3.84.54 \(\int \frac {e^{2 x} (9-5 x)+e^x (54-84 x+30 x^2)+(-5 e^{2 x} x+e^x (78 x-84 x^2+30 x^3)) \log (x)+e^x (162 x-180 x^2+50 x^3) \log ^2(x)}{36 x+e^{2 x} x-72 x^2+36 x^3+e^x (12 x-12 x^2)+(216 x-336 x^2+120 x^3+e^x (36 x-20 x^2)) \log (x)+(324 x-360 x^2+100 x^3) \log ^2(x)} \, dx\)

Optimal. Leaf size=35 \[ \frac {e^x}{2+\frac {e^x-3 (-2+2 x)}{(4+5 (1-x)) \log (x)}} \]

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Rubi [F]  time = 23.27, 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^{2 x} (9-5 x)+e^x \left (54-84 x+30 x^2\right )+\left (-5 e^{2 x} x+e^x \left (78 x-84 x^2+30 x^3\right )\right ) \log (x)+e^x \left (162 x-180 x^2+50 x^3\right ) \log ^2(x)}{36 x+e^{2 x} x-72 x^2+36 x^3+e^x \left (12 x-12 x^2\right )+\left (216 x-336 x^2+120 x^3+e^x \left (36 x-20 x^2\right )\right ) \log (x)+\left (324 x-360 x^2+100 x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x)*(9 - 5*x) + E^x*(54 - 84*x + 30*x^2) + (-5*E^(2*x)*x + E^x*(78*x - 84*x^2 + 30*x^3))*Log[x] + E^x
*(162*x - 180*x^2 + 50*x^3)*Log[x]^2)/(36*x + E^(2*x)*x - 72*x^2 + 36*x^3 + E^x*(12*x - 12*x^2) + (216*x - 336
*x^2 + 120*x^3 + E^x*(36*x - 20*x^2))*Log[x] + (324*x - 360*x^2 + 100*x^3)*Log[x]^2),x]

[Out]

