3.27.44 \(\int \frac {64+(80 x^2+40 x^3) \log (\frac {10}{x})+(-40 x^2-40 x^3) \log ^2(\frac {10}{x})}{256+(320 x^2+160 x^3) \log ^2(\frac {10}{x})+(100 x^4+100 x^5+25 x^6) \log ^4(\frac {10}{x})} \, dx\)

Optimal. Leaf size=24 \[ \frac {x}{4+\frac {5}{4} x^2 (2+x) \log ^2\left (\frac {10}{x}\right )} \]

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Rubi [F]  time = 37.04, 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 {64+\left (80 x^2+40 x^3\right ) \log \left (\frac {10}{x}\right )+\left (-40 x^2-40 x^3\right ) \log ^2\left (\frac {10}{x}\right )}{256+\left (320 x^2+160 x^3\right ) \log ^2\left (\frac {10}{x}\right )+\left (100 x^4+100 x^5+25 x^6\right ) \log ^4\left (\frac {10}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(64 + (80*x^2 + 40*x^3)*Log[10/x] + (-40*x^2 - 40*x^3)*Log[10/x]^2)/(256 + (320*x^2 + 160*x^3)*Log[10/x]^2
+ (100*x^4 + 100*x^5 + 25*x^6)*Log[10/x]^4),x]

[Out]

192*Defer[Int][(16 + 5*x^2*(2 + x)*Log[10/x]^2)^(-2), x] - 128*Defer[Int][1/((2 + x)*(16 + 5*x^2*(2 + x)*Log[1
0/x]^2)^2), x] + 80*Defer[Int][(x^2*Log[10/x])/(16 + 5*x^2*(2 + x)*Log[10/x]^2)^2, x] + 40*Defer[Int][(x^3*Log
[10/x])/(16 + 5*x^2*(2 + x)*Log[10/x]^2)^2, x] - 8*Defer[Int][(16 + 5*x^2*(2 + x)*Log[10/x]^2)^(-1), x] + 8*De
fer[Int][1/((2 + x)*(16 + 5*x^2*(2 + x)*Log[10/x]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 \left (8+5 x^2 (2+x) \log \left (\frac {10}{x}\right )-5 x^2 (1+x) \log ^2\left (\frac {10}{x}\right )\right )}{\left (16+5 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx\\ &=8 \int \frac {8+5 x^2 (2+x) \log \left (\frac {10}{x}\right )-5 x^2 (1+x) \log ^2\left (\frac {10}{x}\right )}{\left (16+5 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx\\ &=8 \int \left (\frac {32+24 x+20 x^2 \log \left (\frac {10}{x}\right )+20 x^3 \log \left (\frac {10}{x}\right )+5 x^4 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2}+\frac {-1-x}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )}\right ) \, dx\\ &=8 \int \frac {32+24 x+20 x^2 \log \left (\frac {10}{x}\right )+20 x^3 \log \left (\frac {10}{x}\right )+5 x^4 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx+8 \int \frac {-1-x}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )} \, dx\\ &=8 \int \frac {8 (4+3 x)+5 x^2 (2+x)^2 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+5 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx+8 \int \frac {-1-x}{(2+x) \left (16+5 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )\right )} \, dx\\ &=8 \int \left (\frac {32}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2}+\frac {24 x}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2}+\frac {20 x^2 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2}+\frac {20 x^3 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2}+\frac {5 x^4 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2}\right ) \, dx+8 \int \left (-\frac {1}{16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )}+\frac {1}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )}\right ) \, dx\\ &=-\left (8 \int \frac {1}{16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )} \, dx\right )+8 \int \frac {1}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )} \, dx+40 \int \frac {x^4 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx+160 \int \frac {x^2 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx+160 \int \frac {x^3 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx+192 \int \frac {x}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx+256 \int \frac {1}{(2+x) \left (16+10 x^2 \log ^2\left (\frac {10}{x}\right )+5 x^3 \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx\\ &=-\left (8 \int \frac {1}{16+5 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )} \, dx\right )+8 \int \frac {1}{(2+x) \left (16+5 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )\right )} \, dx+40 \int \frac {x^4 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+5 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx+160 \int \frac {x^2 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+5 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx+160 \int \frac {x^3 \log \left (\frac {10}{x}\right )}{(2+x) \left (16+5 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx+192 \int \frac {x}{(2+x) \left (16+5 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx+256 \int \frac {1}{(2+x) \left (16+5 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.47, size = 23, normalized size = 0.96 \begin {gather*} \frac {8 x}{32+10 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(64 + (80*x^2 + 40*x^3)*Log[10/x] + (-40*x^2 - 40*x^3)*Log[10/x]^2)/(256 + (320*x^2 + 160*x^3)*Log[1
0/x]^2 + (100*x^4 + 100*x^5 + 25*x^6)*Log[10/x]^4),x]

[Out]

(8*x)/(32 + 10*x^2*(2 + x)*Log[10/x]^2)

