3.13.4 \(\int \frac {e^{4/3} (-x+3 x^2-2 x^3+x^4)+e^{4/3} (-2 x+4 x^2-2 x^3) \log (x)+e^{4/3} (-2 x+x^2) \log ^2(x)}{5-10 x+15 x^2-10 x^3+5 x^4+(10-20 x+20 x^2-10 x^3) \log (x)+(5-10 x+5 x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ \frac {e^{4/3} x^2}{5 \left (-1+x+\frac {1}{x-\log (x)}\right )} \]

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Rubi [F]  time = 1.20, 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^{4/3} \left (-x+3 x^2-2 x^3+x^4\right )+e^{4/3} \left (-2 x+4 x^2-2 x^3\right ) \log (x)+e^{4/3} \left (-2 x+x^2\right ) \log ^2(x)}{5-10 x+15 x^2-10 x^3+5 x^4+\left (10-20 x+20 x^2-10 x^3\right ) \log (x)+\left (5-10 x+5 x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(4/3)*(-x + 3*x^2 - 2*x^3 + x^4) + E^(4/3)*(-2*x + 4*x^2 - 2*x^3)*Log[x] + E^(4/3)*(-2*x + x^2)*Log[x]^
2)/(5 - 10*x + 15*x^2 - 10*x^3 + 5*x^4 + (10 - 20*x + 20*x^2 - 10*x^3)*Log[x] + (5 - 10*x + 5*x^2)*Log[x]^2),x
]

[Out]

-1/5*(E^(4/3)*x^2)/(1 - x) - (E^(4/3)*Defer[Int][(1 - x + x^2 + Log[x] - x*Log[x])^(-2), x])/5 - (E^(4/3)*Defe
r[Int][1/((-1 + x)^2*(1 - x + x^2 + Log[x] - x*Log[x])^2), x])/5 - (2*E^(4/3)*Defer[Int][1/((-1 + x)*(1 - x +
x^2 + Log[x] - x*Log[x])^2), x])/5 - (E^(4/3)*Defer[Int][x/(1 - x + x^2 + Log[x] - x*Log[x])^2, x])/5 + (E^(4/
3)*Defer[Int][x^2/(1 - x + x^2 + Log[x] - x*Log[x])^2, x])/5 + (2*E^(4/3)*Defer[Int][1/((-1 + x)^2*(1 - x + x^
2 + Log[x] - x*Log[x])), x])/5 + (2*E^(4/3)*Defer[Int][1/((-1 + x)*(1 - x + x^2 + Log[x] - x*Log[x])), x])/5

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 43, normalized size = 1.72 \begin {gather*} \frac {1}{5} e^{4/3} \left (\frac {1}{-1+x}+x+\frac {x^2}{(-1+x) \left (-1+x-x^2-\log (x)+x \log (x)\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4/3)*(-x + 3*x^2 - 2*x^3 + x^4) + E^(4/3)*(-2*x + 4*x^2 - 2*x^3)*Log[x] + E^(4/3)*(-2*x + x^2)*L
og[x]^2)/(5 - 10*x + 15*x^2 - 10*x^3 + 5*x^4 + (10 - 20*x + 20*x^2 - 10*x^3)*Log[x] + (5 - 10*x + 5*x^2)*Log[x
]^2),x]

[Out]

(E^(4/3)*((-1 + x)^(-1) + x + x^2/((-1 + x)*(-1 + x - x^2 - Log[x] + x*Log[x]))))/5

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fricas [B]  time = 0.62, size = 48, normalized size = 1.92 \begin {gather*} -\frac {{\left (x^{2} - x + 1\right )} e^{\frac {4}{3}} \log \relax (x) - {\left (x^{3} - x^{2} + x - 1\right )} e^{\frac {4}{3}}}{5 \, {\left (x^{2} - {\left (x - 1\right )} \log \relax (x) - x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-2*x)*exp(2/3)^2*log(x)^2+(-2*x^3+4*x^2-2*x)*exp(2/3)^2*log(x)+(x^4-2*x^3+3*x^2-x)*exp(2/3)^2)/
((5*x^2-10*x+5)*log(x)^2+(-10*x^3+20*x^2-20*x+10)*log(x)+5*x^4-10*x^3+15*x^2-10*x+5),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.33, size = 62, normalized size = 2.48 \begin {gather*} \frac {x^{3} e^{\frac {4}{3}} - x^{2} e^{\frac {4}{3}} \log \relax (x) - x^{2} e^{\frac {4}{3}} + x e^{\frac {4}{3}} \log \relax (x) + x e^{\frac {4}{3}} - e^{\frac {4}{3}} \log \relax (x) - e^{\frac {4}{3}}}{5 \, {\left (x^{2} - x \log \relax (x) - x + \log \relax (x) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-2*x)*exp(2/3)^2*log(x)^2+(-2*x^3+4*x^2-2*x)*exp(2/3)^2*log(x)+(x^4-2*x^3+3*x^2-x)*exp(2/3)^2)/
((5*x^2-10*x+5)*log(x)^2+(-10*x^3+20*x^2-20*x+10)*log(x)+5*x^4-10*x^3+15*x^2-10*x+5),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.10, size = 40, normalized size = 1.60

