3.1.21 \(\int \frac {e^2 (100+40 x+4 x^2)+e^2 (100+40 x+4 x^2) \log (x)+(20 x^3-4 e^5 x^3) \log ^3(x)}{e^2 (-25 x-10 x^2-x^3) \log (x)+(10 x^4+2 x^5+e^5 (10 x^3+2 x^4)) \log ^3(x)} \, dx\)

Optimal. Leaf size=28 \[ \log \left (\left (\frac {2 \left (e^5+x\right )}{5+x}-\frac {e^2}{x^2 \log ^2(x)}\right )^2\right ) \]

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Rubi [F]  time = 15.62, 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 {e^2 \left (100+40 x+4 x^2\right )+e^2 \left (100+40 x+4 x^2\right ) \log (x)+\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)}{e^2 \left (-25 x-10 x^2-x^3\right ) \log (x)+\left (10 x^4+2 x^5+e^5 \left (10 x^3+2 x^4\right )\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^2*(100 + 40*x + 4*x^2) + E^2*(100 + 40*x + 4*x^2)*Log[x] + (20*x^3 - 4*E^5*x^3)*Log[x]^3)/(E^2*(-25*x -
10*x^2 - x^3)*Log[x] + (10*x^4 + 2*x^5 + E^5*(10*x^3 + 2*x^4))*Log[x]^3),x]

[Out]

