3.52.0 ∫ x2+5x4+(−x2−15x4) log(x)+(−5−10x2) log2(x)+5 log3(x) __________________________ 16x4+160x6+400x8+(320x4+1600x6) log(x)+(160x2+2400x4) log2(x)+1600x2 log3(x)+400 log4(x) dx [5133]

Integrand size = 105, antiderivative size = 28 \[ \int \frac {x^2+5 x^4+\left (-x^2-15 x^4\right ) \log (x)+\left (-5-10 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+\left (320 x^4+1600 x^6\right ) \log (x)+\left (160 x^2+2400 x^4\right ) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx=\frac {\log (x)}{4 x \left (4+5 \left (2 x+\frac {2 \log (x)}{x}\right )^2\right )} \]

Rubi [F]

\[ \int \frac {x^2+5 x^4+\left (-x^2-15 x^4\right ) \log (x)+\left (-5-10 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+\left (320 x^4+1600 x^6\right ) \log (x)+\left (160 x^2+2400 x^4\right ) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx=\int \frac {x^2+5 x^4+\left (-x^2-15 x^4\right ) \log (x)+\left (-5-10 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+\left (320 x^4+1600 x^6\right ) \log (x)+\left (160 x^2+2400 x^4\right ) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx \]

Int[(x^2 + 5*x^4 + (-x^2 - 15*x^4)*Log[x] + (-5 - 10*x^2)*Log[x]^2 + 5*Log[x]^3)/(16*x^4 + 160*x^6 + 400*x^8 +

(320*x^4 + 1600*x^6)*Log[x] + (160*x^2 + 2400*x^4)*Log[x]^2 + 1600*x^2*Log[x]^3 + 400*Log[x]^4),x]

Defer[Int][(-x^2 - 5*x^4 - 10*x^2*Log[x] - 5*Log[x]^2)^(-1), x]/16 + Defer[Int][x^2/(x^2 + 5*x^4 + 10*x^2*Log[

x] + 5*Log[x]^2)^2, x]/8 + (7*Defer[Int][x^4/(x^2 + 5*x^4 + 10*x^2*Log[x] + 5*Log[x]^2)^2, x])/8 + (5*Defer[In

t][x^6/(x^2 + 5*x^4 + 10*x^2*Log[x] + 5*Log[x]^2)^2, x])/4 + Defer[Int][(x^2*Log[x])/(x^2 + 5*x^4 + 10*x^2*Log

[x] + 5*Log[x]^2)^2, x]/2 + (5*Defer[Int][(x^4*Log[x])/(x^2 + 5*x^4 + 10*x^2*Log[x] + 5*Log[x]^2)^2, x])/4 - D

efer[Int][x^2/(x^2 + 5*x^4 + 10*x^2*Log[x] + 5*Log[x]^2), x]/4 + Defer[Int][Log[x]/(x^2 + 5*x^4 + 10*x^2*Log[x

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2+5 x^4-\left (x^2+15 x^4\right ) \log (x)-5 \left (1+2 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 \left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx \\ & = \frac {1}{16} \int \frac {x^2+5 x^4-\left (x^2+15 x^4\right ) \log (x)-5 \left (1+2 x^2\right ) \log ^2(x)+5 \log ^3(x)}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx \\ & = \frac {1}{16} \int \left (\frac {2 x^2 \left (1+7 x^2+10 x^4+4 \log (x)+10 x^2 \log (x)\right )}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}+\frac {-1-4 x^2+\log (x)}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)}\right ) \, dx \\ & = \frac {1}{16} \int \frac {-1-4 x^2+\log (x)}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)} \, dx+\frac {1}{8} \int \frac {x^2 \left (1+7 x^2+10 x^4+4 \log (x)+10 x^2 \log (x)\right )}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx \\ & = \frac {1}{16} \int \left (\frac {1}{-x^2-5 x^4-10 x^2 \log (x)-5 \log ^2(x)}-\frac {4 x^2}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)}+\frac {\log (x)}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)}\right ) \, dx+\frac {1}{8} \int \left (\frac {x^2}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}+\frac {7 x^4}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}+\frac {10 x^6}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}+\frac {4 x^2 \log (x)}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}+\frac {10 x^4 \log (x)}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}\right ) \, dx \\ & = \frac {1}{16} \int \frac {1}{-x^2-5 x^4-10 x^2 \log (x)-5 \log ^2(x)} \, dx+\frac {1}{16} \int \frac {\log (x)}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)} \, dx+\frac {1}{8} \int \frac {x^2}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx-\frac {1}{4} \int \frac {x^2}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)} \, dx+\frac {1}{2} \int \frac {x^2 \log (x)}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx+\frac {7}{8} \int \frac {x^4}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx+\frac {5}{4} \int \frac {x^6}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx+\frac {5}{4} \int \frac {x^4 \log (x)}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {x^2+5 x^4+\left (-x^2-15 x^4\right ) \log (x)+\left (-5-10 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+\left (320 x^4+1600 x^6\right ) \log (x)+\left (160 x^2+2400 x^4\right ) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx=\frac {x \log (x)}{16 \left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )} \]

