3.31.74 \(\int \frac {e (1125+7425 x)+e^x (27 x^2+297 x^3+1089 x^4+1331 x^5)+(225 e+e^x (27 x^2+198 x^3+363 x^4)) \log (x)+e^x (9 x^2+33 x^3) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+(27 x^2+198 x^3+363 x^4) \log (x)+(9 x^2+33 x^3) \log ^2(x)+x^2 \log ^3(x)} \, dx\) [3074]

Optimal. Leaf size=25 \[ e^x-\frac {9 e}{x \left (2 x+\frac {1}{5} (3+x+\log (x))\right )^2} \]

[Out]

exp(x)-9/x/(3/5+11/5*x+1/5*ln(x))^2*exp(1)

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Rubi [F]
time = 1.29, 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 (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E*(1125 + 7425*x) + E^x*(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5) + (225*E + E^x*(27*x^2 + 198*x^3 + 363*x
^4))*Log[x] + E^x*(9*x^2 + 33*x^3)*Log[x]^2 + E^x*x^2*Log[x]^3)/(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5 + (27*
x^2 + 198*x^3 + 363*x^4)*Log[x] + (9*x^2 + 33*x^3)*Log[x]^2 + x^2*Log[x]^3),x]

[Out]

E^x + 450*E*Defer[Int][1/(x^2*(3 + 11*x + Log[x])^3), x] + 4950*E*Defer[Int][1/(x*(3 + 11*x + Log[x])^3), x] +
225*E*Defer[Int][1/(x^2*(3 + 11*x + Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x x^2 (3+11 x)^3+225 e (5+33 x)+3 \left (75 e+e^x x^2 (3+11 x)^2\right ) \log (x)+3 e^x x^2 (3+11 x) \log ^2(x)+e^x x^2 \log ^3(x)}{x^2 (3+11 x+\log (x))^3} \, dx\\ &=\int \left (e^x+\frac {225 e (5+33 x+\log (x))}{x^2 (3+11 x+\log (x))^3}\right ) \, dx\\ &=(225 e) \int \frac {5+33 x+\log (x)}{x^2 (3+11 x+\log (x))^3} \, dx+\int e^x \, dx\\ &=e^x+(225 e) \int \left (\frac {2 (1+11 x)}{x^2 (3+11 x+\log (x))^3}+\frac {1}{x^2 (3+11 x+\log (x))^2}\right ) \, dx\\ &=e^x+(225 e) \int \frac {1}{x^2 (3+11 x+\log (x))^2} \, dx+(450 e) \int \frac {1+11 x}{x^2 (3+11 x+\log (x))^3} \, dx\\ &=e^x+(225 e) \int \frac {1}{x^2 (3+11 x+\log (x))^2} \, dx+(450 e) \int \left (\frac {1}{x^2 (3+11 x+\log (x))^3}+\frac {11}{x (3+11 x+\log (x))^3}\right ) \, dx\\ &=e^x+(225 e) \int \frac {1}{x^2 (3+11 x+\log (x))^2} \, dx+(450 e) \int \frac {1}{x^2 (3+11 x+\log (x))^3} \, dx+(4950 e) \int \frac {1}{x (3+11 x+\log (x))^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.06, size = 19, normalized size = 0.76 \begin {gather*} e^x-\frac {225 e}{x (3+11 x+\log (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E*(1125 + 7425*x) + E^x*(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5) + (225*E + E^x*(27*x^2 + 198*x^3 +
363*x^4))*Log[x] + E^x*(9*x^2 + 33*x^3)*Log[x]^2 + E^x*x^2*Log[x]^3)/(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5
+ (27*x^2 + 198*x^3 + 363*x^4)*Log[x] + (9*x^2 + 33*x^3)*Log[x]^2 + x^2*Log[x]^3),x]

[Out]

E^x - (225*E)/(x*(3 + 11*x + Log[x])^2)