-5*Defer[Int][E^x/(6 + E^x - 6*x + 18*Log[x] - 10*x*Log[x]), x] + 288*Defer[Int][(E^x*Log[x])/(-6 - E^x + 6*x
- 18*Log[x] + 10*x*Log[x])^2, x] - 162*Defer[Int][(E^x*Log[x])/(x*(-6 - E^x + 6*x - 18*Log[x] + 10*x*Log[x])^2
), x] - 164*Defer[Int][(E^x*x*Log[x])/(-6 - E^x + 6*x - 18*Log[x] + 10*x*Log[x])^2, x] + 30*Defer[Int][(E^x*x^
2*Log[x])/(-6 - E^x + 6*x - 18*Log[x] + 10*x*Log[x])^2, x] + 252*Defer[Int][(E^x*Log[x]^2)/(-6 - E^x + 6*x - 1
8*Log[x] + 10*x*Log[x])^2, x] - 230*Defer[Int][(E^x*x*Log[x]^2)/(-6 - E^x + 6*x - 18*Log[x] + 10*x*Log[x])^2,
x] + 50*Defer[Int][(E^x*x^2*Log[x]^2)/(-6 - E^x + 6*x - 18*Log[x] + 10*x*Log[x])^2, x] - 9*Defer[Int][E^x/(x*(
-6 - E^x + 6*x - 18*Log[x] + 10*x*Log[x])), x] + 5*Defer[Int][(E^x*Log[x])/(-6 - E^x + 6*x - 18*Log[x] + 10*x*
Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (-\left (\left (6+e^x-6 x\right ) (-9+5 x)\right )+x \left (78-5 e^x-84 x+30 x^2\right ) \log (x)+2 (9-5 x)^2 x \log ^2(x)\right )}{x \left (6+e^x-6 x-2 (-9+5 x) \log (x)\right )^2} \, dx\\ &=\int \left (\frac {e^x (-9+5 x+5 x \log (x))}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )}+\frac {2 e^x (-9+5 x) \log (x) \left (9-11 x+3 x^2-14 x \log (x)+5 x^2 \log (x)\right )}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}\right ) \, dx\\ &=2 \int \frac {e^x (-9+5 x) \log (x) \left (9-11 x+3 x^2-14 x \log (x)+5 x^2 \log (x)\right )}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx+\int \frac {e^x (-9+5 x+5 x \log (x))}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )} \, dx\\ &=2 \int \left (\frac {5 e^x \log (x) \left (9-11 x+3 x^2-14 x \log (x)+5 x^2 \log (x)\right )}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}-\frac {9 e^x \log (x) \left (9-11 x+3 x^2-14 x \log (x)+5 x^2 \log (x)\right )}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}\right ) \, dx+\int \left (-\frac {5 e^x}{6+e^x-6 x+18 \log (x)-10 x \log (x)}-\frac {9 e^x}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )}+\frac {5 e^x \log (x)}{-6-e^x+6 x-18 \log (x)+10 x \log (x)}\right ) \, dx\\ &=-\left (5 \int \frac {e^x}{6+e^x-6 x+18 \log (x)-10 x \log (x)} \, dx\right )+5 \int \frac {e^x \log (x)}{-6-e^x+6 x-18 \log (x)+10 x \log (x)} \, dx-9 \int \frac {e^x}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )} \, dx+10 \int \frac {e^x \log (x) \left (9-11 x+3 x^2-14 x \log (x)+5 x^2 \log (x)\right )}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx-18 \int \frac {e^x \log (x) \left (9-11 x+3 x^2-14 x \log (x)+5 x^2 \log (x)\right )}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx\\ &=-\left (5 \int \frac {e^x}{6+e^x-6 x+18 \log (x)-10 x \log (x)} \, dx\right )+5 \int \frac {e^x \log (x)}{-6-e^x+6 x-18 \log (x)+10 x \log (x)} \, dx-9 \int \frac {e^x}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )} \, dx+10 \int \left (\frac {9 e^x \log (x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}-\frac {11 e^x x \log (x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}+\frac {3 e^x x^2 \log (x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}-\frac {14 e^x x \log ^2(x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}+\frac {5 e^x x^2 \log ^2(x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}\right ) \, dx-18 \int \left (-\frac {11 e^x \log (x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}+\frac {9 e^x \log (x)}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}+\frac {3 e^x x \log (x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}-\frac {14 e^x \log ^2(x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}+\frac {5 e^x x \log ^2(x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2}\right ) \, dx\\ &=-\left (5 \int \frac {e^x}{6+e^x-6 x+18 \log (x)-10 x \log (x)} \, dx\right )+5 \int \frac {e^x \log (x)}{-6-e^x+6 x-18 \log (x)+10 x \log (x)} \, dx-9 \int \frac {e^x}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )} \, dx+30 \int \frac {e^x x^2 \log (x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx+50 \int \frac {e^x x^2 \log ^2(x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx-54 \int \frac {e^x x \log (x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx+90 \int \frac {e^x \log (x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx-90 \int \frac {e^x x \log ^2(x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx-110 \int \frac {e^x x \log (x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx-140 \int \frac {e^x x \log ^2(x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx-162 \int \frac {e^x \log (x)}{x \left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx+198 \int \frac {e^x \log (x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx+252 \int \frac {e^x \log ^2(x)}{\left (-6-e^x+6 x-18 \log (x)+10 x \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 3.63, size = 31, normalized size = 0.89 \begin {gather*} -\frac {e^x (-9+5 x) \log (x)}{6+e^x-6 x-2 (-9+5 x) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(9 - 5*x) + E^x*(54 - 84*x + 30*x^2) + (-5*E^(2*x)*x + E^x*(78*x - 84*x^2 + 30*x^3))*Log[x]
+ E^x*(162*x - 180*x^2 + 50*x^3)*Log[x]^2)/(36*x + E^(2*x)*x - 72*x^2 + 36*x^3 + E^x*(12*x - 12*x^2) + (216*x
- 336*x^2 + 120*x^3 + E^x*(36*x - 20*x^2))*Log[x] + (324*x - 360*x^2 + 100*x^3)*Log[x]^2),x]

