3.46.38 \(\int \frac {-x^2+768 x^3+128 x^4+5 x^5+(512 x^3+96 x^4+4 x^5+e^x (2 x-1024 x^2-192 x^3-8 x^4)) \log (x)+(e^{2 x} (-1+512 x+96 x^2+4 x^3)+e^x (-768 x^2+128 x^3+27 x^4+x^5)) \log ^2(x)}{x^2-2 e^x x \log (x)+e^{2 x} \log ^2(x)} \, dx\)

Optimal. Leaf size=32 \[ 3-x+x^2 (16+x)^2 \left (1+\frac {x}{-e^x+\frac {x}{\log (x)}}\right ) \]

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Rubi [F]  time = 4.70, 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 {-x^2+768 x^3+128 x^4+5 x^5+\left (512 x^3+96 x^4+4 x^5+e^x \left (2 x-1024 x^2-192 x^3-8 x^4\right )\right ) \log (x)+\left (e^{2 x} \left (-1+512 x+96 x^2+4 x^3\right )+e^x \left (-768 x^2+128 x^3+27 x^4+x^5\right )\right ) \log ^2(x)}{x^2-2 e^x x \log (x)+e^{2 x} \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^2 + 768*x^3 + 128*x^4 + 5*x^5 + (512*x^3 + 96*x^4 + 4*x^5 + E^x*(2*x - 1024*x^2 - 192*x^3 - 8*x^4))*Lo
g[x] + (E^(2*x)*(-1 + 512*x + 96*x^2 + 4*x^3) + E^x*(-768*x^2 + 128*x^3 + 27*x^4 + x^5))*Log[x]^2)/(x^2 - 2*E^
x*x*Log[x] + E^(2*x)*Log[x]^2),x]

[Out]

