3.84.2 \(\int \frac {e^7 (-3 x^2+9 x^3)+(x^3-6 x^4+e^3 (4 x^3-12 x^4)) \log (2)+(e^4 (6 x^2-18 x^3)+(-4 x^3+12 x^4) \log (2)) \log (-x+3 x^2)+e (-3 x^2+9 x^3) \log ^2(-x+3 x^2)}{e^6 (-1+3 x)+e^3 (2-6 x) \log (-x+3 x^2)+(-1+3 x) \log ^2(-x+3 x^2)} \, dx\)

Optimal. Leaf size=28 \[ x^3 \left (e+\frac {x \log (2)}{-e^3+\log \left (-x+3 x^2\right )}\right ) \]

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Rubi [F]  time = 1.55, 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^7 \left (-3 x^2+9 x^3\right )+\left (x^3-6 x^4+e^3 \left (4 x^3-12 x^4\right )\right ) \log (2)+\left (e^4 \left (6 x^2-18 x^3\right )+\left (-4 x^3+12 x^4\right ) \log (2)\right ) \log \left (-x+3 x^2\right )+e \left (-3 x^2+9 x^3\right ) \log ^2\left (-x+3 x^2\right )}{e^6 (-1+3 x)+e^3 (2-6 x) \log \left (-x+3 x^2\right )+(-1+3 x) \log ^2\left (-x+3 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^7*(-3*x^2 + 9*x^3) + (x^3 - 6*x^4 + E^3*(4*x^3 - 12*x^4))*Log[2] + (E^4*(6*x^2 - 18*x^3) + (-4*x^3 + 12
*x^4)*Log[2])*Log[-x + 3*x^2] + E*(-3*x^2 + 9*x^3)*Log[-x + 3*x^2]^2)/(E^6*(-1 + 3*x) + E^3*(2 - 6*x)*Log[-x +
3*x^2] + (-1 + 3*x)*Log[-x + 3*x^2]^2),x]

[Out]

E*x^3 - (Log[8]*Defer[Int][(E^3 - Log[x*(-1 + 3*x)])^(-2), x])/81 - (Log[8]*Defer[Int][x/(E^3 - Log[x*(-1 + 3*
x)])^2, x])/27 - (Log[8]*Defer[Int][x^2/(E^3 - Log[x*(-1 + 3*x)])^2, x])/9 - (Log[64]*Defer[Int][x^3/(E^3 - Lo
g[x*(-1 + 3*x)])^2, x])/3 - (Log[8]*Defer[Int][1/((-1 + 3*x)*(E^3 - Log[x*(-1 + 3*x)])^2), x])/81 - 2*Log[4]*D
efer[Int][x^3/(E^3 - Log[x*(-1 + 3*x)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (-e^7 (-3+9 x)-(1-6 x) x \log (2)-e^3 x (-12 x \log (2)+\log (16))-2 (-1+3 x) \left (-3 e^4+x \log (4)\right ) \log (x (-1+3 x))-3 e (-1+3 x) \log ^2(x (-1+3 x))\right )}{(1-3 x) \left (e^3-\log (x (-1+3 x))\right )^2} \, dx\\ &=\int \left (3 e x^2-\frac {x^3 (-\log (2)+x \log (64))}{(-1+3 x) \left (e^3-\log (x (-1+3 x))\right )^2}-\frac {2 x^3 \log (4)}{e^3-\log (x (-1+3 x))}\right ) \, dx\\ &=e x^3-(2 \log (4)) \int \frac {x^3}{e^3-\log (x (-1+3 x))} \, dx-\int \frac {x^3 (-\log (2)+x \log (64))}{(-1+3 x) \left (e^3-\log (x (-1+3 x))\right )^2} \, dx\\ &=e x^3-(2 \log (4)) \int \frac {x^3}{e^3-\log (x (-1+3 x))} \, dx-\int \left (\frac {\log (8)}{81 \left (e^3-\log (x (-1+3 x))\right )^2}+\frac {x \log (8)}{27 \left (e^3-\log (x (-1+3 x))\right )^2}+\frac {x^2 \log (8)}{9 \left (e^3-\log (x (-1+3 x))\right )^2}+\frac {\log (8)}{81 (-1+3 x) \left (e^3-\log (x (-1+3 x))\right )^2}+\frac {x^3 \log (64)}{3 \left (e^3-\log (x (-1+3 x))\right )^2}\right ) \, dx\\ &=e x^3-(2 \log (4)) \int \frac {x^3}{e^3-\log (x (-1+3 x))} \, dx-\frac {1}{81} \log (8) \int \frac {1}{\left (e^3-\log (x (-1+3 x))\right )^2} \, dx-\frac {1}{81} \log (8) \int \frac {1}{(-1+3 x) \left (e^3-\log (x (-1+3 x))\right )^2} \, dx-\frac {1}{27} \log (8) \int \frac {x}{\left (e^3-\log (x (-1+3 x))\right )^2} \, dx-\frac {1}{9} \log (8) \int \frac {x^2}{\left (e^3-\log (x (-1+3 x))\right )^2} \, dx-\frac {1}{3} \log (64) \int \frac {x^3}{\left (e^3-\log (x (-1+3 x))\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.11, size = 62, normalized size = 2.21 \begin {gather*} \frac {x^3 \left (e^4 (-1+6 x)+x (\log (2)-x \log (64))+(e-6 e x) \log (x (-1+3 x))\right )}{(-1+6 x) \left (e^3-\log (x (-1+3 x))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^7*(-3*x^2 + 9*x^3) + (x^3 - 6*x^4 + E^3*(4*x^3 - 12*x^4))*Log[2] + (E^4*(6*x^2 - 18*x^3) + (-4*x^
3 + 12*x^4)*Log[2])*Log[-x + 3*x^2] + E*(-3*x^2 + 9*x^3)*Log[-x + 3*x^2]^2)/(E^6*(-1 + 3*x) + E^3*(2 - 6*x)*Lo
g[-x + 3*x^2] + (-1 + 3*x)*Log[-x + 3*x^2]^2),x]