Integrate[(x^2 + 5*x^4 + (-x^2 - 15*x^4)*Log[x] + (-5 - 10*x^2)*Log[x]^2 + 5*Log[x]^3)/(16*x^4 + 160*x^6 + 400

*x^8 + (320*x^4 + 1600*x^6)*Log[x] + (160*x^2 + 2400*x^4)*Log[x]^2 + 1600*x^2*Log[x]^3 + 400*Log[x]^4),x]

Maple [A] (veriﬁed)

Time = 0.50 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07


method	result	size

default	\(\frac {x \ln \left (x \right )}{80 x^{4}+160 x^{2} \ln \left (x \right )+80 \ln \left (x \right )^{2}+16 x^{2}}\)	\(30\)
risch	\(\frac {x \ln \left (x \right )}{80 x^{4}+160 x^{2} \ln \left (x \right )+80 \ln \left (x \right )^{2}+16 x^{2}}\)	\(30\)
parallelrisch	\(\frac {x \ln \left (x \right )}{80 x^{4}+160 x^{2} \ln \left (x \right )+80 \ln \left (x \right )^{2}+16 x^{2}}\)	\(30\)

int((5*ln(x)^3+(-10*x^2-5)*ln(x)^2+(-15*x^4-x^2)*ln(x)+5*x^4+x^2)/(400*ln(x)^4+1600*x^2*ln(x)^3+(2400*x^4+160*

x^2)*ln(x)^2+(1600*x^6+320*x^4)*ln(x)+400*x^8+160*x^6+16*x^4),x,method=_RETURNVERBOSE)

Fricas [A] (veriﬁcation not implemented)

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x^2+5 x^4+\left (-x^2-15 x^4\right ) \log (x)+\left (-5-10 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+\left (320 x^4+1600 x^6\right ) \log (x)+\left (160 x^2+2400 x^4\right ) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx=\frac {x \log \left (x\right )}{16 \, {\left (5 \, x^{4} + 10 \, x^{2} \log \left (x\right ) + x^{2} + 5 \, \log \left (x\right )^{2}\right )}} \]

integrate((5*log(x)^3+(-10*x^2-5)*log(x)^2+(-15*x^4-x^2)*log(x)+5*x^4+x^2)/(400*log(x)^4+1600*x^2*log(x)^3+(24

00*x^4+160*x^2)*log(x)^2+(1600*x^6+320*x^4)*log(x)+400*x^8+160*x^6+16*x^4),x, algorithm="fricas")

Sympy [A] (veriﬁcation not implemented)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x^2+5 x^4+\left (-x^2-15 x^4\right ) \log (x)+\left (-5-10 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+\left (320 x^4+1600 x^6\right ) \log (x)+\left (160 x^2+2400 x^4\right ) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx=\frac {x \log {\left (x \right )}}{80 x^{4} + 160 x^{2} \log {\left (x \right )} + 16 x^{2} + 80 \log {\left (x \right )}^{2}} \]

integrate((5*ln(x)**3+(-10*x**2-5)*ln(x)**2+(-15*x**4-x**2)*ln(x)+5*x**4+x**2)/(400*ln(x)**4+1600*x**2*ln(x)**