-x + 256*x^2 + 32*x^3 + x^4 + 256*Defer[Int][x^3/(x - E^x*Log[x])^2, x] + 32*Defer[Int][x^4/(x - E^x*Log[x])^2
, x] + Defer[Int][x^5/(x - E^x*Log[x])^2, x] - 256*Defer[Int][(x^3*Log[x])/(x - E^x*Log[x])^2, x] + 224*Defer[
Int][(x^4*Log[x])/(x - E^x*Log[x])^2, x] + 31*Defer[Int][(x^5*Log[x])/(x - E^x*Log[x])^2, x] + Defer[Int][(x^6
*Log[x])/(x - E^x*Log[x])^2, x] + 768*Defer[Int][(x^2*Log[x])/(x - E^x*Log[x]), x] - 128*Defer[Int][(x^3*Log[x
])/(x - E^x*Log[x]), x] - 27*Defer[Int][(x^4*Log[x])/(x - E^x*Log[x]), x] - Defer[Int][(x^5*Log[x])/(x - E^x*L
og[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^2+768 x^3+128 x^4+5 x^5+\left (512 x^3+96 x^4+4 x^5+e^x \left (2 x-1024 x^2-192 x^3-8 x^4\right )\right ) \log (x)+\left (e^{2 x} \left (-1+512 x+96 x^2+4 x^3\right )+e^x \left (-768 x^2+128 x^3+27 x^4+x^5\right )\right ) \log ^2(x)}{\left (x-e^x \log (x)\right )^2} \, dx\\ &=\int \left (-1+512 x+96 x^2+4 x^3-\frac {x^2 \left (-768+128 x+27 x^2+x^3\right ) \log (x)}{x-e^x \log (x)}+\frac {x^3 (16+x)^2 (1-\log (x)+x \log (x))}{\left (x-e^x \log (x)\right )^2}\right ) \, dx\\ &=-x+256 x^2+32 x^3+x^4-\int \frac {x^2 \left (-768+128 x+27 x^2+x^3\right ) \log (x)}{x-e^x \log (x)} \, dx+\int \frac {x^3 (16+x)^2 (1-\log (x)+x \log (x))}{\left (x-e^x \log (x)\right )^2} \, dx\\ &=-x+256 x^2+32 x^3+x^4-\int \left (-\frac {768 x^2 \log (x)}{x-e^x \log (x)}+\frac {128 x^3 \log (x)}{x-e^x \log (x)}+\frac {27 x^4 \log (x)}{x-e^x \log (x)}+\frac {x^5 \log (x)}{x-e^x \log (x)}\right ) \, dx+\int \left (\frac {256 x^3 (1-\log (x)+x \log (x))}{\left (x-e^x \log (x)\right )^2}+\frac {32 x^4 (1-\log (x)+x \log (x))}{\left (x-e^x \log (x)\right )^2}+\frac {x^5 (1-\log (x)+x \log (x))}{\left (x-e^x \log (x)\right )^2}\right ) \, dx\\ &=-x+256 x^2+32 x^3+x^4-27 \int \frac {x^4 \log (x)}{x-e^x \log (x)} \, dx+32 \int \frac {x^4 (1-\log (x)+x \log (x))}{\left (x-e^x \log (x)\right )^2} \, dx-128 \int \frac {x^3 \log (x)}{x-e^x \log (x)} \, dx+256 \int \frac {x^3 (1-\log (x)+x \log (x))}{\left (x-e^x \log (x)\right )^2} \, dx+768 \int \frac {x^2 \log (x)}{x-e^x \log (x)} \, dx-\int \frac {x^5 \log (x)}{x-e^x \log (x)} \, dx+\int \frac {x^5 (1-\log (x)+x \log (x))}{\left (x-e^x \log (x)\right )^2} \, dx\\ &=-x+256 x^2+32 x^3+x^4-27 \int \frac {x^4 \log (x)}{x-e^x \log (x)} \, dx+32 \int \left (\frac {x^4}{\left (x-e^x \log (x)\right )^2}-\frac {x^4 \log (x)}{\left (x-e^x \log (x)\right )^2}+\frac {x^5 \log (x)}{\left (x-e^x \log (x)\right )^2}\right ) \, dx-128 \int \frac {x^3 \log (x)}{x-e^x \log (x)} \, dx+256 \int \left (\frac {x^3}{\left (x-e^x \log (x)\right )^2}-\frac {x^3 \log (x)}{\left (x-e^x \log (x)\right )^2}+\frac {x^4 \log (x)}{\left (x-e^x \log (x)\right )^2}\right ) \, dx+768 \int \frac {x^2 \log (x)}{x-e^x \log (x)} \, dx-\int \frac {x^5 \log (x)}{x-e^x \log (x)} \, dx+\int \left (\frac {x^5}{\left (x-e^x \log (x)\right )^2}-\frac {x^5 \log (x)}{\left (x-e^x \log (x)\right )^2}+\frac {x^6 \log (x)}{\left (x-e^x \log (x)\right )^2}\right ) \, dx\\ &=-x+256 x^2+32 x^3+x^4-27 \int \frac {x^4 \log (x)}{x-e^x \log (x)} \, dx+32 \int \frac {x^4}{\left (x-e^x \log (x)\right )^2} \, dx-32 \int \frac {x^4 \log (x)}{\left (x-e^x \log (x)\right )^2} \, dx+32 \int \frac {x^5 \log (x)}{\left (x-e^x \log (x)\right )^2} \, dx-128 \int \frac {x^3 \log (x)}{x-e^x \log (x)} \, dx+256 \int \frac {x^3}{\left (x-e^x \log (x)\right )^2} \, dx-256 \int \frac {x^3 \log (x)}{\left (x-e^x \log (x)\right )^2} \, dx+256 \int \frac {x^4 \log (x)}{\left (x-e^x \log (x)\right )^2} \, dx+768 \int \frac {x^2 \log (x)}{x-e^x \log (x)} \, dx+\int \frac {x^5}{\left (x-e^x \log (x)\right )^2} \, dx-\int \frac {x^5 \log (x)}{\left (x-e^x \log (x)\right )^2} \, dx+\int \frac {x^6 \log (x)}{\left (x-e^x \log (x)\right )^2} \, dx-\int \frac {x^5 \log (x)}{x-e^x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 60, normalized size = 1.88 \begin {gather*} \frac {x \left (x \left (-1+256 x+32 x^2+x^3\right )+\left (x^2 (16+x)^2-e^x \left (-1+256 x+32 x^2+x^3\right )\right ) \log (x)\right )}{x-e^x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + 768*x^3 + 128*x^4 + 5*x^5 + (512*x^3 + 96*x^4 + 4*x^5 + E^x*(2*x - 1024*x^2 - 192*x^3 - 8*x^
4))*Log[x] + (E^(2*x)*(-1 + 512*x + 96*x^2 + 4*x^3) + E^x*(-768*x^2 + 128*x^3 + 27*x^4 + x^5))*Log[x]^2)/(x^2
- 2*E^x*x*Log[x] + E^(2*x)*Log[x]^2),x]

