3.21.51 \(\int \frac {125 x-425 x^2+315 x^3+81 x^4+e^x (-25 x+90 x^2-81 x^3)+e^{3-e^x} (25-90 x+e^x (-5+48 x-54 x^2)+e^{2 x} (-5 x+9 x^2))+e^x (-25 x^2+90 x^3-81 x^4) \log (x)}{25 x^2-90 x^3+81 x^4} \, dx\)

Optimal. Leaf size=34 \[ -2+x+\left (5-e^x\right ) \left (\frac {e^{3-e^x}}{x (-5+9 x)}+\log (x)\right ) \]

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Rubi [F]  time = 22.59, 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 {125 x-425 x^2+315 x^3+81 x^4+e^x \left (-25 x+90 x^2-81 x^3\right )+e^{3-e^x} \left (25-90 x+e^x \left (-5+48 x-54 x^2\right )+e^{2 x} \left (-5 x+9 x^2\right )\right )+e^x \left (-25 x^2+90 x^3-81 x^4\right ) \log (x)}{25 x^2-90 x^3+81 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(125*x - 425*x^2 + 315*x^3 + 81*x^4 + E^x*(-25*x + 90*x^2 - 81*x^3) + E^(3 - E^x)*(25 - 90*x + E^x*(-5 + 4
8*x - 54*x^2) + E^(2*x)*(-5*x + 9*x^2)) + E^x*(-25*x^2 + 90*x^3 - 81*x^4)*Log[x])/(25*x^2 - 90*x^3 + 81*x^4),x
]

[Out]