-2*Log[5 + x] + 2*Log[E^5 + x] - 4*Log[Log[x]] - 20*E^2*Defer[Int][1/((E^5 + x)*(-5*E^2 - E^2*x + 2*E^5*x^2*Lo
g[x]^2 + 2*x^3*Log[x]^2)), x] - 4*E^7*Defer[Int][1/((E^5 + x)*(-5*E^2 - E^2*x + 2*E^5*x^2*Log[x]^2 + 2*x^3*Log
[x]^2)), x] + 2*E^2*(15 + E^5)*Defer[Int][1/((E^5 + x)*(-5*E^2 - E^2*x + 2*E^5*x^2*Log[x]^2 + 2*x^3*Log[x]^2))
, x] + 4*E^2*Defer[Int][(-(E^2*(5 + x)) + 2*x^2*(E^5 + x)*Log[x]^2)^(-1), x] + 8*E^5*Defer[Int][(x*Log[x])/(-(
E^2*(5 + x)) + 2*x^2*(E^5 + x)*Log[x]^2), x] + 8*Defer[Int][(x^2*Log[x])/(-(E^2*(5 + x)) + 2*x^2*(E^5 + x)*Log
[x]^2), x] + 20*E^2*Defer[Int][(-(E^2*x*(5 + x)) + 2*x^3*(E^5 + x)*Log[x]^2)^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (-e^2 (5+x)^2-e^2 (5+x)^2 \log (x)+\left (-5+e^5\right ) x^3 \log ^3(x)\right )}{x (5+x) \log (x) \left (e^2 (5+x)-2 x^2 \left (e^5+x\right ) \log ^2(x)\right )} \, dx\\ &=4 \int \frac {-e^2 (5+x)^2-e^2 (5+x)^2 \log (x)+\left (-5+e^5\right ) x^3 \log ^3(x)}{x (5+x) \log (x) \left (e^2 (5+x)-2 x^2 \left (e^5+x\right ) \log ^2(x)\right )} \, dx\\ &=4 \int \left (\frac {5-e^5}{2 (5+x) \left (e^5+x\right )}-\frac {1}{x \log (x)}+\frac {-10 e^7-15 e^2 \left (1+\frac {e^5}{15}\right ) x-2 e^2 x^2-4 e^{10} x^2 \log (x)-8 e^5 x^3 \log (x)-4 x^4 \log (x)}{2 x \left (e^5+x\right ) \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )}\right ) \, dx\\ &=2 \int \frac {-10 e^7-15 e^2 \left (1+\frac {e^5}{15}\right ) x-2 e^2 x^2-4 e^{10} x^2 \log (x)-8 e^5 x^3 \log (x)-4 x^4 \log (x)}{x \left (e^5+x\right ) \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )} \, dx-4 \int \frac {1}{x \log (x)} \, dx+\left (2 \left (5-e^5\right )\right ) \int \frac {1}{(5+x) \left (e^5+x\right )} \, dx\\ &=-\left (2 \int \frac {1}{5+x} \, dx\right )+2 \int \frac {1}{e^5+x} \, dx+2 \int \frac {-e^7 (10+x)-e^2 x (15+2 x)-4 x^2 \left (e^5+x\right )^2 \log (x)}{x \left (e^5+x\right ) \left (e^2 (5+x)-2 x^2 \left (e^5+x\right ) \log ^2(x)\right )} \, dx-4 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-2 \log (5+x)+2 \log \left (e^5+x\right )-4 \log (\log (x))+2 \int \left (\frac {-10 e^7-15 e^2 \left (1+\frac {e^5}{15}\right ) x-2 e^2 x^2-4 e^{10} x^2 \log (x)-8 e^5 x^3 \log (x)-4 x^4 \log (x)}{e^5 x \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )}+\frac {10 e^7+15 e^2 \left (1+\frac {e^5}{15}\right ) x+2 e^2 x^2+4 e^{10} x^2 \log (x)+8 e^5 x^3 \log (x)+4 x^4 \log (x)}{e^5 \left (e^5+x\right ) \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )}\right ) \, dx\\ &=-2 \log (5+x)+2 \log \left (e^5+x\right )-4 \log (\log (x))+\frac {2 \int \frac {-10 e^7-15 e^2 \left (1+\frac {e^5}{15}\right ) x-2 e^2 x^2-4 e^{10} x^2 \log (x)-8 e^5 x^3 \log (x)-4 x^4 \log (x)}{x \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )} \, dx}{e^5}+\frac {2 \int \frac {10 e^7+15 e^2 \left (1+\frac {e^5}{15}\right ) x+2 e^2 x^2+4 e^{10} x^2 \log (x)+8 e^5 x^3 \log (x)+4 x^4 \log (x)}{\left (e^5+x\right ) \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )} \, dx}{e^5}\\ &=-2 \log (5+x)+2 \log \left (e^5+x\right )-4 \log (\log (x))+\frac {2 \int \frac {e^7 (10+x)+e^2 x (15+2 x)+4 x^2 \left (e^5+x\right )^2 \log (x)}{\left (e^5+x\right ) \left (e^2 (5+x)-2 x^2 \left (e^5+x\right ) \log ^2(x)\right )} \, dx}{e^5}+\frac {2 \int \frac {-e^7 (10+x)-e^2 x (15+2 x)-4 x^2 \left (e^5+x\right )^2 \log (x)}{e^2 x (5+x)-2 x^3 \left (e^5+x\right ) \log ^2(x)} \, dx}{e^5}\\ &=-2 \log (5+x)+2 \log \left (e^5+x\right )-4 \log (\log (x))+\frac {2 \int \left (\frac {15 e^2 \left (1+\frac {e^5}{15}\right )}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)}+\frac {10 e^7}{x \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}+\frac {2 e^2 x}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)}+\frac {4 e^{10} x \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)}+\frac {8 e^5 x^2 \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)}+\frac {4 x^3 \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)}\right ) \, dx}{e^5}+\frac {2 \int \left (-\frac {10 e^7}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}-\frac {15 e^2 \left (1+\frac {e^5}{15}\right ) x}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}-\frac {2 e^2 x^2}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}-\frac {4 e^{10} x^2 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}-\frac {8 e^5 x^3 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}-\frac {4 x^4 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}\right ) \, dx}{e^5}\\ &=-2 \log (5+x)+2 \log \left (e^5+x\right )-4 \log (\log (x))+16 \int \frac {x^2 \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)} \, dx-16 \int \frac {x^3 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx+\frac {8 \int \frac {x^3 \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)} \, dx}{e^5}-\frac {8 \int \frac {x^4 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx}{e^5}+\frac {4 \int \frac {x}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)} \, dx}{e^3}-\frac {4 \int \frac {x^2}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx}{e^3}+\left (20 e^2\right ) \int \frac {1}{x \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx-\left (20 e^2\right ) \int \frac {1}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx+\left (8 e^5\right ) \int \frac {x \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)} \, dx-\left (8 e^5\right ) \int \frac {x^2 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx+\frac {\left (2 \left (15+e^5\right )\right ) \int \frac {1}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)} \, dx}{e^3}-\frac {\left (2 \left (15+e^5\right )\right ) \int \frac {x}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx}{e^3}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 74.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^2 \left (100+40 x+4 x^2\right )+e^2 \left (100+40 x+4 x^2\right ) \log (x)+\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)}{e^2 \left (-25 x-10 x^2-x^3\right ) \log (x)+\left (10 x^4+2 x^5+e^5 \left (10 x^3+2 x^4\right )\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^2*(100 + 40*x + 4*x^2) + E^2*(100 + 40*x + 4*x^2)*Log[x] + (20*x^3 - 4*E^5*x^3)*Log[x]^3)/(E^2*(-
25*x - 10*x^2 - x^3)*Log[x] + (10*x^4 + 2*x^5 + E^5*(10*x^3 + 2*x^4))*Log[x]^3),x]

