3.8.90 \(\int \frac {19 x^3-3 x^4+e^8 (729 x-81 x^2)+e^4 (252 x^2-36 x^3)+(162 e^8 x+54 e^4 x^2+4 x^3) \log (x)}{25-10 x+x^2+(10-2 x) \log (x)+\log ^2(x)} \, dx\) [790]

Optimal. Leaf size=22 \[ \frac {x^2 \left (9 e^4+x\right )^2}{5-x+\log (x)} \]

[Out]

x^2*(9*exp(4)+x)^2/(ln(x)-x+5)

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Rubi [F]
time = 0.60, 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 {19 x^3-3 x^4+e^8 \left (729 x-81 x^2\right )+e^4 \left (252 x^2-36 x^3\right )+\left (162 e^8 x+54 e^4 x^2+4 x^3\right ) \log (x)}{25-10 x+x^2+(10-2 x) \log (x)+\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(19*x^3 - 3*x^4 + E^8*(729*x - 81*x^2) + E^4*(252*x^2 - 36*x^3) + (162*E^8*x + 54*E^4*x^2 + 4*x^3)*Log[x])
/(25 - 10*x + x^2 + (10 - 2*x)*Log[x] + Log[x]^2),x]

[Out]

-81*E^8*Defer[Int][x/(-5 + x - Log[x])^2, x] - 9*E^4*(2 - 9*E^4)*Defer[Int][x^2/(-5 + x - Log[x])^2, x] - (1 -
18*E^4)*Defer[Int][x^3/(-5 + x - Log[x])^2, x] + Defer[Int][x^4/(-5 + x - Log[x])^2, x] - 162*E^8*Defer[Int][
x/(-5 + x - Log[x]), x] - 54*E^4*Defer[Int][x^2/(-5 + x - Log[x]), x] - 4*Defer[Int][x^3/(-5 + x - Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (9 e^4+x\right ) \left (-9 e^4 (-9+x)-x (-19+3 x)+2 \left (9 e^4+2 x\right ) \log (x)\right )}{(5-x+\log (x))^2} \, dx\\ &=\int \left (\frac {(-1+x) x \left (9 e^4+x\right )^2}{(-5+x-\log (x))^2}-\frac {2 x \left (81 e^8+27 e^4 x+2 x^2\right )}{-5+x-\log (x)}\right ) \, dx\\ &=-\left (2 \int \frac {x \left (81 e^8+27 e^4 x+2 x^2\right )}{-5+x-\log (x)} \, dx\right )+\int \frac {(-1+x) x \left (9 e^4+x\right )^2}{(-5+x-\log (x))^2} \, dx\\ &=-\left (2 \int \left (\frac {81 e^8 x}{-5+x-\log (x)}+\frac {27 e^4 x^2}{-5+x-\log (x)}+\frac {2 x^3}{-5+x-\log (x)}\right ) \, dx\right )+\int \left (-\frac {81 e^8 x}{(-5+x-\log (x))^2}+\frac {9 e^4 \left (-2+9 e^4\right ) x^2}{(-5+x-\log (x))^2}+\frac {\left (-1+18 e^4\right ) x^3}{(-5+x-\log (x))^2}+\frac {x^4}{(-5+x-\log (x))^2}\right ) \, dx\\ &=-\left (4 \int \frac {x^3}{-5+x-\log (x)} \, dx\right )-\left (54 e^4\right ) \int \frac {x^2}{-5+x-\log (x)} \, dx-\left (81 e^8\right ) \int \frac {x}{(-5+x-\log (x))^2} \, dx-\left (162 e^8\right ) \int \frac {x}{-5+x-\log (x)} \, dx-\left (9 e^4 \left (2-9 e^4\right )\right ) \int \frac {x^2}{(-5+x-\log (x))^2} \, dx+\left (-1+18 e^4\right ) \int \frac {x^3}{(-5+x-\log (x))^2} \, dx+\int \frac {x^4}{(-5+x-\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.09, size = 23, normalized size = 1.05 \begin {gather*} -\frac {x^2 \left (9 e^4+x\right )^2}{-5+x-\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(19*x^3 - 3*x^4 + E^8*(729*x - 81*x^2) + E^4*(252*x^2 - 36*x^3) + (162*E^8*x + 54*E^4*x^2 + 4*x^3)*L
og[x])/(25 - 10*x + x^2 + (10 - 2*x)*Log[x] + Log[x]^2),x]

[Out]

-((x^2*(9*E^4 + x)^2)/(-5 + x - Log[x]))

