3.14.88 \(\int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+(8 x^2-16 x^3) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+(2 x^3-4 x^4) \log (x)+x^3 \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ \log \left (25+e^x-\frac {1}{5} x (-x+x (2 x-\log (x)))^2\right ) \]

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Rubi [F]  time = 6.71, 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 {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5*E^x + 5*x^2 - 20*x^3 + 20*x^4 + (8*x^2 - 16*x^3)*Log[x] + 3*x^2*Log[x]^2)/(-125 - 5*E^x + x^3 - 4*x^4
+ 4*x^5 + (2*x^3 - 4*x^4)*Log[x] + x^3*Log[x]^2),x]

[Out]

x - 125*Defer[Int][(125 + 5*E^x - x^3 + 4*x^4 - 4*x^5 - 2*x^3*Log[x] + 4*x^4*Log[x] - x^3*Log[x]^2)^(-1), x] +
5*Defer[Int][x^2/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + 2*x^3*Log[x] - 4*x^4*Log[x] + x^3*Log[x]^2), x] - 21*D
efer[Int][x^3/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + 2*x^3*Log[x] - 4*x^4*Log[x] + x^3*Log[x]^2), x] + 24*Defer
[Int][x^4/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + 2*x^3*Log[x] - 4*x^4*Log[x] + x^3*Log[x]^2), x] - 4*Defer[Int]
[x^5/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + 2*x^3*Log[x] - 4*x^4*Log[x] + x^3*Log[x]^2), x] + 8*Defer[Int][(x^2
*Log[x])/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + 2*x^3*Log[x] - 4*x^4*Log[x] + x^3*Log[x]^2), x] - 18*Defer[Int]
[(x^3*Log[x])/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + 2*x^3*Log[x] - 4*x^4*Log[x] + x^3*Log[x]^2), x] + 4*Defer[
Int][(x^4*Log[x])/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + 2*x^3*Log[x] - 4*x^4*Log[x] + x^3*Log[x]^2), x] + 3*De
fer[Int][(x^2*Log[x]^2)/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + 2*x^3*Log[x] - 4*x^4*Log[x] + x^3*Log[x]^2), x]
- Defer[Int][(x^3*Log[x]^2)/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + 2*x^3*Log[x] - 4*x^4*Log[x] + x^3*Log[x]^2),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {-125-5 x^2+21 x^3-24 x^4+4 x^5-8 x^2 \log (x)+18 x^3 \log (x)-4 x^4 \log (x)-3 x^2 \log ^2(x)+x^3 \log ^2(x)}{125+5 e^x-x^3+4 x^4-4 x^5-2 x^3 \log (x)+4 x^4 \log (x)-x^3 \log ^2(x)}\right ) \, dx\\ &=x+\int \frac {-125-5 x^2+21 x^3-24 x^4+4 x^5-8 x^2 \log (x)+18 x^3 \log (x)-4 x^4 \log (x)-3 x^2 \log ^2(x)+x^3 \log ^2(x)}{125+5 e^x-x^3+4 x^4-4 x^5-2 x^3 \log (x)+4 x^4 \log (x)-x^3 \log ^2(x)} \, dx\\ &=x+\int \left (-\frac {125}{125+5 e^x-x^3+4 x^4-4 x^5-2 x^3 \log (x)+4 x^4 \log (x)-x^3 \log ^2(x)}+\frac {5 x^2}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}-\frac {21 x^3}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}+\frac {24 x^4}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}-\frac {4 x^5}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}+\frac {8 x^2 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}-\frac {18 x^3 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}+\frac {4 x^4 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}+\frac {3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}-\frac {x^3 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}\right ) \, dx\\ &=x+3 \int \frac {x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx-4 \int \frac {x^5}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx+4 \int \frac {x^4 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx+5 \int \frac {x^2}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx+8 \int \frac {x^2 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx-18 \int \frac {x^3 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx-21 \int \frac {x^3}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx+24 \int \frac {x^4}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx-125 \int \frac {1}{125+5 e^x-x^3+4 x^4-4 x^5-2 x^3 \log (x)+4 x^4 \log (x)-x^3 \log ^2(x)} \, dx-\int \frac {x^3 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-5*E^x + 5*x^2 - 20*x^3 + 20*x^4 + (8*x^2 - 16*x^3)*Log[x] + 3*x^2*Log[x]^2)/(-125 - 5*E^x + x^3 -
4*x^4 + 4*x^5 + (2*x^3 - 4*x^4)*Log[x] + x^3*Log[x]^2),x]

[Out]

Integrate[(-5*E^x + 5*x^2 - 20*x^3 + 20*x^4 + (8*x^2 - 16*x^3)*Log[x] + 3*x^2*Log[x]^2)/(-125 - 5*E^x + x^3 -
4*x^4 + 4*x^5 + (2*x^3 - 4*x^4)*Log[x] + x^3*Log[x]^2), x]