[Out]

(x^3*(E^4*(-1 + 6*x) + x*(Log[2] - x*Log[64]) + (E - 6*E*x)*Log[x*(-1 + 3*x)]))/((-1 + 6*x)*(E^3 - Log[x*(-1 +
3*x)]))

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fricas [A]  time = 0.55, size = 49, normalized size = 1.75 \begin {gather*} -\frac {x^{4} \log \relax (2) + x^{3} e \log \left (3 \, x^{2} - x\right ) - x^{3} e^{4}}{e^{3} - \log \left (3 \, x^{2} - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^3-3*x^2)*exp(1)*log(3*x^2-x)^2+((12*x^4-4*x^3)*log(2)+(-18*x^3+6*x^2)*exp(1)*exp(3))*log(3*x^2
-x)+((-12*x^4+4*x^3)*exp(3)-6*x^4+x^3)*log(2)+(9*x^3-3*x^2)*exp(1)*exp(3)^2)/((3*x-1)*log(3*x^2-x)^2+(-6*x+2)*
exp(3)*log(3*x^2-x)+(3*x-1)*exp(3)^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.27, size = 49, normalized size = 1.75 \begin {gather*} -\frac {x^{4} \log \relax (2) + x^{3} e \log \left (3 \, x^{2} - x\right ) - x^{3} e^{4}}{e^{3} - \log \left (3 \, x^{2} - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^3-3*x^2)*exp(1)*log(3*x^2-x)^2+((12*x^4-4*x^3)*log(2)+(-18*x^3+6*x^2)*exp(1)*exp(3))*log(3*x^2
-x)+((-12*x^4+4*x^3)*exp(3)-6*x^4+x^3)*log(2)+(9*x^3-3*x^2)*exp(1)*exp(3)^2)/((3*x-1)*log(3*x^2-x)^2+(-6*x+2)*
exp(3)*log(3*x^2-x)+(3*x-1)*exp(3)^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.35, size = 32, normalized size = 1.14

method	result	size
default	\(x^{3} {\mathrm e}-\frac {x^{4} \ln \relax (2)}{{\mathrm e}^{3}-\ln \left (3 x^{2}-x \right )}\)	\(32\)
risch	\(x^{3} {\mathrm e}-\frac {x^{4} \ln \relax (2)}{{\mathrm e}^{3}-\ln \left (3 x^{2}-x \right )}\)	\(32\)
norman	\(\frac {{\mathrm e} \,{\mathrm e}^{3} x^{3}-x^{4} \ln \relax (2)-x^{3} {\mathrm e} \ln \left (3 x^{2}-x \right )}{{\mathrm e}^{3}-\ln \left (3 x^{2}-x \right )}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9*x^3-3*x^2)*exp(1)*ln(3*x^2-x)^2+((12*x^4-4*x^3)*ln(2)+(-18*x^3+6*x^2)*exp(1)*exp(3))*ln(3*x^2-x)+((-12
*x^4+4*x^3)*exp(3)-6*x^4+x^3)*ln(2)+(9*x^3-3*x^2)*exp(1)*exp(3)^2)/((3*x-1)*ln(3*x^2-x)^2+(-6*x+2)*exp(3)*ln(3
*x^2-x)+(3*x-1)*exp(3)^2),x,method=_RETURNVERBOSE)