x + 5*Log[x] - E^x*Log[x] + Defer[Int][E^(3 - E^x)/x^2, x] - Defer[Int][E^(3 - E^x + x)/x^2, x]/5 + (6*Defer[I
nt][E^(3 - E^x + x)/x, x])/5 - Defer[Int][E^(3 - E^x + 2*x)/x, x]/5 - 81*Defer[Int][E^(3 - E^x)/(-5 + 9*x)^2,
x] + (81*Defer[Int][E^(3 - E^x + x)/(-5 + 9*x)^2, x])/5 - (54*Defer[Int][E^(3 - E^x + x)/(-5 + 9*x), x])/5 + (
9*Defer[Int][E^(3 - E^x + 2*x)/(-5 + 9*x), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {125 x-425 x^2+315 x^3+81 x^4+e^x \left (-25 x+90 x^2-81 x^3\right )+e^{3-e^x} \left (25-90 x+e^x \left (-5+48 x-54 x^2\right )+e^{2 x} \left (-5 x+9 x^2\right )\right )+e^x \left (-25 x^2+90 x^3-81 x^4\right ) \log (x)}{x^2 \left (25-90 x+81 x^2\right )} \, dx\\ &=\int \frac {125 x-425 x^2+315 x^3+81 x^4+e^x \left (-25 x+90 x^2-81 x^3\right )+e^{3-e^x} \left (25-90 x+e^x \left (-5+48 x-54 x^2\right )+e^{2 x} \left (-5 x+9 x^2\right )\right )+e^x \left (-25 x^2+90 x^3-81 x^4\right ) \log (x)}{x^2 (-5+9 x)^2} \, dx\\ &=\int \left (-\frac {425}{(-5+9 x)^2}+\frac {25 e^{3-e^x}}{x^2 (-5+9 x)^2}+\frac {125}{x (-5+9 x)^2}-\frac {90 e^{3-e^x}}{x (-5+9 x)^2}+\frac {315 x}{(-5+9 x)^2}+\frac {81 x^2}{(-5+9 x)^2}+\frac {e^{3-e^x+2 x}}{x (-5+9 x)}-\frac {e^{-e^x+x} \left (5 e^3-48 e^3 x+25 e^{e^x} x+54 e^3 x^2-90 e^{e^x} x^2+81 e^{e^x} x^3+25 e^{e^x} x^2 \log (x)-90 e^{e^x} x^3 \log (x)+81 e^{e^x} x^4 \log (x)\right )}{x^2 (-5+9 x)^2}\right ) \, dx\\ &=-\frac {425}{9 (5-9 x)}+25 \int \frac {e^{3-e^x}}{x^2 (-5+9 x)^2} \, dx+81 \int \frac {x^2}{(-5+9 x)^2} \, dx-90 \int \frac {e^{3-e^x}}{x (-5+9 x)^2} \, dx+125 \int \frac {1}{x (-5+9 x)^2} \, dx+315 \int \frac {x}{(-5+9 x)^2} \, dx+\int \frac {e^{3-e^x+2 x}}{x (-5+9 x)} \, dx-\int \frac {e^{-e^x+x} \left (5 e^3-48 e^3 x+25 e^{e^x} x+54 e^3 x^2-90 e^{e^x} x^2+81 e^{e^x} x^3+25 e^{e^x} x^2 \log (x)-90 e^{e^x} x^3 \log (x)+81 e^{e^x} x^4 \log (x)\right )}{x^2 (-5+9 x)^2} \, dx\\ &=-\frac {425}{9 (5-9 x)}+25 \int \left (\frac {e^{3-e^x}}{25 x^2}+\frac {18 e^{3-e^x}}{125 x}+\frac {81 e^{3-e^x}}{25 (-5+9 x)^2}-\frac {162 e^{3-e^x}}{125 (-5+9 x)}\right ) \, dx+81 \int \left (\frac {1}{81}+\frac {25}{81 (-5+9 x)^2}+\frac {10}{81 (-5+9 x)}\right ) \, dx-90 \int \left (\frac {e^{3-e^x}}{25 x}+\frac {9 e^{3-e^x}}{5 (-5+9 x)^2}-\frac {9 e^{3-e^x}}{25 (-5+9 x)}\right ) \, dx+125 \int \left (\frac {1}{25 x}+\frac {9}{5 (-5+9 x)^2}-\frac {9}{25 (-5+9 x)}\right ) \, dx+315 \int \left (\frac {5}{9 (-5+9 x)^2}+\frac {1}{9 (-5+9 x)}\right ) \, dx+\int \left (-\frac {e^{3-e^x+2 x}}{5 x}+\frac {9 e^{3-e^x+2 x}}{5 (-5+9 x)}\right ) \, dx-\int e^x \left (\frac {x+\frac {e^{3-e^x} \left (5-48 x+54 x^2\right )}{(5-9 x)^2}}{x^2}+\log (x)\right ) \, dx\\ &=x+5 \log (x)-\frac {1}{5} \int \frac {e^{3-e^x+2 x}}{x} \, dx+\frac {9}{5} \int \frac {e^{3-e^x+2 x}}{-5+9 x} \, dx+81 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx-162 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx+\int \frac {e^{3-e^x}}{x^2} \, dx-\int \left (\frac {e^{-e^x+x} \left (5 e^3-48 e^3 x+25 e^{e^x} x+54 e^3 x^2-90 e^{e^x} x^2+81 e^{e^x} x^3\right )}{x^2 (-5+9 x)^2}+e^x \log (x)\right ) \, dx\\ &=x+5 \log (x)-\frac {1}{5} \int \frac {e^{3-e^x+2 x}}{x} \, dx+\frac {9}{5} \int \frac {e^{3-e^x+2 x}}{-5+9 x} \, dx+81 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx-162 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx+\int \frac {e^{3-e^x}}{x^2} \, dx-\int \frac {e^{-e^x+x} \left (5 e^3-48 e^3 x+25 e^{e^x} x+54 e^3 x^2-90 e^{e^x} x^2+81 e^{e^x} x^3\right )}{x^2 (-5+9 x)^2} \, dx-\int e^x \log (x) \, dx\\ &=x+5 \log (x)-e^x \log (x)-\frac {1}{5} \int \frac {e^{3-e^x+2 x}}{x} \, dx+\frac {9}{5} \int \frac {e^{3-e^x+2 x}}{-5+9 x} \, dx+81 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx-162 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx+\int \frac {e^{3-e^x}}{x^2} \, dx+\int \frac {e^x}{x} \, dx-\int \frac {e^x \left (x+\frac {e^{3-e^x} \left (5-48 x+54 x^2\right )}{(5-9 x)^2}\right )}{x^2} \, dx\\ &=x+\text {Ei}(x)+5 \log (x)-e^x \log (x)-\frac {1}{5} \int \frac {e^{3-e^x+2 x}}{x} \, dx+\frac {9}{5} \int \frac {e^{3-e^x+2 x}}{-5+9 x} \, dx+81 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx-162 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx+\int \frac {e^{3-e^x}}{x^2} \, dx-\int \left (\frac {e^x}{x}+\frac {e^{3-e^x+x} \left (5-48 x+54 x^2\right )}{x^2 (-5+9 x)^2}\right ) \, dx\\ &=x+\text {Ei}(x)+5 \log (x)-e^x \log (x)-\frac {1}{5} \int \frac {e^{3-e^x+2 x}}{x} \, dx+\frac {9}{5} \int \frac {e^{3-e^x+2 x}}{-5+9 x} \, dx+81 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx-162 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx+\int \frac {e^{3-e^x}}{x^2} \, dx-\int \frac {e^x}{x} \, dx-\int \frac {e^{3-e^x+x} \left (5-48 x+54 x^2\right )}{x^2 (-5+9 x)^2} \, dx\\ &=x+5 \log (x)-e^x \log (x)-\frac {1}{5} \int \frac {e^{3-e^x+2 x}}{x} \, dx+\frac {9}{5} \int \frac {e^{3-e^x+2 x}}{-5+9 x} \, dx+81 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx-162 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx+\int \frac {e^{3-e^x}}{x^2} \, dx-\int \left (\frac {e^{3-e^x+x}}{5 x^2}-\frac {6 e^{3-e^x+x}}{5 x}-\frac {81 e^{3-e^x+x}}{5 (-5+9 x)^2}+\frac {54 e^{3-e^x+x}}{5 (-5+9 x)}\right ) \, dx\\ &=x+5 \log (x)-e^x \log (x)-\frac {1}{5} \int \frac {e^{3-e^x+x}}{x^2} \, dx-\frac {1}{5} \int \frac {e^{3-e^x+2 x}}{x} \, dx+\frac {6}{5} \int \frac {e^{3-e^x+x}}{x} \, dx+\frac {9}{5} \int \frac {e^{3-e^x+2 x}}{-5+9 x} \, dx-\frac {54}{5} \int \frac {e^{3-e^x+x}}{-5+9 x} \, dx+\frac {81}{5} \int \frac {e^{3-e^x+x}}{(-5+9 x)^2} \, dx+81 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx-162 \int \frac {e^{3-e^x}}{(-5+9 x)^2} \, dx+\int \frac {e^{3-e^x}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 39, normalized size = 1.15 \begin {gather*} x-\frac {e^{3-e^x} \left (-5+e^x\right )}{x (-5+9 x)}+5 \log (x)-e^x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(125*x - 425*x^2 + 315*x^3 + 81*x^4 + E^x*(-25*x + 90*x^2 - 81*x^3) + E^(3 - E^x)*(25 - 90*x + E^x*(
-5 + 48*x - 54*x^2) + E^(2*x)*(-5*x + 9*x^2)) + E^x*(-25*x^2 + 90*x^3 - 81*x^4)*Log[x])/(25*x^2 - 90*x^3 + 81*
x^4),x]