[Out]

(x*(x*(-1 + 256*x + 32*x^2 + x^3) + (x^2*(16 + x)^2 - E^x*(-1 + 256*x + 32*x^2 + x^3))*Log[x]))/(x - E^x*Log[x
])

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fricas [B]  time = 0.59, size = 70, normalized size = 2.19 \begin {gather*} -\frac {x^{5} + 32 \, x^{4} + 256 \, x^{3} - x^{2} + {\left (x^{5} + 32 \, x^{4} + 256 \, x^{3} - {\left (x^{4} + 32 \, x^{3} + 256 \, x^{2} - x\right )} e^{x}\right )} \log \relax (x)}{e^{x} \log \relax (x) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3+96*x^2+512*x-1)*exp(x)^2+(x^5+27*x^4+128*x^3-768*x^2)*exp(x))*log(x)^2+((-8*x^4-192*x^3-102
4*x^2+2*x)*exp(x)+4*x^5+96*x^4+512*x^3)*log(x)+5*x^5+128*x^4+768*x^3-x^2)/(exp(x)^2*log(x)^2-2*x*exp(x)*log(x)
+x^2),x, algorithm="fricas")

[Out]

-(x^5 + 32*x^4 + 256*x^3 - x^2 + (x^5 + 32*x^4 + 256*x^3 - (x^4 + 32*x^3 + 256*x^2 - x)*e^x)*log(x))/(e^x*log(
x) - x)

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giac [B]  time = 0.26, size = 85, normalized size = 2.66 \begin {gather*} -\frac {x^{5} \log \relax (x) - x^{4} e^{x} \log \relax (x) + x^{5} + 32 \, x^{4} \log \relax (x) - 32 \, x^{3} e^{x} \log \relax (x) + 32 \, x^{4} + 256 \, x^{3} \log \relax (x) - 256 \, x^{2} e^{x} \log \relax (x) + 256 \, x^{3} + x e^{x} \log \relax (x) - x^{2}}{e^{x} \log \relax (x) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3+96*x^2+512*x-1)*exp(x)^2+(x^5+27*x^4+128*x^3-768*x^2)*exp(x))*log(x)^2+((-8*x^4-192*x^3-102
4*x^2+2*x)*exp(x)+4*x^5+96*x^4+512*x^3)*log(x)+5*x^5+128*x^4+768*x^3-x^2)/(exp(x)^2*log(x)^2-2*x*exp(x)*log(x)
+x^2),x, algorithm="giac")

[Out]

-(x^5*log(x) - x^4*e^x*log(x) + x^5 + 32*x^4*log(x) - 32*x^3*e^x*log(x) + 32*x^4 + 256*x^3*log(x) - 256*x^2*e^
x*log(x) + 256*x^3 + x*e^x*log(x) - x^2)/(e^x*log(x) - x)

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maple [B]  time = 0.04, size = 70, normalized size = 2.19

method	result	size
risch	\(-x \left (x^{4}-{\mathrm e}^{x} x^{3}+32 x^{3}-32 \,{\mathrm e}^{x} x^{2}+256 x^{2}-256 \,{\mathrm e}^{x} x +{\mathrm e}^{x}\right ) {\mathrm e}^{-x}+\frac {\left (x^{2}+32 x +256\right ) x^{4} {\mathrm e}^{-x}}{-{\mathrm e}^{x} \ln \relax (x )+x}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x^3+96*x^2+512*x-1)*exp(x)^2+(x^5+27*x^4+128*x^3-768*x^2)*exp(x))*ln(x)^2+((-8*x^4-192*x^3-1024*x^2+2
*x)*exp(x)+4*x^5+96*x^4+512*x^3)*ln(x)+5*x^5+128*x^4+768*x^3-x^2)/(exp(x)^2*ln(x)^2-2*x*exp(x)*ln(x)+x^2),x,me
thod=_RETURNVERBOSE)