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Maple [A]
time = 0.58, size = 20, normalized size = 0.80

method	result	size
default	\(-\frac {225 \,{\mathrm e}}{x \left (\ln \left (x \right )+11 x +3\right )^{2}}+{\mathrm e}^{x}\)	\(20\)
risch	\(-\frac {225 \,{\mathrm e}}{x \left (\ln \left (x \right )+11 x +3\right )^{2}}+{\mathrm e}^{x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(x)*ln(x)^3+(33*x^3+9*x^2)*exp(x)*ln(x)^2+((363*x^4+198*x^3+27*x^2)*exp(x)+225*exp(1))*ln(x)+(1331
*x^5+1089*x^4+297*x^3+27*x^2)*exp(x)+(7425*x+1125)*exp(1))/(x^2*ln(x)^3+(33*x^3+9*x^2)*ln(x)^2+(363*x^4+198*x^
3+27*x^2)*ln(x)+1331*x^5+1089*x^4+297*x^3+27*x^2),x,method=_RETURNVERBOSE)

[Out]

-225*exp(1)/x/(ln(x)+11*x+3)^2+exp(x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (19) = 38\).
time = 0.30, size = 77, normalized size = 3.08 \begin {gather*} \frac {{\left (121 \, x^{3} + x \log \left (x\right )^{2} + 66 \, x^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} \log \left (x\right ) + 9 \, x\right )} e^{x} - 225 \, e}{121 \, x^{3} + x \log \left (x\right )^{2} + 66 \, x^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} \log \left (x\right ) + 9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(x)*log(x)^3+(33*x^3+9*x^2)*exp(x)*log(x)^2+((363*x^4+198*x^3+27*x^2)*exp(x)+225*exp(1))*log
(x)+(1331*x^5+1089*x^4+297*x^3+27*x^2)*exp(x)+(7425*x+1125)*exp(1))/(x^2*log(x)^3+(33*x^3+9*x^2)*log(x)^2+(363
*x^4+198*x^3+27*x^2)*log(x)+1331*x^5+1089*x^4+297*x^3+27*x^2),x, algorithm="maxima")

[Out]

((121*x^3 + x*log(x)^2 + 66*x^2 + 2*(11*x^2 + 3*x)*log(x) + 9*x)*e^x - 225*e)/(121*x^3 + x*log(x)^2 + 66*x^2 +
2*(11*x^2 + 3*x)*log(x) + 9*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (19) = 38\).
time = 0.39, size = 81, normalized size = 3.24 \begin {gather*} \frac {x e^{x} \log \left (x\right )^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} e^{x} \log \left (x\right ) + {\left (121 \, x^{3} + 66 \, x^{2} + 9 \, x\right )} e^{x} - 225 \, e}{121 \, x^{3} + x \log \left (x\right )^{2} + 66 \, x^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} \log \left (x\right ) + 9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(x)*log(x)^3+(33*x^3+9*x^2)*exp(x)*log(x)^2+((363*x^4+198*x^3+27*x^2)*exp(x)+225*exp(1))*log
(x)+(1331*x^5+1089*x^4+297*x^3+27*x^2)*exp(x)+(7425*x+1125)*exp(1))/(x^2*log(x)^3+(33*x^3+9*x^2)*log(x)^2+(363
*x^4+198*x^3+27*x^2)*log(x)+1331*x^5+1089*x^4+297*x^3+27*x^2),x, algorithm="fricas")

[Out]

(x*e^x*log(x)^2 + 2*(11*x^2 + 3*x)*e^x*log(x) + (121*x^3 + 66*x^2 + 9*x)*e^x - 225*e)/(121*x^3 + x*log(x)^2 +
66*x^2 + 2*(11*x^2 + 3*x)*log(x) + 9*x)