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fricas [B]  time = 0.92, size = 52, normalized size = 1.93 \begin {gather*} 3 \, \log \relax (x) + \log \left (\frac {4 \, x^{5} + x^{3} \log \relax (x)^{2} - 4 \, x^{4} + x^{3} - 2 \, {\left (2 \, x^{4} - x^{3}\right )} \log \relax (x) - 5 \, e^{x} - 125}{x^{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(x)^2+(-16*x^3+8*x^2)*log(x)-5*exp(x)+20*x^4-20*x^3+5*x^2)/(x^3*log(x)^2+(-4*x^4+2*x^3)*lo
g(x)-5*exp(x)+4*x^5-4*x^4+x^3-125),x, algorithm="fricas")

[Out]

3*log(x) + log((4*x^5 + x^3*log(x)^2 - 4*x^4 + x^3 - 2*(2*x^4 - x^3)*log(x) - 5*e^x - 125)/x^3)

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giac [A]  time = 0.67, size = 45, normalized size = 1.67 \begin {gather*} \log \left (-4 \, x^{5} + 4 \, x^{4} \log \relax (x) - x^{3} \log \relax (x)^{2} + 4 \, x^{4} - 2 \, x^{3} \log \relax (x) - x^{3} + 5 \, e^{x} + 125\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(x)^2+(-16*x^3+8*x^2)*log(x)-5*exp(x)+20*x^4-20*x^3+5*x^2)/(x^3*log(x)^2+(-4*x^4+2*x^3)*lo
g(x)-5*exp(x)+4*x^5-4*x^4+x^3-125),x, algorithm="giac")

[Out]

log(-4*x^5 + 4*x^4*log(x) - x^3*log(x)^2 + 4*x^4 - 2*x^3*log(x) - x^3 + 5*e^x + 125)

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maple [A]  time = 0.03, size = 43, normalized size = 1.59

method	result	size
risch	\(3 \ln \relax (x )+\ln \left (\ln \relax (x )^{2}+\left (-4 x +2\right ) \ln \relax (x )+\frac {4 x^{5}-4 x^{4}+x^{3}-5 \,{\mathrm e}^{x}-125}{x^{3}}\right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2*ln(x)^2+(-16*x^3+8*x^2)*ln(x)-5*exp(x)+20*x^4-20*x^3+5*x^2)/(x^3*ln(x)^2+(-4*x^4+2*x^3)*ln(x)-5*exp
(x)+4*x^5-4*x^4+x^3-125),x,method=_RETURNVERBOSE)

[Out]

3*ln(x)+ln(ln(x)^2+(-4*x+2)*ln(x)+(4*x^5-4*x^4+x^3-5*exp(x)-125)/x^3)

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maxima [A]  time = 0.50, size = 44, normalized size = 1.63 \begin {gather*} \log \left (-\frac {4}{5} \, x^{5} - \frac {1}{5} \, x^{3} \log \relax (x)^{2} + \frac {4}{5} \, x^{4} - \frac {1}{5} \, x^{3} + \frac {2}{5} \, {\left (2 \, x^{4} - x^{3}\right )} \log \relax (x) + e^{x} + 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(x)^2+(-16*x^3+8*x^2)*log(x)-5*exp(x)+20*x^4-20*x^3+5*x^2)/(x^3*log(x)^2+(-4*x^4+2*x^3)*lo
g(x)-5*exp(x)+4*x^5-4*x^4+x^3-125),x, algorithm="maxima")

[Out]

log(-4/5*x^5 - 1/5*x^3*log(x)^2 + 4/5*x^4 - 1/5*x^3 + 2/5*(2*x^4 - x^3)*log(x) + e^x + 25)

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mupad [B]  time = 1.31, size = 42, normalized size = 1.56 \begin {gather*} \ln \left (\ln \relax (x)\,\left (2\,x^3-4\,x^4\right )-5\,{\mathrm {e}}^x+x^3\,{\ln \relax (x)}^2+x^3-4\,x^4+4\,x^5-125\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(8*x^2 - 16*x^3) - 5*exp(x) + 3*x^2*log(x)^2 + 5*x^2 - 20*x^3 + 20*x^4)/(log(x)*(2*x^3 - 4*x^4) -
5*exp(x) + x^3*log(x)^2 + x^3 - 4*x^4 + 4*x^5 - 125),x)

[Out]

log(log(x)*(2*x^3 - 4*x^4) - 5*exp(x) + x^3*log(x)^2 + x^3 - 4*x^4 + 4*x^5 - 125)

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sympy [B]  time = 0.43, size = 54, normalized size = 2.00 \begin {gather*} \log {\left (- \frac {4 x^{5}}{5} + \frac {4 x^{4} \log {\relax (x )}}{5} + \frac {4 x^{4}}{5} - \frac {x^{3} \log {\relax (x )}^{2}}{5} - \frac {2 x^{3} \log {\relax (x )}}{5} - \frac {x^{3}}{5} + e^{x} + 25 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2*ln(x)**2+(-16*x**3+8*x**2)*ln(x)-5*exp(x)+20*x**4-20*x**3+5*x**2)/(x**3*ln(x)**2+(-4*x**4+2*
x**3)*ln(x)-5*exp(x)+4*x**5-4*x**4+x**3-125),x)

[Out]

log(-4*x**5/5 + 4*x**4*log(x)/5 + 4*x**4/5 - x**3*log(x)**2/5 - 2*x**3*log(x)/5 - x**3/5 + exp(x) + 25)

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