method	result	size
norman	\(\frac {\frac {{\mathrm e}^{\frac {4}{3}} x^{3}}{5}-\frac {{\mathrm e}^{\frac {4}{3}} x^{2} \ln \relax (x )}{5}}{x^{2}-x \ln \relax (x )-x +\ln \relax (x )+1}\)	\(40\)
risch	\(\frac {{\mathrm e}^{\frac {4}{3}} \left (x^{2}-x +1\right )}{5 x -5}-\frac {{\mathrm e}^{\frac {4}{3}} x^{2}}{5 \left (x -1\right ) \left (x^{2}-x \ln \relax (x )-x +\ln \relax (x )+1\right )}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-2*x)*exp(2/3)^2*ln(x)^2+(-2*x^3+4*x^2-2*x)*exp(2/3)^2*ln(x)+(x^4-2*x^3+3*x^2-x)*exp(2/3)^2)/((5*x^2-
10*x+5)*ln(x)^2+(-10*x^3+20*x^2-20*x+10)*ln(x)+5*x^4-10*x^3+15*x^2-10*x+5),x,method=_RETURNVERBOSE)

[Out]

(1/5*exp(2/3)^2*x^3-1/5*exp(2/3)^2*x^2*ln(x))/(x^2-x*ln(x)-x+ln(x)+1)

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maxima [B]  time = 0.53, size = 59, normalized size = 2.36 \begin {gather*} \frac {x^{3} e^{\frac {4}{3}} - x^{2} e^{\frac {4}{3}} + x e^{\frac {4}{3}} - {\left (x^{2} e^{\frac {4}{3}} - x e^{\frac {4}{3}} + e^{\frac {4}{3}}\right )} \log \relax (x) - e^{\frac {4}{3}}}{5 \, {\left (x^{2} - {\left (x - 1\right )} \log \relax (x) - x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-2*x)*exp(2/3)^2*log(x)^2+(-2*x^3+4*x^2-2*x)*exp(2/3)^2*log(x)+(x^4-2*x^3+3*x^2-x)*exp(2/3)^2)/
((5*x^2-10*x+5)*log(x)^2+(-10*x^3+20*x^2-20*x+10)*log(x)+5*x^4-10*x^3+15*x^2-10*x+5),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.11, size = 51, normalized size = 2.04 \begin {gather*} -\frac {{\mathrm {e}}^{4/3}\,\left (\ln \relax (x)-x+x^2\,\ln \relax (x)-x\,\ln \relax (x)+x^2-x^3+1\right )}{5\,\left (\ln \relax (x)-x-x\,\ln \relax (x)+x^2+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4/3)*(x - 3*x^2 + 2*x^3 - x^4) + exp(4/3)*log(x)*(2*x - 4*x^2 + 2*x^3) + exp(4/3)*log(x)^2*(2*x - x^
2))/(log(x)^2*(5*x^2 - 10*x + 5) - 10*x + 15*x^2 - 10*x^3 + 5*x^4 - log(x)*(20*x - 20*x^2 + 10*x^3 - 10) + 5),
x)

[Out]

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

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sympy [B]  time = 0.30, size = 54, normalized size = 2.16 \begin {gather*} \frac {x^{2} e^{\frac {4}{3}}}{- 5 x^{3} + 10 x^{2} - 10 x + \left (5 x^{2} - 10 x + 5\right ) \log {\relax (x )} + 5} + \frac {x e^{\frac {4}{3}}}{5} + \frac {e^{\frac {4}{3}}}{5 x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-2*x)*exp(2/3)**2*ln(x)**2+(-2*x**3+4*x**2-2*x)*exp(2/3)**2*ln(x)+(x**4-2*x**3+3*x**2-x)*exp(2
/3)**2)/((5*x**2-10*x+5)*ln(x)**2+(-10*x**3+20*x**2-20*x+10)*ln(x)+5*x**4-10*x**3+15*x**2-10*x+5),x)

[Out]

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

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