[Out]

-((E^x*(-9 + 5*x)*Log[x])/(6 + E^x - 6*x - 2*(-9 + 5*x)*Log[x]))

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fricas [A]  time = 0.59, size = 30, normalized size = 0.86 \begin {gather*} \frac {{\left (5 \, x - 9\right )} e^{x} \log \relax (x)}{2 \, {\left (5 \, x - 9\right )} \log \relax (x) + 6 \, x - e^{x} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^3-180*x^2+162*x)*exp(x)*log(x)^2+(-5*x*exp(x)^2+(30*x^3-84*x^2+78*x)*exp(x))*log(x)+(-5*x+9)*
exp(x)^2+(30*x^2-84*x+54)*exp(x))/((100*x^3-360*x^2+324*x)*log(x)^2+((-20*x^2+36*x)*exp(x)+120*x^3-336*x^2+216
*x)*log(x)+x*exp(x)^2+(-12*x^2+12*x)*exp(x)+36*x^3-72*x^2+36*x),x, algorithm="fricas")

[Out]

(5*x - 9)*e^x*log(x)/(2*(5*x - 9)*log(x) + 6*x - e^x - 6)

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giac [A]  time = 0.28, size = 35, normalized size = 1.00 \begin {gather*} \frac {5 \, x e^{x} \log \relax (x) - 9 \, e^{x} \log \relax (x)}{10 \, x \log \relax (x) + 6 \, x - e^{x} - 18 \, \log \relax (x) - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^3-180*x^2+162*x)*exp(x)*log(x)^2+(-5*x*exp(x)^2+(30*x^3-84*x^2+78*x)*exp(x))*log(x)+(-5*x+9)*
exp(x)^2+(30*x^2-84*x+54)*exp(x))/((100*x^3-360*x^2+324*x)*log(x)^2+((-20*x^2+36*x)*exp(x)+120*x^3-336*x^2+216
*x)*log(x)+x*exp(x)^2+(-12*x^2+12*x)*exp(x)+36*x^3-72*x^2+36*x),x, algorithm="giac")

[Out]

(5*x*e^x*log(x) - 9*e^x*log(x))/(10*x*log(x) + 6*x - e^x - 18*log(x) - 6)

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maple [A]  time = 0.06, size = 39, normalized size = 1.11

method	result	size
risch	\(\frac {{\mathrm e}^{x}}{2}-\frac {\left (6 x -{\mathrm e}^{x}-6\right ) {\mathrm e}^{x}}{2 \left (10 x \ln \relax (x )-18 \ln \relax (x )-{\mathrm e}^{x}+6 x -6\right )}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((50*x^3-180*x^2+162*x)*exp(x)*ln(x)^2+(-5*x*exp(x)^2+(30*x^3-84*x^2+78*x)*exp(x))*ln(x)+(-5*x+9)*exp(x)^2
+(30*x^2-84*x+54)*exp(x))/((100*x^3-360*x^2+324*x)*ln(x)^2+((-20*x^2+36*x)*exp(x)+120*x^3-336*x^2+216*x)*ln(x)
+x*exp(x)^2+(-12*x^2+12*x)*exp(x)+36*x^3-72*x^2+36*x),x,method=_RETURNVERBOSE)

[Out]

1/2*exp(x)-1/2*(6*x-exp(x)-6)*exp(x)/(10*x*ln(x)-18*ln(x)-exp(x)+6*x-6)

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maxima [A]  time = 0.42, size = 30, normalized size = 0.86 \begin {gather*} \frac {{\left (5 \, x - 9\right )} e^{x} \log \relax (x)}{2 \, {\left (5 \, x - 9\right )} \log \relax (x) + 6 \, x - e^{x} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^3-180*x^2+162*x)*exp(x)*log(x)^2+(-5*x*exp(x)^2+(30*x^3-84*x^2+78*x)*exp(x))*log(x)+(-5*x+9)*
exp(x)^2+(30*x^2-84*x+54)*exp(x))/((100*x^3-360*x^2+324*x)*log(x)^2+((-20*x^2+36*x)*exp(x)+120*x^3-336*x^2+216
*x)*log(x)+x*exp(x)^2+(-12*x^2+12*x)*exp(x)+36*x^3-72*x^2+36*x),x, algorithm="maxima")