[Out]

-x*(x^4-exp(x)*x^3+32*x^3-32*exp(x)*x^2+256*x^2-256*exp(x)*x+exp(x))*exp(-x)+(x^2+32*x+256)*x^4*exp(-x)/(-exp(
x)*ln(x)+x)

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maxima [B]  time = 0.42, size = 72, normalized size = 2.25 \begin {gather*} -\frac {x^{5} + 32 \, x^{4} + 256 \, x^{3} - {\left (x^{4} + 32 \, x^{3} + 256 \, x^{2} - x\right )} e^{x} \log \relax (x) - x^{2} + {\left (x^{5} + 32 \, x^{4} + 256 \, x^{3}\right )} \log \relax (x)}{e^{x} \log \relax (x) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3+96*x^2+512*x-1)*exp(x)^2+(x^5+27*x^4+128*x^3-768*x^2)*exp(x))*log(x)^2+((-8*x^4-192*x^3-102
4*x^2+2*x)*exp(x)+4*x^5+96*x^4+512*x^3)*log(x)+5*x^5+128*x^4+768*x^3-x^2)/(exp(x)^2*log(x)^2-2*x*exp(x)*log(x)
+x^2),x, algorithm="maxima")

[Out]

-(x^5 + 32*x^4 + 256*x^3 - (x^4 + 32*x^3 + 256*x^2 - x)*e^x*log(x) - x^2 + (x^5 + 32*x^4 + 256*x^3)*log(x))/(e
^x*log(x) - x)

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mupad [B]  time = 3.46, size = 109, normalized size = 3.41 \begin {gather*} 256\,x^2-{\mathrm {e}}^{-x}\,\left (x^5+32\,x^4+256\,x^3\right )-x+32\,x^3+x^4+\frac {256\,x^5\,{\mathrm {e}}^x+32\,x^6\,{\mathrm {e}}^x+x^7\,{\mathrm {e}}^x-256\,x^6+224\,x^7+31\,x^8+x^9}{\left (x-{\mathrm {e}}^x\,\ln \relax (x)\right )\,\left (x\,{\mathrm {e}}^{2\,x}-x^2\,{\mathrm {e}}^x+x^3\,{\mathrm {e}}^x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(exp(x)*(128*x^3 - 768*x^2 + 27*x^4 + x^5) + exp(2*x)*(512*x + 96*x^2 + 4*x^3 - 1)) + log(x)*(51
2*x^3 - exp(x)*(1024*x^2 - 2*x + 192*x^3 + 8*x^4) + 96*x^4 + 4*x^5) - x^2 + 768*x^3 + 128*x^4 + 5*x^5)/(x^2 +
exp(2*x)*log(x)^2 - 2*x*exp(x)*log(x)),x)

[Out]

256*x^2 - exp(-x)*(256*x^3 + 32*x^4 + x^5) - x + 32*x^3 + x^4 + (256*x^5*exp(x) + 32*x^6*exp(x) + x^7*exp(x) -
256*x^6 + 224*x^7 + 31*x^8 + x^9)/((x - exp(x)*log(x))*(x*exp(2*x) - x^2*exp(x) + x^3*exp(x)))

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sympy [B]  time = 0.31, size = 48, normalized size = 1.50 \begin {gather*} x^{4} + 32 x^{3} + 256 x^{2} - x + \frac {- x^{5} \log {\relax (x )} - 32 x^{4} \log {\relax (x )} - 256 x^{3} \log {\relax (x )}}{- x + e^{x} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x**3+96*x**2+512*x-1)*exp(x)**2+(x**5+27*x**4+128*x**3-768*x**2)*exp(x))*ln(x)**2+((-8*x**4-192
*x**3-1024*x**2+2*x)*exp(x)+4*x**5+96*x**4+512*x**3)*ln(x)+5*x**5+128*x**4+768*x**3-x**2)/(exp(x)**2*ln(x)**2-
2*x*exp(x)*ln(x)+x**2),x)

[Out]

x**4 + 32*x**3 + 256*x**2 - x + (-x**5*log(x) - 32*x**4*log(x) - 256*x**3*log(x))/(-x + exp(x)*log(x))

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