3.1.81 \(\int \frac {e^{3 x} (2 x-x^2)+e^{2 x} (-x+625 x^2)+e^x (x^2+x^3)+2 e^{2 x} x \log (x)}{e^{4 x}+1250 e^{3 x} x+x^2+1250 e^x x^2+e^{2 x} (2 x+390625 x^2)+(2 e^{3 x}+2 e^x x+1250 e^{2 x} x) \log (x)+e^{2 x} \log ^2(x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {x^2}{e^x+625 x+e^{-x} x+\log (x)} \]

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Rubi [F]  time = 6.95, 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^{3 x} \left (2 x-x^2\right )+e^{2 x} \left (-x+625 x^2\right )+e^x \left (x^2+x^3\right )+2 e^{2 x} x \log (x)}{e^{4 x}+1250 e^{3 x} x+x^2+1250 e^x x^2+e^{2 x} \left (2 x+390625 x^2\right )+\left (2 e^{3 x}+2 e^x x+1250 e^{2 x} x\right ) \log (x)+e^{2 x} \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(3*x)*(2*x - x^2) + E^(2*x)*(-x + 625*x^2) + E^x*(x^2 + x^3) + 2*E^(2*x)*x*Log[x])/(E^(4*x) + 1250*E^(3
*x)*x + x^2 + 1250*E^x*x^2 + E^(2*x)*(2*x + 390625*x^2) + (2*E^(3*x) + 2*E^x*x + 1250*E^(2*x)*x)*Log[x] + E^(2
*x)*Log[x]^2),x]

[Out]

-Defer[Int][(E^(2*x)*x)/(E^(2*x) + x + 625*E^x*x + E^x*Log[x])^2, x] - Defer[Int][(E^x*x^2)/(E^(2*x) + x + 625
*E^x*x + E^x*Log[x])^2, x] - 625*Defer[Int][(E^(2*x)*x^2)/(E^(2*x) + x + 625*E^x*x + E^x*Log[x])^2, x] + 2*Def
er[Int][(E^x*x^3)/(E^(2*x) + x + 625*E^x*x + E^x*Log[x])^2, x] + 625*Defer[Int][(E^(2*x)*x^3)/(E^(2*x) + x + 6
25*E^x*x + E^x*Log[x])^2, x] + Defer[Int][(E^(2*x)*x^2*Log[x])/(E^(2*x) + x + 625*E^x*x + E^x*Log[x])^2, x] +
2*Defer[Int][(E^x*x)/(E^(2*x) + x + 625*E^x*x + E^x*Log[x]), x] - Defer[Int][(E^x*x^2)/(E^(2*x) + x + 625*E^x*
x + E^x*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x x \left (-e^x (1-625 x)-e^{2 x} (-2+x)+x (1+x)+2 e^x \log (x)\right )}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx\\ &=\int \left (-\frac {e^x (-2+x) x}{e^{2 x}+x+625 e^x x+e^x \log (x)}+\frac {e^x x \left (-e^x-x-625 e^x x+2 x^2+625 e^x x^2+e^x x \log (x)\right )}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}\right ) \, dx\\ &=-\int \frac {e^x (-2+x) x}{e^{2 x}+x+625 e^x x+e^x \log (x)} \, dx+\int \frac {e^x x \left (-e^x-x-625 e^x x+2 x^2+625 e^x x^2+e^x x \log (x)\right )}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx\\ &=\int \left (-\frac {e^{2 x} x}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}-\frac {e^x x^2}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}-\frac {625 e^{2 x} x^2}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}+\frac {2 e^x x^3}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}+\frac {625 e^{2 x} x^3}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}+\frac {e^{2 x} x^2 \log (x)}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}\right ) \, dx-\int \left (-\frac {2 e^x x}{e^{2 x}+x+625 e^x x+e^x \log (x)}+\frac {e^x x^2}{e^{2 x}+x+625 e^x x+e^x \log (x)}\right ) \, dx\\ &=2 \int \frac {e^x x^3}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx+2 \int \frac {e^x x}{e^{2 x}+x+625 e^x x+e^x \log (x)} \, dx-625 \int \frac {e^{2 x} x^2}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx+625 \int \frac {e^{2 x} x^3}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx-\int \frac {e^{2 x} x}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx-\int \frac {e^x x^2}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx+\int \frac {e^{2 x} x^2 \log (x)}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx-\int \frac {e^x x^2}{e^{2 x}+x+625 e^x x+e^x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.95, size = 28, normalized size = 1.27 \begin {gather*} \frac {e^x x^2}{e^{2 x}+x+625 e^x x+e^x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*x)*(2*x - x^2) + E^(2*x)*(-x + 625*x^2) + E^x*(x^2 + x^3) + 2*E^(2*x)*x*Log[x])/(E^(4*x) + 125
0*E^(3*x)*x + x^2 + 1250*E^x*x^2 + E^(2*x)*(2*x + 390625*x^2) + (2*E^(3*x) + 2*E^x*x + 1250*E^(2*x)*x)*Log[x]
+ E^(2*x)*Log[x]^2),x]

[Out]

(E^x*x^2)/(E^(2*x) + x + 625*E^x*x + E^x*Log[x])