[Out]

x - (E^(3 - E^x)*(-5 + E^x))/(x*(-5 + 9*x)) + 5*Log[x] - E^x*Log[x]

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fricas [B]  time = 0.56, size = 61, normalized size = 1.79 \begin {gather*} \frac {9 \, x^{3} - 5 \, x^{2} - {\left (e^{x} - 5\right )} e^{\left (-e^{x} + 3\right )} + {\left (45 \, x^{2} - {\left (9 \, x^{2} - 5 \, x\right )} e^{x} - 25 \, x\right )} \log \relax (x)}{9 \, x^{2} - 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((9*x^2-5*x)*exp(x)^2+(-54*x^2+48*x-5)*exp(x)-90*x+25)*exp(-exp(x)+3)+(-81*x^4+90*x^3-25*x^2)*exp(x
)*log(x)+(-81*x^3+90*x^2-25*x)*exp(x)+81*x^4+315*x^3-425*x^2+125*x)/(81*x^4-90*x^3+25*x^2),x, algorithm="frica
s")

[Out]

(9*x^3 - 5*x^2 - (e^x - 5)*e^(-e^x + 3) + (45*x^2 - (9*x^2 - 5*x)*e^x - 25*x)*log(x))/(9*x^2 - 5*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {81 \, x^{4} + 315 \, x^{3} - {\left (81 \, x^{4} - 90 \, x^{3} + 25 \, x^{2}\right )} e^{x} \log \relax (x) - 425 \, x^{2} - {\left (81 \, x^{3} - 90 \, x^{2} + 25 \, x\right )} e^{x} + {\left ({\left (9 \, x^{2} - 5 \, x\right )} e^{\left (2 \, x\right )} - {\left (54 \, x^{2} - 48 \, x + 5\right )} e^{x} - 90 \, x + 25\right )} e^{\left (-e^{x} + 3\right )} + 125 \, x}{81 \, x^{4} - 90 \, x^{3} + 25 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((9*x^2-5*x)*exp(x)^2+(-54*x^2+48*x-5)*exp(x)-90*x+25)*exp(-exp(x)+3)+(-81*x^4+90*x^3-25*x^2)*exp(x
)*log(x)+(-81*x^3+90*x^2-25*x)*exp(x)+81*x^4+315*x^3-425*x^2+125*x)/(81*x^4-90*x^3+25*x^2),x, algorithm="giac"
)

