2*x)*
Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^2), x])/221 + (14800*(73 - 5*Sqrt[219])*Defer[Int][1/((30 - 2*Sqrt[21
9] + 6*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^2), x])/73 - 1000*Sqrt[3/73]*Defer[Int][1/((30 + 2*Sqrt[219
] + 6*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^2), x] + (14800*(73 + 5*Sqrt[219])*Defer[Int][1/((30 + 2*Sqr
t[219] + 6*x)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^2), x])/73 - 100*Defer[Int][x/Log[(2 + 30*x + 3*x^2)/(1
+ 15*x + x^2)], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {50 x^5 (2+15 x)}{\left (1+15 x+x^2\right ) \left (2+30 x+3 x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}+\frac {50 x^3 \left (6+135 x+910 x^2+150 x^3+6 x^4\right )}{\left (1+15 x+x^2\right ) \left (2+30 x+3 x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}-\frac {100 x}{\log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}\right ) \, dx\\ &=-\left (50 \int \frac {x^5 (2+15 x)}{\left (1+15 x+x^2\right ) \left (2+30 x+3 x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx\right )+50 \int \frac {x^3 \left (6+135 x+910 x^2+150 x^3+6 x^4\right )}{\left (1+15 x+x^2\right ) \left (2+30 x+3 x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-100 \int \frac {x}{\log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx\\ &=-\left (50 \int \left (\frac {2350}{\log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}-\frac {373 x}{3 \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}+\frac {5 x^2}{\log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}+\frac {-3330-49727 x}{\left (1+15 x+x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}+\frac {8 (735+10951 x)}{3 \left (2+30 x+3 x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}\right ) \, dx\right )+50 \int \left (\frac {5}{\log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}+\frac {2 x^3}{\log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}+\frac {-15-223 x}{\left (1+15 x+x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}+\frac {4 (5+74 x)}{\left (2+30 x+3 x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}\right ) \, dx-100 \int \frac {x}{\log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx\\ &=-\left (50 \int \frac {-3330-49727 x}{\left (1+15 x+x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx\right )+50 \int \frac {-15-223 x}{\left (1+15 x+x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+100 \int \frac {x^3}{\log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-100 \int \frac {x}{\log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-\frac {400}{3} \int \frac {735+10951 x}{\left (2+30 x+3 x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+200 \int \frac {5+74 x}{\left (2+30 x+3 x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-250 \int \frac {x^2}{\log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+250 \int \frac {1}{\log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+\frac {18650}{3} \int \frac {x}{\log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-117500 \int \frac {1}{\log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx\\ &=-\left (50 \int \left (-\frac {3330}{\left (1+15 x+x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}-\frac {49727 x}{\left (1+15 x+x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}\right ) \, dx\right )+50 \int \left (-\frac {15}{\left (1+15 x+x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}-\frac {223 x}{\left (1+15 x+x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}\right ) \, dx+100 \int \frac {x^3}{\log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-100 \int \frac {x}{\log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-\frac {400}{3} \int \left (\frac {735}{\left (2+30 x+3 x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}+\frac {10951 x}{\left (2+30 x+3 x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}\right ) \, dx+200 \int \left (\frac {5}{\left (2+30 x+3 x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}+\frac {74 x}{\left (2+30 x+3 x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}\right ) \, dx-250 \int \frac {x^2}{\log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+250 \int \frac {1}{\log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+\frac {18650}{3} \int \frac {x}{\log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-117500 \int \frac {1}{\log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx\\ &=100 \int \frac {x^3}{\log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-100 \int \frac {x}{\log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-250 \int \frac {x^2}{\log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+250 \int \frac {1}{\log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-750 \int \frac {1}{\left (1+15 x+x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+1000 \int \frac {1}{\left (2+30 x+3 x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+\frac {18650}{3} \int \frac {x}{\log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-11150 \int \frac {x}{\left (1+15 x+x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+14800 \int \frac {x}{\left (2+30 x+3 x^2\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-98000 \int \frac {1}{\left (2+30 x+3 x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-117500 \int \frac {1}{\log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+166500 \int \frac {1}{\left (1+15 x+x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx-\frac {4380400}{3} \int \frac {x}{\left (2+30 x+3 x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx+2486350 \int \frac {x}{\left (1+15 x+x^2\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 63, normalized size = 2.62 \begin {gather*} 50 \left (\frac {x^4}{2 \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}-\frac {x^2}{\log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-100*x^5 - 750*x^6 + (300*x^3 + 6750*x^4 + 45500*x^5 + 7500*x^6 + 300*x^7)*Log[(2 + 30*x + 3*x^2)/(
1 + 15*x + x^2)] + (-200*x - 6000*x^2 - 45500*x^3 - 7500*x^4 - 300*x^5)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2
)]^2)/((2 + 60*x + 455*x^2 + 75*x^3 + 3*x^4)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^3),x]

