3.9.82 \(\int \frac {150 x^2-500 x^4+100 e^x x^4+30 \log (x)+(-15-100 x^2+20 e^x x^2) \log ^2(x)+(-5+e^x) \log ^4(x)}{-150 x^3+500 x^4+100 e^x x^4-500 x^5+(-15 x+100 x^2+20 e^x x^2-100 x^3) \log ^2(x)+(5+e^x-5 x) \log ^4(x)} \, dx\)

Optimal. Leaf size=33 \[ \log \left (-1+x+\frac {1}{5} \left (-e^x+\frac {3}{2 x+\frac {\log ^2(x)}{5 x}}\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 9.44, 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 {150 x^2-500 x^4+100 e^x x^4+30 \log (x)+\left (-15-100 x^2+20 e^x x^2\right ) \log ^2(x)+\left (-5+e^x\right ) \log ^4(x)}{-150 x^3+500 x^4+100 e^x x^4-500 x^5+\left (-15 x+100 x^2+20 e^x x^2-100 x^3\right ) \log ^2(x)+\left (5+e^x-5 x\right ) \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(150*x^2 - 500*x^4 + 100*E^x*x^4 + 30*Log[x] + (-15 - 100*x^2 + 20*E^x*x^2)*Log[x]^2 + (-5 + E^x)*Log[x]^4
)/(-150*x^3 + 500*x^4 + 100*E^x*x^4 - 500*x^5 + (-15*x + 100*x^2 + 20*E^x*x^2 - 100*x^3)*Log[x]^2 + (5 + E^x -
5*x)*Log[x]^4),x]

[Out]

x - 150*Defer[Int][x^2/((10*x^2 + Log[x]^2)*(15*x - 50*x^2 - 10*E^x*x^2 + 50*x^3 - 5*Log[x]^2 - E^x*Log[x]^2 +
5*x*Log[x]^2)), x] - 150*Defer[Int][x^3/((10*x^2 + Log[x]^2)*(15*x - 50*x^2 - 10*E^x*x^2 + 50*x^3 - 5*Log[x]^
2 - E^x*Log[x]^2 + 5*x*Log[x]^2)), x] + 1000*Defer[Int][x^4/((10*x^2 + Log[x]^2)*(15*x - 50*x^2 - 10*E^x*x^2 +
50*x^3 - 5*Log[x]^2 - E^x*Log[x]^2 + 5*x*Log[x]^2)), x] - 500*Defer[Int][x^5/((10*x^2 + Log[x]^2)*(15*x - 50*
x^2 - 10*E^x*x^2 + 50*x^3 - 5*Log[x]^2 - E^x*Log[x]^2 + 5*x*Log[x]^2)), x] - 30*Defer[Int][Log[x]/((10*x^2 + L
og[x]^2)*(15*x - 50*x^2 - 10*E^x*x^2 + 50*x^3 - 5*Log[x]^2 - E^x*Log[x]^2 + 5*x*Log[x]^2)), x] + 15*Defer[Int]
[Log[x]^2/((10*x^2 + Log[x]^2)*(15*x - 50*x^2 - 10*E^x*x^2 + 50*x^3 - 5*Log[x]^2 - E^x*Log[x]^2 + 5*x*Log[x]^2
)), x] - 15*Defer[Int][(x*Log[x]^2)/((10*x^2 + Log[x]^2)*(15*x - 50*x^2 - 10*E^x*x^2 + 50*x^3 - 5*Log[x]^2 - E
^x*Log[x]^2 + 5*x*Log[x]^2)), x] + 200*Defer[Int][(x^2*Log[x]^2)/((10*x^2 + Log[x]^2)*(15*x - 50*x^2 - 10*E^x*
x^2 + 50*x^3 - 5*Log[x]^2 - E^x*Log[x]^2 + 5*x*Log[x]^2)), x] - 100*Defer[Int][(x^3*Log[x]^2)/((10*x^2 + Log[x
]^2)*(15*x - 50*x^2 - 10*E^x*x^2 + 50*x^3 - 5*Log[x]^2 - E^x*Log[x]^2 + 5*x*Log[x]^2)), x] + 10*Defer[Int][Log
[x]^4/((10*x^2 + Log[x]^2)*(15*x - 50*x^2 - 10*E^x*x^2 + 50*x^3 - 5*Log[x]^2 - E^x*Log[x]^2 + 5*x*Log[x]^2)),
x] - 5*Defer[Int][(x*Log[x]^4)/((10*x^2 + Log[x]^2)*(15*x - 50*x^2 - 10*E^x*x^2 + 50*x^3 - 5*Log[x]^2 - E^x*Lo
g[x]^2 + 5*x*Log[x]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-150 x^2+500 x^4-100 e^x x^4-30 \log (x)-\left (-15-100 x^2+20 e^x x^2\right ) \log ^2(x)-\left (-5+e^x\right ) \log ^4(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx\\ &=\int \left (1-\frac {5 \left (30 x^2+30 x^3-200 x^4+100 x^5+6 \log (x)-3 \log ^2(x)+3 x \log ^2(x)-40 x^2 \log ^2(x)+20 x^3 \log ^2(x)-2 \log ^4(x)+x \log ^4(x)\right )}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}\right ) \, dx\\ &=x-5 \int \frac {30 x^2+30 x^3-200 x^4+100 x^5+6 \log (x)-3 \log ^2(x)+3 x \log ^2(x)-40 x^2 \log ^2(x)+20 x^3 \log ^2(x)-2 \log ^4(x)+x \log ^4(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx\\ &=x-5 \int \left (\frac {30 x^2}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}+\frac {30 x^3}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}-\frac {200 x^4}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}+\frac {100 x^5}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}+\frac {6 \log (x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}-\frac {3 \log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}+\frac {3 x \log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}-\frac {40 x^2 \log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}+\frac {20 x^3 \log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}-\frac {2 \log ^4(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}+\frac {x \log ^4(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )}\right ) \, dx\\ &=x-5 \int \frac {x \log ^4(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx+10 \int \frac {\log ^4(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx+15 \int \frac {\log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx-15 \int \frac {x \log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx-30 \int \frac {\log (x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx-100 \int \frac {x^3 \log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx-150 \int \frac {x^2}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx-150 \int \frac {x^3}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx+200 \int \frac {x^2 \log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx-500 \int \frac {x^5}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx+1000 \int \frac {x^4}{\left (10 x^2+\log ^2(x)\right ) \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.17, size = 59, normalized size = 1.79 \begin {gather*} -\log \left (10 x^2+\log ^2(x)\right )+\log \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(150*x^2 - 500*x^4 + 100*E^x*x^4 + 30*Log[x] + (-15 - 100*x^2 + 20*E^x*x^2)*Log[x]^2 + (-5 + E^x)*Lo
g[x]^4)/(-150*x^3 + 500*x^4 + 100*E^x*x^4 - 500*x^5 + (-15*x + 100*x^2 + 20*E^x*x^2 - 100*x^3)*Log[x]^2 + (5 +
E^x - 5*x)*Log[x]^4),x]