3+(2400*x**4+160*x**2)*ln(x)**2+(1600*x**6+320*x**4)*ln(x)+400*x**8+160*x**6+16*x**4),x)

Maxima [A] (veriﬁcation not implemented)

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x^2+5 x^4+\left (-x^2-15 x^4\right ) \log (x)+\left (-5-10 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+\left (320 x^4+1600 x^6\right ) \log (x)+\left (160 x^2+2400 x^4\right ) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx=\frac {x \log \left (x\right )}{16 \, {\left (5 \, x^{4} + 10 \, x^{2} \log \left (x\right ) + x^{2} + 5 \, \log \left (x\right )^{2}\right )}} \]

integrate((5*log(x)^3+(-10*x^2-5)*log(x)^2+(-15*x^4-x^2)*log(x)+5*x^4+x^2)/(400*log(x)^4+1600*x^2*log(x)^3+(24

00*x^4+160*x^2)*log(x)^2+(1600*x^6+320*x^4)*log(x)+400*x^8+160*x^6+16*x^4),x, algorithm="maxima")

Giac [A] (veriﬁcation not implemented)

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x^2+5 x^4+\left (-x^2-15 x^4\right ) \log (x)+\left (-5-10 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+\left (320 x^4+1600 x^6\right ) \log (x)+\left (160 x^2+2400 x^4\right ) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx=\frac {x \log \left (x\right )}{16 \, {\left (5 \, x^{4} + 10 \, x^{2} \log \left (x\right ) + x^{2} + 5 \, \log \left (x\right )^{2}\right )}} \]

integrate((5*log(x)^3+(-10*x^2-5)*log(x)^2+(-15*x^4-x^2)*log(x)+5*x^4+x^2)/(400*log(x)^4+1600*x^2*log(x)^3+(24

00*x^4+160*x^2)*log(x)^2+(1600*x^6+320*x^4)*log(x)+400*x^8+160*x^6+16*x^4),x, algorithm="giac")

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2+5 x^4+\left (-x^2-15 x^4\right ) \log (x)+\left (-5-10 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+\left (320 x^4+1600 x^6\right ) \log (x)+\left (160 x^2+2400 x^4\right ) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx=\int \frac {5\,{\ln \left (x\right )}^3-{\ln \left (x\right )}^2\,\left (10\,x^2+5\right )-\ln \left (x\right )\,\left (15\,x^4+x^2\right )+x^2+5\,x^4}{\ln \left (x\right )\,\left (1600\,x^6+320\,x^4\right )+400\,{\ln \left (x\right )}^4+{\ln \left (x\right )}^2\,\left (2400\,x^4+160\,x^2\right )+1600\,x^2\,{\ln \left (x\right )}^3+16\,x^4+160\,x^6+400\,x^8} \,d x \]

int((5*log(x)^3 - log(x)^2*(10*x^2 + 5) - log(x)*(x^2 + 15*x^4) + x^2 + 5*x^4)/(log(x)*(320*x^4 + 1600*x^6) +

400*log(x)^4 + log(x)^2*(160*x^2 + 2400*x^4) + 1600*x^2*log(x)^3 + 16*x^4 + 160*x^6 + 400*x^8),x)

int((5*log(x)^3 - log(x)^2*(10*x^2 + 5) - log(x)*(x^2 + 15*x^4) + x^2 + 5*x^4)/(log(x)*(320*x^4 + 1600*x^6) +

400*log(x)^4 + log(x)^2*(160*x^2 + 2400*x^4) + 1600*x^2*log(x)^3 + 16*x^4 + 160*x^6 + 400*x^8), x)

\(\int \frac {x^2+5 x^4+(-x^2-15 x^4) \log (x)+(-5-10 x^2) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+(320 x^4+1600 x^6) \log (x)+(160 x^2+2400 x^4) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx\) [5133]

Optimal result

Rubi [F]

Mathematica [A] (veriﬁed)

Maple [A] (veriﬁed)

Fricas [A] (veriﬁcation not implemented)

Sympy [A] (veriﬁcation not implemented)

Maxima [A] (veriﬁcation not implemented)

Giac [A] (veriﬁcation not implemented)

Mupad [F(-1)]