[Out]

50*(x^4/(2*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^2) - x^2/Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)])

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fricas [B]  time = 0.54, size = 57, normalized size = 2.38 \begin {gather*} \frac {25 \, {\left (x^{4} - 2 \, x^{2} \log \left (\frac {3 \, x^{2} + 30 \, x + 2}{x^{2} + 15 \, x + 1}\right )\right )}}{\log \left (\frac {3 \, x^{2} + 30 \, x + 2}{x^{2} + 15 \, x + 1}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-300*x^5-7500*x^4-45500*x^3-6000*x^2-200*x)*log((3*x^2+30*x+2)/(x^2+15*x+1))^2+(300*x^7+7500*x^6+4
5500*x^5+6750*x^4+300*x^3)*log((3*x^2+30*x+2)/(x^2+15*x+1))-750*x^6-100*x^5)/(3*x^4+75*x^3+455*x^2+60*x+2)/log
((3*x^2+30*x+2)/(x^2+15*x+1))^3,x, algorithm="fricas")

[Out]

25*(x^4 - 2*x^2*log((3*x^2 + 30*x + 2)/(x^2 + 15*x + 1)))/log((3*x^2 + 30*x + 2)/(x^2 + 15*x + 1))^2

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giac [B]  time = 0.61, size = 57, normalized size = 2.38 \begin {gather*} \frac {25 \, {\left (x^{4} - 2 \, x^{2} \log \left (\frac {3 \, x^{2} + 30 \, x + 2}{x^{2} + 15 \, x + 1}\right )\right )}}{\log \left (\frac {3 \, x^{2} + 30 \, x + 2}{x^{2} + 15 \, x + 1}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-300*x^5-7500*x^4-45500*x^3-6000*x^2-200*x)*log((3*x^2+30*x+2)/(x^2+15*x+1))^2+(300*x^7+7500*x^6+4
5500*x^5+6750*x^4+300*x^3)*log((3*x^2+30*x+2)/(x^2+15*x+1))-750*x^6-100*x^5)/(3*x^4+75*x^3+455*x^2+60*x+2)/log
((3*x^2+30*x+2)/(x^2+15*x+1))^3,x, algorithm="giac")

[Out]

25*(x^4 - 2*x^2*log((3*x^2 + 30*x + 2)/(x^2 + 15*x + 1)))/log((3*x^2 + 30*x + 2)/(x^2 + 15*x + 1))^2

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maple [B]  time = 0.05, size = 58, normalized size = 2.42

method	result	size
risch	\(\frac {25 \left (x^{2}-2 \ln \left (\frac {3 x^{2}+30 x +2}{x^{2}+15 x +1}\right )\right ) x^{2}}{\ln \left (\frac {3 x^{2}+30 x +2}{x^{2}+15 x +1}\right )^{2}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-300*x^5-7500*x^4-45500*x^3-6000*x^2-200*x)*ln((3*x^2+30*x+2)/(x^2+15*x+1))^2+(300*x^7+7500*x^6+45500*x^
5+6750*x^4+300*x^3)*ln((3*x^2+30*x+2)/(x^2+15*x+1))-750*x^6-100*x^5)/(3*x^4+75*x^3+455*x^2+60*x+2)/ln((3*x^2+3
0*x+2)/(x^2+15*x+1))^3,x,method=_RETURNVERBOSE)

[Out]