[Out]

Integrate[(E^2*(100 + 40*x + 4*x^2) + E^2*(100 + 40*x + 4*x^2)*Log[x] + (20*x^3 - 4*E^5*x^3)*Log[x]^3)/(E^2*(-
25*x - 10*x^2 - x^3)*Log[x] + (10*x^4 + 2*x^5 + E^5*(10*x^3 + 2*x^4))*Log[x]^3), x]

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fricas [B]  time = 0.83, size = 59, normalized size = 2.11 \begin {gather*} 2 \, \log \left (x + e^{5}\right ) - 2 \, \log \left (x + 5\right ) + 2 \, \log \left (\frac {2 \, {\left (x^{3} + x^{2} e^{5}\right )} \log \relax (x)^{2} - {\left (x + 5\right )} e^{2}}{x^{3} + x^{2} e^{5}}\right ) - 4 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*exp(5)+20*x^3)*log(x)^3+(4*x^2+40*x+100)*exp(2)*log(x)+(4*x^2+40*x+100)*exp(2))/(((2*x^4+10
*x^3)*exp(5)+2*x^5+10*x^4)*log(x)^3+(-x^3-10*x^2-25*x)*exp(2)*log(x)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.86, size = 49, normalized size = 1.75 \begin {gather*} 2 \, \log \left (2 \, x^{3} \log \relax (x)^{2} + 2 \, x^{2} e^{5} \log \relax (x)^{2} - x e^{2} - 5 \, e^{2}\right ) - 2 \, \log \left (x + 5\right ) - 4 \, \log \relax (x) - 4 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*exp(5)+20*x^3)*log(x)^3+(4*x^2+40*x+100)*exp(2)*log(x)+(4*x^2+40*x+100)*exp(2))/(((2*x^4+10
*x^3)*exp(5)+2*x^5+10*x^4)*log(x)^3+(-x^3-10*x^2-25*x)*exp(2)*log(x)),x, algorithm="giac")

[Out]

2*log(2*x^3*log(x)^2 + 2*x^2*e^5*log(x)^2 - x*e^2 - 5*e^2) - 2*log(x + 5) - 4*log(x) - 4*log(log(x))