[Out]

x^3*exp(1)-x^4*ln(2)/(exp(3)-ln(3*x^2-x))

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maxima [A]  time = 0.49, size = 53, normalized size = 1.89 \begin {gather*} -\frac {x^{4} \log \relax (2) + x^{3} e \log \left (3 \, x - 1\right ) + x^{3} e \log \relax (x) - x^{3} e^{4}}{e^{3} - \log \left (3 \, x - 1\right ) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^3-3*x^2)*exp(1)*log(3*x^2-x)^2+((12*x^4-4*x^3)*log(2)+(-18*x^3+6*x^2)*exp(1)*exp(3))*log(3*x^2
-x)+((-12*x^4+4*x^3)*exp(3)-6*x^4+x^3)*log(2)+(9*x^3-3*x^2)*exp(1)*exp(3)^2)/((3*x-1)*log(3*x^2-x)^2+(-6*x+2)*
exp(3)*log(3*x^2-x)+(3*x-1)*exp(3)^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.01, size = 129, normalized size = 4.61 \begin {gather*} \frac {\frac {x^3\,\left (x\,\ln \relax (2)-6\,x^2\,\ln \relax (2)-12\,x^2\,{\mathrm {e}}^3\,\ln \relax (2)+4\,x\,{\mathrm {e}}^3\,\ln \relax (2)\right )}{6\,x-1}+\frac {4\,x^4\,\ln \relax (2)\,\ln \left (3\,x^2-x\right )\,\left (3\,x-1\right )}{6\,x-1}}{{\mathrm {e}}^3-\ln \left (3\,x^2-x\right )}-\frac {x\,\ln \relax (2)}{108}-\frac {\ln \relax (2)}{6\,\left (648\,x-108\right )}-\frac {x^2\,\ln \relax (2)}{18}+2\,x^4\,\ln \relax (2)+x^3\,\left (\mathrm {e}-\frac {\ln \relax (2)}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(7)*(3*x^2 - 9*x^3) - log(3*x^2 - x)*(exp(4)*(6*x^2 - 18*x^3) - log(2)*(4*x^3 - 12*x^4)) - log(2)*(ex
p(3)*(4*x^3 - 12*x^4) + x^3 - 6*x^4) + exp(1)*log(3*x^2 - x)^2*(3*x^2 - 9*x^3))/(log(3*x^2 - x)^2*(3*x - 1) +
exp(6)*(3*x - 1) - exp(3)*log(3*x^2 - x)*(6*x - 2)),x)

[Out]

((x^3*(x*log(2) - 6*x^2*log(2) - 12*x^2*exp(3)*log(2) + 4*x*exp(3)*log(2)))/(6*x - 1) + (4*x^4*log(2)*log(3*x^
2 - x)*(3*x - 1))/(6*x - 1))/(exp(3) - log(3*x^2 - x)) - (x*log(2))/108 - log(2)/(6*(648*x - 108)) - (x^2*log(
2))/18 + 2*x^4*log(2) + x^3*(exp(1) - log(2)/3)

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sympy [A]  time = 0.21, size = 24, normalized size = 0.86 \begin {gather*} \frac {x^{4} \log {\relax (2 )}}{\log {\left (3 x^{2} - x \right )} - e^{3}} + e x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x**3-3*x**2)*exp(1)*ln(3*x**2-x)**2+((12*x**4-4*x**3)*ln(2)+(-18*x**3+6*x**2)*exp(1)*exp(3))*ln(
3*x**2-x)+((-12*x**4+4*x**3)*exp(3)-6*x**4+x**3)*ln(2)+(9*x**3-3*x**2)*exp(1)*exp(3)**2)/((3*x-1)*ln(3*x**2-x)
**2+(-6*x+2)*exp(3)*ln(3*x**2-x)+(3*x-1)*exp(3)**2),x)

[Out]

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

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