[Out]

-Log[10*x^2 + Log[x]^2] + Log[15*x - 50*x^2 - 10*E^x*x^2 + 50*x^3 - 5*Log[x]^2 - E^x*Log[x]^2 + 5*x*Log[x]^2]

________________________________________________________________________________________

fricas [B]  time = 0.85, size = 70, normalized size = 2.12 \begin {gather*} -\log \left (10 \, x^{2} + \log \relax (x)^{2}\right ) + \log \left (-5 \, x + e^{x} + 5\right ) + \log \left (\frac {50 \, x^{3} - 10 \, x^{2} e^{x} + {\left (5 \, x - e^{x} - 5\right )} \log \relax (x)^{2} - 50 \, x^{2} + 15 \, x}{5 \, x - e^{x} - 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)-5)*log(x)^4+(20*exp(x)*x^2-100*x^2-15)*log(x)^2+30*log(x)+100*exp(x)*x^4-500*x^4+150*x^2)/(
(exp(x)-5*x+5)*log(x)^4+(20*exp(x)*x^2-100*x^3+100*x^2-15*x)*log(x)^2+100*exp(x)*x^4-500*x^5+500*x^4-150*x^3),
x, algorithm="fricas")

[Out]

-log(10*x^2 + log(x)^2) + log(-5*x + e^x + 5) + log((50*x^3 - 10*x^2*e^x + (5*x - e^x - 5)*log(x)^2 - 50*x^2 +
15*x)/(5*x - e^x - 5))

________________________________________________________________________________________

giac [B]  time = 9.02, size = 56, normalized size = 1.70 \begin {gather*} \log \left (-50 \, x^{3} + 10 \, x^{2} e^{x} - 5 \, x \log \relax (x)^{2} + e^{x} \log \relax (x)^{2} + 50 \, x^{2} + 5 \, \log \relax (x)^{2} - 15 \, x\right ) - \log \left (10 \, x^{2} + \log \relax (x)^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)-5)*log(x)^4+(20*exp(x)*x^2-100*x^2-15)*log(x)^2+30*log(x)+100*exp(x)*x^4-500*x^4+150*x^2)/(
(exp(x)-5*x+5)*log(x)^4+(20*exp(x)*x^2-100*x^3+100*x^2-15*x)*log(x)^2+100*exp(x)*x^4-500*x^5+500*x^4-150*x^3),
x, algorithm="giac")