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Sympy [A]
time = 0.15, size = 39, normalized size = 1.56 \begin {gather*} e^{x} - \frac {225 e}{121 x^{3} + 66 x^{2} + x \log {\left (x \right )}^{2} + 9 x + \left (22 x^{2} + 6 x\right ) \log {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*exp(x)*ln(x)**3+(33*x**3+9*x**2)*exp(x)*ln(x)**2+((363*x**4+198*x**3+27*x**2)*exp(x)+225*exp(1
))*ln(x)+(1331*x**5+1089*x**4+297*x**3+27*x**2)*exp(x)+(7425*x+1125)*exp(1))/(x**2*ln(x)**3+(33*x**3+9*x**2)*l
n(x)**2+(363*x**4+198*x**3+27*x**2)*ln(x)+1331*x**5+1089*x**4+297*x**3+27*x**2),x)

[Out]

exp(x) - 225*E/(121*x**3 + 66*x**2 + x*log(x)**2 + 9*x + (22*x**2 + 6*x)*log(x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (19) = 38\).
time = 0.41, size = 83, normalized size = 3.32 \begin {gather*} \frac {121 \, x^{3} e^{x} + 22 \, x^{2} e^{x} \log \left (x\right ) + x e^{x} \log \left (x\right )^{2} + 66 \, x^{2} e^{x} + 6 \, x e^{x} \log \left (x\right ) + 9 \, x e^{x} - 225 \, e}{121 \, x^{3} + 22 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + 66 \, x^{2} + 6 \, x \log \left (x\right ) + 9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(x)*log(x)^3+(33*x^3+9*x^2)*exp(x)*log(x)^2+((363*x^4+198*x^3+27*x^2)*exp(x)+225*exp(1))*log
(x)+(1331*x^5+1089*x^4+297*x^3+27*x^2)*exp(x)+(7425*x+1125)*exp(1))/(x^2*log(x)^3+(33*x^3+9*x^2)*log(x)^2+(363
*x^4+198*x^3+27*x^2)*log(x)+1331*x^5+1089*x^4+297*x^3+27*x^2),x, algorithm="giac")

[Out]

(121*x^3*e^x + 22*x^2*e^x*log(x) + x*e^x*log(x)^2 + 66*x^2*e^x + 6*x*e^x*log(x) + 9*x*e^x - 225*e)/(121*x^3 +
22*x^2*log(x) + x*log(x)^2 + 66*x^2 + 6*x*log(x) + 9*x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (1331\,x^5+1089\,x^4+297\,x^3+27\,x^2\right )+\ln \left (x\right )\,\left (225\,\mathrm {e}+{\mathrm {e}}^x\,\left (363\,x^4+198\,x^3+27\,x^2\right )\right )+\mathrm {e}\,\left (7425\,x+1125\right )+{\mathrm {e}}^x\,{\ln \left (x\right )}^2\,\left (33\,x^3+9\,x^2\right )+x^2\,{\mathrm {e}}^x\,{\ln \left (x\right )}^3}{\ln \left (x\right )\,\left (363\,x^4+198\,x^3+27\,x^2\right )+{\ln \left (x\right )}^2\,\left (33\,x^3+9\,x^2\right )+x^2\,{\ln \left (x\right )}^3+27\,x^2+297\,x^3+1089\,x^4+1331\,x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5) + log(x)*(225*exp(1) + exp(x)*(27*x^2 + 198*x^3 + 363*x^4
)) + exp(1)*(7425*x + 1125) + exp(x)*log(x)^2*(9*x^2 + 33*x^3) + x^2*exp(x)*log(x)^3)/(log(x)*(27*x^2 + 198*x^
3 + 363*x^4) + log(x)^2*(9*x^2 + 33*x^3) + x^2*log(x)^3 + 27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5),x)

[Out]

int((exp(x)*(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5) + log(x)*(225*exp(1) + exp(x)*(27*x^2 + 198*x^3 + 363*x^4
)) + exp(1)*(7425*x + 1125) + exp(x)*log(x)^2*(9*x^2 + 33*x^3) + x^2*exp(x)*log(x)^3)/(log(x)*(27*x^2 + 198*x^
3 + 363*x^4) + log(x)^2*(9*x^2 + 33*x^3) + x^2*log(x)^3 + 27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5), x)

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