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Maple [A]
time = 0.47, size = 31, normalized size = 1.41

method	result	size
risch	\(-\frac {\left (81 \,{\mathrm e}^{8}+18 x \,{\mathrm e}^{4}+x^{2}\right ) x^{2}}{-\ln \left (x \right )+x -5}\)	\(28\)
default	\(\frac {x^{4}+18 x^{3} {\mathrm e}^{4}+81 x^{2} {\mathrm e}^{8}}{\ln \left (x \right )-x +5}\)	\(31\)
norman	\(\frac {-x^{4}-81 x^{2} {\mathrm e}^{8}-18 x^{3} {\mathrm e}^{4}}{-\ln \left (x \right )+x -5}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((162*x*exp(4)^2+54*x^2*exp(4)+4*x^3)*ln(x)+(-81*x^2+729*x)*exp(4)^2+(-36*x^3+252*x^2)*exp(4)-3*x^4+19*x^3
)/(ln(x)^2+(-2*x+10)*ln(x)+x^2-10*x+25),x,method=_RETURNVERBOSE)

[Out]

(x^4+18*x^3*exp(4)+81*x^2*exp(4)^2)/(ln(x)-x+5)

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Maxima [A]
time = 0.29, size = 29, normalized size = 1.32 \begin {gather*} -\frac {x^{4} + 18 \, x^{3} e^{4} + 81 \, x^{2} e^{8}}{x - \log \left (x\right ) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x*exp(4)^2+54*x^2*exp(4)+4*x^3)*log(x)+(-81*x^2+729*x)*exp(4)^2+(-36*x^3+252*x^2)*exp(4)-3*x^4
+19*x^3)/(log(x)^2+(-2*x+10)*log(x)+x^2-10*x+25),x, algorithm="maxima")

[Out]

-(x^4 + 18*x^3*e^4 + 81*x^2*e^8)/(x - log(x) - 5)

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Fricas [A]
time = 0.41, size = 29, normalized size = 1.32 \begin {gather*} -\frac {x^{4} + 18 \, x^{3} e^{4} + 81 \, x^{2} e^{8}}{x - \log \left (x\right ) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x*exp(4)^2+54*x^2*exp(4)+4*x^3)*log(x)+(-81*x^2+729*x)*exp(4)^2+(-36*x^3+252*x^2)*exp(4)-3*x^4
+19*x^3)/(log(x)^2+(-2*x+10)*log(x)+x^2-10*x+25),x, algorithm="fricas")

[Out]

-(x^4 + 18*x^3*e^4 + 81*x^2*e^8)/(x - log(x) - 5)

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Sympy [A]
time = 0.05, size = 26, normalized size = 1.18 \begin {gather*} \frac {x^{4} + 18 x^{3} e^{4} + 81 x^{2} e^{8}}{- x + \log {\left (x \right )} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x*exp(4)**2+54*x**2*exp(4)+4*x**3)*ln(x)+(-81*x**2+729*x)*exp(4)**2+(-36*x**3+252*x**2)*exp(4)
-3*x**4+19*x**3)/(ln(x)**2+(-2*x+10)*ln(x)+x**2-10*x+25),x)

[Out]

(x**4 + 18*x**3*exp(4) + 81*x**2*exp(8))/(-x + log(x) + 5)

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Giac [A]
time = 0.41, size = 29, normalized size = 1.32 \begin {gather*} -\frac {x^{4} + 18 \, x^{3} e^{4} + 81 \, x^{2} e^{8}}{x - \log \left (x\right ) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x*exp(4)^2+54*x^2*exp(4)+4*x^3)*log(x)+(-81*x^2+729*x)*exp(4)^2+(-36*x^3+252*x^2)*exp(4)-3*x^4
+19*x^3)/(log(x)^2+(-2*x+10)*log(x)+x^2-10*x+25),x, algorithm="giac")

[Out]

-(x^4 + 18*x^3*e^4 + 81*x^2*e^8)/(x - log(x) - 5)

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Mupad [B]
time = 0.72, size = 21, normalized size = 0.95 \begin {gather*} \frac {x^2\,{\left (x+9\,{\mathrm {e}}^4\right )}^2}{\ln \left (x\right )-x+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(8)*(729*x - 81*x^2) + exp(4)*(252*x^2 - 36*x^3) + log(x)*(162*x*exp(8) + 54*x^2*exp(4) + 4*x^3) + 19*
x^3 - 3*x^4)/(log(x)^2 - 10*x - log(x)*(2*x - 10) + x^2 + 25),x)

[Out]

(x^2*(x + 9*exp(4))^2)/(log(x) - x + 5)

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