[Out]

(5*x - 9)*e^x*log(x)/(2*(5*x - 9)*log(x) + 6*x - e^x - 6)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {-{\mathrm {e}}^x\,\left (50\,x^3-180\,x^2+162\,x\right )\,{\ln \relax (x)}^2+\left (5\,x\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (30\,x^3-84\,x^2+78\,x\right )\right )\,\ln \relax (x)-{\mathrm {e}}^x\,\left (30\,x^2-84\,x+54\right )+{\mathrm {e}}^{2\,x}\,\left (5\,x-9\right )}{36\,x+x\,{\mathrm {e}}^{2\,x}+{\ln \relax (x)}^2\,\left (100\,x^3-360\,x^2+324\,x\right )+\ln \relax (x)\,\left (216\,x+{\mathrm {e}}^x\,\left (36\,x-20\,x^2\right )-336\,x^2+120\,x^3\right )+{\mathrm {e}}^x\,\left (12\,x-12\,x^2\right )-72\,x^2+36\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(5*x*exp(2*x) - exp(x)*(78*x - 84*x^2 + 30*x^3)) - exp(x)*(30*x^2 - 84*x + 54) + exp(2*x)*(5*x -
9) - exp(x)*log(x)^2*(162*x - 180*x^2 + 50*x^3))/(36*x + x*exp(2*x) + log(x)^2*(324*x - 360*x^2 + 100*x^3) + l
og(x)*(216*x + exp(x)*(36*x - 20*x^2) - 336*x^2 + 120*x^3) + exp(x)*(12*x - 12*x^2) - 72*x^2 + 36*x^3),x)

[Out]

int(-(log(x)*(5*x*exp(2*x) - exp(x)*(78*x - 84*x^2 + 30*x^3)) - exp(x)*(30*x^2 - 84*x + 54) + exp(2*x)*(5*x -
9) - exp(x)*log(x)^2*(162*x - 180*x^2 + 50*x^3))/(36*x + x*exp(2*x) + log(x)^2*(324*x - 360*x^2 + 100*x^3) + l
og(x)*(216*x + exp(x)*(36*x - 20*x^2) - 336*x^2 + 120*x^3) + exp(x)*(12*x - 12*x^2) - 72*x^2 + 36*x^3), x)

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sympy [B]  time = 0.40, size = 76, normalized size = 2.17 \begin {gather*} - 5 x \log {\relax (x )} + 9 \log {\relax (x )} + \frac {- 50 x^{2} \log {\relax (x )}^{2} - 30 x^{2} \log {\relax (x )} + 180 x \log {\relax (x )}^{2} + 84 x \log {\relax (x )} - 162 \log {\relax (x )}^{2} - 54 \log {\relax (x )}}{- 10 x \log {\relax (x )} - 6 x + e^{x} + 18 \log {\relax (x )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x**3-180*x**2+162*x)*exp(x)*ln(x)**2+(-5*x*exp(x)**2+(30*x**3-84*x**2+78*x)*exp(x))*ln(x)+(-5*x
+9)*exp(x)**2+(30*x**2-84*x+54)*exp(x))/((100*x**3-360*x**2+324*x)*ln(x)**2+((-20*x**2+36*x)*exp(x)+120*x**3-3
36*x**2+216*x)*ln(x)+x*exp(x)**2+(-12*x**2+12*x)*exp(x)+36*x**3-72*x**2+36*x),x)

[Out]

-5*x*log(x) + 9*log(x) + (-50*x**2*log(x)**2 - 30*x**2*log(x) + 180*x*log(x)**2 + 84*x*log(x) - 162*log(x)**2
- 54*log(x))/(-10*x*log(x) - 6*x + exp(x) + 18*log(x) + 6)

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