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maple [A]  time = 0.29, size = 44, normalized size = 1.57

method	result	size
risch	\(-2 \ln \left (5+x \right )+2 \ln \left ({\mathrm e}^{5}+x \right )+2 \ln \left (\ln \relax (x )^{2}-\frac {{\mathrm e}^{2} \left (5+x \right )}{2 \left ({\mathrm e}^{5}+x \right ) x^{2}}\right )-4 \ln \left (\ln \relax (x )\right )\)	\(44\)
norman	\(-4 \ln \relax (x )-4 \ln \left (\ln \relax (x )\right )-2 \ln \left (5+x \right )+2 \ln \left (2 x^{2} {\mathrm e}^{5} \ln \relax (x )^{2}+2 x^{3} \ln \relax (x )^{2}-{\mathrm e}^{2} x -5 \,{\mathrm e}^{2}\right )\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3*exp(5)+20*x^3)*ln(x)^3+(4*x^2+40*x+100)*exp(2)*ln(x)+(4*x^2+40*x+100)*exp(2))/(((2*x^4+10*x^3)*ex
p(5)+2*x^5+10*x^4)*ln(x)^3+(-x^3-10*x^2-25*x)*exp(2)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-2*ln(5+x)+2*ln(exp(5)+x)+2*ln(ln(x)^2-1/2*exp(2)*(5+x)/(exp(5)+x)/x^2)-4*ln(ln(x))

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maxima [B]  time = 0.43, size = 62, normalized size = 2.21 \begin {gather*} 2 \, \log \left (x + e^{5}\right ) - 2 \, \log \left (x + 5\right ) + 2 \, \log \left (\frac {2 \, {\left (x^{3} + x^{2} e^{5}\right )} \log \relax (x)^{2} - x e^{2} - 5 \, e^{2}}{2 \, {\left (x^{3} + x^{2} e^{5}\right )}}\right ) - 4 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*exp(5)+20*x^3)*log(x)^3+(4*x^2+40*x+100)*exp(2)*log(x)+(4*x^2+40*x+100)*exp(2))/(((2*x^4+10
*x^3)*exp(5)+2*x^5+10*x^4)*log(x)^3+(-x^3-10*x^2-25*x)*exp(2)*log(x)),x, algorithm="maxima")

[Out]

2*log(x + e^5) - 2*log(x + 5) + 2*log(1/2*(2*(x^3 + x^2*e^5)*log(x)^2 - x*e^2 - 5*e^2)/(x^3 + x^2*e^5)) - 4*lo
g(log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\left (20\,x^3-4\,x^3\,{\mathrm {e}}^5\right )\,{\ln \relax (x)}^3+{\mathrm {e}}^2\,\left (4\,x^2+40\,x+100\right )\,\ln \relax (x)+{\mathrm {e}}^2\,\left (4\,x^2+40\,x+100\right )}{{\ln \relax (x)}^3\,\left ({\mathrm {e}}^5\,\left (2\,x^4+10\,x^3\right )+10\,x^4+2\,x^5\right )-{\mathrm {e}}^2\,\ln \relax (x)\,\left (x^3+10\,x^2+25\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2)*(40*x + 4*x^2 + 100) - log(x)^3*(4*x^3*exp(5) - 20*x^3) + exp(2)*log(x)*(40*x + 4*x^2 + 100))/(log
(x)^3*(exp(5)*(10*x^3 + 2*x^4) + 10*x^4 + 2*x^5) - exp(2)*log(x)*(25*x + 10*x^2 + x^3)),x)

[Out]

int((exp(2)*(40*x + 4*x^2 + 100) - log(x)^3*(4*x^3*exp(5) - 20*x^3) + exp(2)*log(x)*(40*x + 4*x^2 + 100))/(log
(x)^3*(exp(5)*(10*x^3 + 2*x^4) + 10*x^4 + 2*x^5) - exp(2)*log(x)*(25*x + 10*x^2 + x^3)), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3*exp(5)+20*x**3)*ln(x)**3+(4*x**2+40*x+100)*exp(2)*ln(x)+(4*x**2+40*x+100)*exp(2))/(((2*x**
4+10*x**3)*exp(5)+2*x**5+10*x**4)*ln(x)**3+(-x**3-10*x**2-25*x)*exp(2)*ln(x)),x)

[Out]

Exception raised: PolynomialError

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