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fricas [A]  time = 0.54, size = 24, normalized size = 1.09 \begin {gather*} \frac {x^{2} e^{x}}{625 \, x e^{x} + e^{x} \log \relax (x) + x + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)^2*log(x)+(-x^2+2*x)*exp(x)^3+(625*x^2-x)*exp(x)^2+(x^3+x^2)*exp(x))/(exp(x)^2*log(x)^2+(
2*exp(x)^3+1250*x*exp(x)^2+2*exp(x)*x)*log(x)+exp(x)^4+1250*x*exp(x)^3+(390625*x^2+2*x)*exp(x)^2+1250*exp(x)*x
^2+x^2),x, algorithm="fricas")

[Out]

x^2*e^x/(625*x*e^x + e^x*log(x) + x + e^(2*x))

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giac [A]  time = 0.48, size = 24, normalized size = 1.09 \begin {gather*} \frac {x^{2} e^{x}}{625 \, x e^{x} + e^{x} \log \relax (x) + x + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)^2*log(x)+(-x^2+2*x)*exp(x)^3+(625*x^2-x)*exp(x)^2+(x^3+x^2)*exp(x))/(exp(x)^2*log(x)^2+(
2*exp(x)^3+1250*x*exp(x)^2+2*exp(x)*x)*log(x)+exp(x)^4+1250*x*exp(x)^3+(390625*x^2+2*x)*exp(x)^2+1250*exp(x)*x
^2+x^2),x, algorithm="giac")

[Out]

x^2*e^x/(625*x*e^x + e^x*log(x) + x + e^(2*x))

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maple [A]  time = 0.03, size = 25, normalized size = 1.14

method	result	size
risch	\(\frac {{\mathrm e}^{x} x^{2}}{{\mathrm e}^{x} \ln \relax (x )+{\mathrm e}^{2 x}+625 \,{\mathrm e}^{x} x +x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(x)^2*ln(x)+(-x^2+2*x)*exp(x)^3+(625*x^2-x)*exp(x)^2+(x^3+x^2)*exp(x))/(exp(x)^2*ln(x)^2+(2*exp(x)
^3+1250*x*exp(x)^2+2*exp(x)*x)*ln(x)+exp(x)^4+1250*x*exp(x)^3+(390625*x^2+2*x)*exp(x)^2+1250*exp(x)*x^2+x^2),x
,method=_RETURNVERBOSE)

[Out]

exp(x)*x^2/(exp(x)*ln(x)+exp(2*x)+625*exp(x)*x+x)

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maxima [A]  time = 0.81, size = 23, normalized size = 1.05 \begin {gather*} \frac {x^{2} e^{x}}{{\left (625 \, x + \log \relax (x)\right )} e^{x} + x + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)^2*log(x)+(-x^2+2*x)*exp(x)^3+(625*x^2-x)*exp(x)^2+(x^3+x^2)*exp(x))/(exp(x)^2*log(x)^2+(
2*exp(x)^3+1250*x*exp(x)^2+2*exp(x)*x)*log(x)+exp(x)^4+1250*x*exp(x)^3+(390625*x^2+2*x)*exp(x)^2+1250*exp(x)*x
^2+x^2),x, algorithm="maxima")

[Out]

x^2*e^x/((625*x + log(x))*e^x + x + e^(2*x))

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mupad [B]  time = 0.55, size = 98, normalized size = 4.45 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}\,\left (x^4-x^5\right )+x^3\,{\mathrm {e}}^{3\,x}+625\,x^4\,{\mathrm {e}}^{3\,x}+x^4\,{\mathrm {e}}^{4\,x}}{\left (x+{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\ln \relax (x)+625\,x\,{\mathrm {e}}^x\right )\,\left (x\,{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^x-x^3\,{\mathrm {e}}^x+625\,x^2\,{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^{3\,x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3*x)*(2*x - x^2) + exp(x)*(x^2 + x^3) - exp(2*x)*(x - 625*x^2) + 2*x*exp(2*x)*log(x))/(exp(4*x) + log
(x)*(2*exp(3*x) + 1250*x*exp(2*x) + 2*x*exp(x)) + exp(2*x)*(2*x + 390625*x^2) + 1250*x*exp(3*x) + 1250*x^2*exp
(x) + x^2 + exp(2*x)*log(x)^2),x)

[Out]

(exp(2*x)*(x^4 - x^5) + x^3*exp(3*x) + 625*x^4*exp(3*x) + x^4*exp(4*x))/((x + exp(2*x) + exp(x)*log(x) + 625*x
*exp(x))*(x*exp(2*x) + x^2*exp(x) - x^3*exp(x) + 625*x^2*exp(2*x) + x^2*exp(3*x)))

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sympy [A]  time = 0.32, size = 22, normalized size = 1.00 \begin {gather*} \frac {x^{2} e^{x}}{x + \left (625 x + \log {\relax (x )}\right ) e^{x} + e^{2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)**2*ln(x)+(-x**2+2*x)*exp(x)**3+(625*x**2-x)*exp(x)**2+(x**3+x**2)*exp(x))/(exp(x)**2*ln(
x)**2+(2*exp(x)**3+1250*x*exp(x)**2+2*exp(x)*x)*ln(x)+exp(x)**4+1250*x*exp(x)**3+(390625*x**2+2*x)*exp(x)**2+1
250*exp(x)*x**2+x**2),x)

[Out]

x**2*exp(x)/(x + (625*x + log(x))*exp(x) + exp(2*x))

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