25*(x^2-2*ln((3*x^2+30*x+2)/(x^2+15*x+1)))*x^2/ln((3*x^2+30*x+2)/(x^2+15*x+1))^2

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maxima [B]  time = 0.50, size = 85, normalized size = 3.54 \begin {gather*} \frac {25 \, {\left (x^{4} - 2 \, x^{2} \log \left (3 \, x^{2} + 30 \, x + 2\right ) + 2 \, x^{2} \log \left (x^{2} + 15 \, x + 1\right )\right )}}{\log \left (3 \, x^{2} + 30 \, x + 2\right )^{2} - 2 \, \log \left (3 \, x^{2} + 30 \, x + 2\right ) \log \left (x^{2} + 15 \, x + 1\right ) + \log \left (x^{2} + 15 \, x + 1\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-300*x^5-7500*x^4-45500*x^3-6000*x^2-200*x)*log((3*x^2+30*x+2)/(x^2+15*x+1))^2+(300*x^7+7500*x^6+4
5500*x^5+6750*x^4+300*x^3)*log((3*x^2+30*x+2)/(x^2+15*x+1))-750*x^6-100*x^5)/(3*x^4+75*x^3+455*x^2+60*x+2)/log
((3*x^2+30*x+2)/(x^2+15*x+1))^3,x, algorithm="maxima")

[Out]

25*(x^4 - 2*x^2*log(3*x^2 + 30*x + 2) + 2*x^2*log(x^2 + 15*x + 1))/(log(3*x^2 + 30*x + 2)^2 - 2*log(3*x^2 + 30
*x + 2)*log(x^2 + 15*x + 1) + log(x^2 + 15*x + 1)^2)

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mupad [B]  time = 7.98, size = 113, normalized size = 4.71 \begin {gather*} \frac {2531672218\,\ln \left (\frac {3\,x^2+30\,x+2}{x^2+15\,x+1}\right )}{50625}+\frac {25\,x^4-50\,x^2\,\ln \left (\frac {3\,x^2+30\,x+2}{x^2+15\,x+1}\right )}{{\ln \left (\frac {3\,x^2+30\,x+2}{x^2+15\,x+1}\right )}^2}+\frac {\mathrm {atan}\left (\frac {x^2\,371{}\mathrm {i}+x\,2250{}\mathrm {i}+150{}\mathrm {i}}{955\,x^2+11010\,x+734}\right )\,5063344436{}\mathrm {i}}{50625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((30*x + 3*x^2 + 2)/(15*x + x^2 + 1))^2*(200*x + 6000*x^2 + 45500*x^3 + 7500*x^4 + 300*x^5) - log((30
*x + 3*x^2 + 2)/(15*x + x^2 + 1))*(300*x^3 + 6750*x^4 + 45500*x^5 + 7500*x^6 + 300*x^7) + 100*x^5 + 750*x^6)/(
log((30*x + 3*x^2 + 2)/(15*x + x^2 + 1))^3*(60*x + 455*x^2 + 75*x^3 + 3*x^4 + 2)),x)

[Out]

(2531672218*log((30*x + 3*x^2 + 2)/(15*x + x^2 + 1)))/50625 + (atan((x*2250i + x^2*371i + 150i)/(11010*x + 955
*x^2 + 734))*5063344436i)/50625 + (25*x^4 - 50*x^2*log((30*x + 3*x^2 + 2)/(15*x + x^2 + 1)))/log((30*x + 3*x^2
+ 2)/(15*x + x^2 + 1))^2

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sympy [B]  time = 0.24, size = 51, normalized size = 2.12 \begin {gather*} \frac {25 x^{4} - 50 x^{2} \log {\left (\frac {3 x^{2} + 30 x + 2}{x^{2} + 15 x + 1} \right )}}{\log {\left (\frac {3 x^{2} + 30 x + 2}{x^{2} + 15 x + 1} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-300*x**5-7500*x**4-45500*x**3-6000*x**2-200*x)*ln((3*x**2+30*x+2)/(x**2+15*x+1))**2+(300*x**7+750
0*x**6+45500*x**5+6750*x**4+300*x**3)*ln((3*x**2+30*x+2)/(x**2+15*x+1))-750*x**6-100*x**5)/(3*x**4+75*x**3+455
*x**2+60*x+2)/ln((3*x**2+30*x+2)/(x**2+15*x+1))**3,x)

[Out]

(25*x**4 - 50*x**2*log((3*x**2 + 30*x + 2)/(x**2 + 15*x + 1)))/log((3*x**2 + 30*x + 2)/(x**2 + 15*x + 1))**2

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