[Out]

integrate((81*x^4 + 315*x^3 - (81*x^4 - 90*x^3 + 25*x^2)*e^x*log(x) - 425*x^2 - (81*x^3 - 90*x^2 + 25*x)*e^x +
((9*x^2 - 5*x)*e^(2*x) - (54*x^2 - 48*x + 5)*e^x - 90*x + 25)*e^(-e^x + 3) + 125*x)/(81*x^4 - 90*x^3 + 25*x^2
), x)

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maple [A]  time = 0.05, size = 36, normalized size = 1.06

method	result	size
risch	\(-{\mathrm e}^{x} \ln \relax (x )+x +5 \ln \relax (x )-\frac {\left ({\mathrm e}^{x}-5\right ) {\mathrm e}^{-{\mathrm e}^{x}+3}}{x \left (9 x -5\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((9*x^2-5*x)*exp(x)^2+(-54*x^2+48*x-5)*exp(x)-90*x+25)*exp(-exp(x)+3)+(-81*x^4+90*x^3-25*x^2)*exp(x)*ln(x
)+(-81*x^3+90*x^2-25*x)*exp(x)+81*x^4+315*x^3-425*x^2+125*x)/(81*x^4-90*x^3+25*x^2),x,method=_RETURNVERBOSE)

[Out]

-exp(x)*ln(x)+x+5*ln(x)-(exp(x)-5)/x/(9*x-5)*exp(-exp(x)+3)

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maxima [A]  time = 0.49, size = 52, normalized size = 1.53 \begin {gather*} x - \frac {{\left (9 \, x^{2} - 5 \, x\right )} e^{x} \log \relax (x) - {\left (5 \, e^{3} - e^{\left (x + 3\right )}\right )} e^{\left (-e^{x}\right )}}{9 \, x^{2} - 5 \, x} + 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((9*x^2-5*x)*exp(x)^2+(-54*x^2+48*x-5)*exp(x)-90*x+25)*exp(-exp(x)+3)+(-81*x^4+90*x^3-25*x^2)*exp(x
)*log(x)+(-81*x^3+90*x^2-25*x)*exp(x)+81*x^4+315*x^3-425*x^2+125*x)/(81*x^4-90*x^3+25*x^2),x, algorithm="maxim
a")

[Out]

x - ((9*x^2 - 5*x)*e^x*log(x) - (5*e^3 - e^(x + 3))*e^(-e^x))/(9*x^2 - 5*x) + 5*log(x)

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mupad [B]  time = 1.80, size = 45, normalized size = 1.32 \begin {gather*} x+5\,\ln \left (x-\frac {5}{9}\right )+10\,\mathrm {atanh}\left (\frac {18\,x}{5}-1\right )-{\mathrm {e}}^x\,\ln \relax (x)-\frac {{\mathrm {e}}^{3-{\mathrm {e}}^x}\,\left ({\mathrm {e}}^x-5\right )}{x\,\left (9\,x-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3 - exp(x))*(90*x + exp(2*x)*(5*x - 9*x^2) + exp(x)*(54*x^2 - 48*x + 5) - 25) - 125*x + 425*x^2 - 31
5*x^3 - 81*x^4 + exp(x)*(25*x - 90*x^2 + 81*x^3) + exp(x)*log(x)*(25*x^2 - 90*x^3 + 81*x^4))/(25*x^2 - 90*x^3
+ 81*x^4),x)

[Out]

x + 5*log(x - 5/9) + 10*atanh((18*x)/5 - 1) - exp(x)*log(x) - (exp(3 - exp(x))*(exp(x) - 5))/(x*(9*x - 5))

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sympy [A]  time = 0.49, size = 32, normalized size = 0.94 \begin {gather*} x + \frac {\left (5 - e^{x}\right ) e^{3 - e^{x}}}{9 x^{2} - 5 x} - e^{x} \log {\relax (x )} + 5 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((9*x**2-5*x)*exp(x)**2+(-54*x**2+48*x-5)*exp(x)-90*x+25)*exp(-exp(x)+3)+(-81*x**4+90*x**3-25*x**2)
*exp(x)*ln(x)+(-81*x**3+90*x**2-25*x)*exp(x)+81*x**4+315*x**3-425*x**2+125*x)/(81*x**4-90*x**3+25*x**2),x)

[Out]

x + (5 - exp(x))*exp(3 - exp(x))/(9*x**2 - 5*x) - exp(x)*log(x) + 5*log(x)

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