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fricas [A]  time = 0.58, size = 26, normalized size = 1.08 \begin {gather*} \frac {4 \, x}{5 \, {\left (x^{3} + 2 \, x^{2}\right )} \log \left (\frac {10}{x}\right )^{2} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^3-40*x^2)*log(10/x)^2+(40*x^3+80*x^2)*log(10/x)+64)/((25*x^6+100*x^5+100*x^4)*log(10/x)^4+(1
60*x^3+320*x^2)*log(10/x)^2+256),x, algorithm="fricas")

[Out]

4*x/(5*(x^3 + 2*x^2)*log(10/x)^2 + 16)

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giac [A]  time = 0.69, size = 36, normalized size = 1.50 \begin {gather*} \frac {4}{{\left (5 \, \log \left (\frac {10}{x}\right )^{2} + \frac {10 \, \log \left (\frac {10}{x}\right )^{2}}{x} + \frac {16}{x^{3}}\right )} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^3-40*x^2)*log(10/x)^2+(40*x^3+80*x^2)*log(10/x)+64)/((25*x^6+100*x^5+100*x^4)*log(10/x)^4+(1
60*x^3+320*x^2)*log(10/x)^2+256),x, algorithm="giac")

[Out]

4/((5*log(10/x)^2 + 10*log(10/x)^2/x + 16/x^3)*x^2)

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maple [A]  time = 0.06, size = 34, normalized size = 1.42

method	result	size
risch	\(\frac {4 x}{5 \ln \left (\frac {10}{x}\right )^{2} x^{3}+10 x^{2} \ln \left (\frac {10}{x}\right )^{2}+16}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-40*x^3-40*x^2)*ln(10/x)^2+(40*x^3+80*x^2)*ln(10/x)+64)/((25*x^6+100*x^5+100*x^4)*ln(10/x)^4+(160*x^3+32
0*x^2)*ln(10/x)^2+256),x,method=_RETURNVERBOSE)

[Out]

4*x/(5*ln(10/x)^2*x^3+10*x^2*ln(10/x)^2+16)

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maxima [B]  time = 1.01, size = 86, normalized size = 3.58 \begin {gather*} \frac {4 \, x}{5 \, {\left (\log \relax (5)^{2} + 2 \, \log \relax (5) \log \relax (2) + \log \relax (2)^{2}\right )} x^{3} + 10 \, {\left (\log \relax (5)^{2} + 2 \, \log \relax (5) \log \relax (2) + \log \relax (2)^{2}\right )} x^{2} + 5 \, {\left (x^{3} + 2 \, x^{2}\right )} \log \relax (x)^{2} - 10 \, {\left (x^{3} {\left (\log \relax (5) + \log \relax (2)\right )} + 2 \, x^{2} {\left (\log \relax (5) + \log \relax (2)\right )}\right )} \log \relax (x) + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^3-40*x^2)*log(10/x)^2+(40*x^3+80*x^2)*log(10/x)+64)/((25*x^6+100*x^5+100*x^4)*log(10/x)^4+(1
60*x^3+320*x^2)*log(10/x)^2+256),x, algorithm="maxima")

[Out]

4*x/(5*(log(5)^2 + 2*log(5)*log(2) + log(2)^2)*x^3 + 10*(log(5)^2 + 2*log(5)*log(2) + log(2)^2)*x^2 + 5*(x^3 +
2*x^2)*log(x)^2 - 10*(x^3*(log(5) + log(2)) + 2*x^2*(log(5) + log(2)))*log(x) + 16)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\left (-40\,x^3-40\,x^2\right )\,{\ln \left (\frac {10}{x}\right )}^2+\left (40\,x^3+80\,x^2\right )\,\ln \left (\frac {10}{x}\right )+64}{\left (25\,x^6+100\,x^5+100\,x^4\right )\,{\ln \left (\frac {10}{x}\right )}^4+\left (160\,x^3+320\,x^2\right )\,{\ln \left (\frac {10}{x}\right )}^2+256} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(10/x)*(80*x^2 + 40*x^3) - log(10/x)^2*(40*x^2 + 40*x^3) + 64)/(log(10/x)^2*(320*x^2 + 160*x^3) + log(
10/x)^4*(100*x^4 + 100*x^5 + 25*x^6) + 256),x)

[Out]

int((log(10/x)*(80*x^2 + 40*x^3) - log(10/x)^2*(40*x^2 + 40*x^3) + 64)/(log(10/x)^2*(320*x^2 + 160*x^3) + log(
10/x)^4*(100*x^4 + 100*x^5 + 25*x^6) + 256), x)

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sympy [A]  time = 0.25, size = 20, normalized size = 0.83 \begin {gather*} \frac {4 x}{\left (5 x^{3} + 10 x^{2}\right ) \log {\left (\frac {10}{x} \right )}^{2} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x**3-40*x**2)*ln(10/x)**2+(40*x**3+80*x**2)*ln(10/x)+64)/((25*x**6+100*x**5+100*x**4)*ln(10/x)
**4+(160*x**3+320*x**2)*ln(10/x)**2+256),x)

[Out]

4*x/((5*x**3 + 10*x**2)*log(10/x)**2 + 16)

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