[Out]

log(-50*x^3 + 10*x^2*e^x - 5*x*log(x)^2 + e^x*log(x)^2 + 50*x^2 + 5*log(x)^2 - 15*x) - log(10*x^2 + log(x)^2)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 58, normalized size = 1.76

method	result	size
risch	\(\ln \left ({\mathrm e}^{x}-5 x +5\right )+\ln \left (\ln \relax (x )^{2}+\frac {5 x \left (10 x^{2}-2 \,{\mathrm e}^{x} x -10 x +3\right )}{5 x -{\mathrm e}^{x}-5}\right )-\ln \left (10 x^{2}+\ln \relax (x )^{2}\right )\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)-5)*ln(x)^4+(20*exp(x)*x^2-100*x^2-15)*ln(x)^2+30*ln(x)+100*exp(x)*x^4-500*x^4+150*x^2)/((exp(x)-5
*x+5)*ln(x)^4+(20*exp(x)*x^2-100*x^3+100*x^2-15*x)*ln(x)^2+100*exp(x)*x^4-500*x^5+500*x^4-150*x^3),x,method=_R
ETURNVERBOSE)

[Out]

ln(exp(x)-5*x+5)+ln(ln(x)^2+5*x*(10*x^2-2*exp(x)*x-10*x+3)/(5*x-exp(x)-5))-ln(10*x^2+ln(x)^2)

________________________________________________________________________________________

maxima [B]  time = 0.48, size = 52, normalized size = 1.58 \begin {gather*} \log \left (-\frac {50 \, x^{3} + 5 \, {\left (x - 1\right )} \log \relax (x)^{2} - 50 \, x^{2} - {\left (10 \, x^{2} + \log \relax (x)^{2}\right )} e^{x} + 15 \, x}{10 \, x^{2} + \log \relax (x)^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)-5)*log(x)^4+(20*exp(x)*x^2-100*x^2-15)*log(x)^2+30*log(x)+100*exp(x)*x^4-500*x^4+150*x^2)/(
(exp(x)-5*x+5)*log(x)^4+(20*exp(x)*x^2-100*x^3+100*x^2-15*x)*log(x)^2+100*exp(x)*x^4-500*x^5+500*x^4-150*x^3),
x, algorithm="maxima")

[Out]

log(-(50*x^3 + 5*(x - 1)*log(x)^2 - 50*x^2 - (10*x^2 + log(x)^2)*e^x + 15*x)/(10*x^2 + log(x)^2))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {30\,\ln \relax (x)+100\,x^4\,{\mathrm {e}}^x-{\ln \relax (x)}^2\,\left (100\,x^2-20\,x^2\,{\mathrm {e}}^x+15\right )+150\,x^2-500\,x^4+{\ln \relax (x)}^4\,\left ({\mathrm {e}}^x-5\right )}{100\,x^4\,{\mathrm {e}}^x-{\ln \relax (x)}^2\,\left (15\,x-20\,x^2\,{\mathrm {e}}^x-100\,x^2+100\,x^3\right )+{\ln \relax (x)}^4\,\left ({\mathrm {e}}^x-5\,x+5\right )-150\,x^3+500\,x^4-500\,x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((30*log(x) + 100*x^4*exp(x) - log(x)^2*(100*x^2 - 20*x^2*exp(x) + 15) + 150*x^2 - 500*x^4 + log(x)^4*(exp(
x) - 5))/(100*x^4*exp(x) - log(x)^2*(15*x - 20*x^2*exp(x) - 100*x^2 + 100*x^3) + log(x)^4*(exp(x) - 5*x + 5) -
150*x^3 + 500*x^4 - 500*x^5),x)

[Out]

int((30*log(x) + 100*x^4*exp(x) - log(x)^2*(100*x^2 - 20*x^2*exp(x) + 15) + 150*x^2 - 500*x^4 + log(x)^4*(exp(
x) - 5))/(100*x^4*exp(x) - log(x)^2*(15*x - 20*x^2*exp(x) - 100*x^2 + 100*x^3) + log(x)^4*(exp(x) - 5*x + 5) -
150*x^3 + 500*x^4 - 500*x^5), x)

________________________________________________________________________________________

sympy [B]  time = 1.07, size = 42, normalized size = 1.27 \begin {gather*} \log {\left (e^{x} + \frac {- 50 x^{3} + 50 x^{2} - 5 x \log {\relax (x )}^{2} - 15 x + 5 \log {\relax (x )}^{2}}{10 x^{2} + \log {\relax (x )}^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)-5)*ln(x)**4+(20*exp(x)*x**2-100*x**2-15)*ln(x)**2+30*ln(x)+100*exp(x)*x**4-500*x**4+150*x**
2)/((exp(x)-5*x+5)*ln(x)**4+(20*exp(x)*x**2-100*x**3+100*x**2-15*x)*ln(x)**2+100*exp(x)*x**4-500*x**5+500*x**4
-150*x**3),x)

[Out]

log(exp(x) + (-50*x**3 + 50*x**2 - 5*x*log(x)**2 - 15*x + 5*log(x)**2)/(10*x**2 + log(x)**2))

________________________________________________________________________________________