3.14.40 \(\int \frac {162 x+e^{e (20 x-20 x^2+5 x^3)} (x^2+e (-20 x^3+40 x^4-15 x^5))}{6561+162 e^{e (20 x-20 x^2+5 x^3)} x+e^{2 e (20 x-20 x^2+5 x^3)} x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {x}{e^{5 e (2-x)^2 x}+\frac {81}{x}} \]

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Rubi [F]  time = 1.93, 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 {162 x+e^{e \left (20 x-20 x^2+5 x^3\right )} \left (x^2+e \left (-20 x^3+40 x^4-15 x^5\right )\right )}{6561+162 e^{e \left (20 x-20 x^2+5 x^3\right )} x+e^{2 e \left (20 x-20 x^2+5 x^3\right )} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(162*x + E^(E*(20*x - 20*x^2 + 5*x^3))*(x^2 + E*(-20*x^3 + 40*x^4 - 15*x^5)))/(6561 + 162*E^(E*(20*x - 20*
x^2 + 5*x^3))*x + E^(2*E*(20*x - 20*x^2 + 5*x^3))*x^2),x]

[Out]

81*Defer[Int][x/(81 + E^(5*E*(-2 + x)^2*x)*x)^2, x] + 1620*E*Defer[Int][x^2/(81 + E^(5*E*(-2 + x)^2*x)*x)^2, x
] - 3240*E*Defer[Int][x^3/(81 + E^(5*E*(-2 + x)^2*x)*x)^2, x] + 1215*E*Defer[Int][x^4/(81 + E^(5*E*(-2 + x)^2*
x)*x)^2, x] + Defer[Int][x/(81 + E^(5*E*(-2 + x)^2*x)*x), x] - 20*E*Defer[Int][x^2/(81 + E^(5*E*(-2 + x)^2*x)*
x), x] + 40*E*Defer[Int][x^3/(81 + E^(5*E*(-2 + x)^2*x)*x), x] - 15*E*Defer[Int][x^4/(81 + E^(5*E*(-2 + x)^2*x
)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (162+e^{5 e (-2+x)^2 x} x-5 e^{1+5 e (-2+x)^2 x} x^2 \left (4-8 x+3 x^2\right )\right )}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx\\ &=\int \left (-\frac {x \left (-1+20 e x-40 e x^2+15 e x^3\right )}{81+e^{5 e (-2+x)^2 x} x}+\frac {81 x \left (1+20 e x-40 e x^2+15 e x^3\right )}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2}\right ) \, dx\\ &=81 \int \frac {x \left (1+20 e x-40 e x^2+15 e x^3\right )}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx-\int \frac {x \left (-1+20 e x-40 e x^2+15 e x^3\right )}{81+e^{5 e (-2+x)^2 x} x} \, dx\\ &=81 \int \left (\frac {x}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2}+\frac {20 e x^2}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2}-\frac {40 e x^3}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2}+\frac {15 e x^4}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2}\right ) \, dx-\int \left (-\frac {x}{81+e^{5 e (-2+x)^2 x} x}+\frac {20 e x^2}{81+e^{5 e (-2+x)^2 x} x}-\frac {40 e x^3}{81+e^{5 e (-2+x)^2 x} x}+\frac {15 e x^4}{81+e^{5 e (-2+x)^2 x} x}\right ) \, dx\\ &=81 \int \frac {x}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx-(15 e) \int \frac {x^4}{81+e^{5 e (-2+x)^2 x} x} \, dx-(20 e) \int \frac {x^2}{81+e^{5 e (-2+x)^2 x} x} \, dx+(40 e) \int \frac {x^3}{81+e^{5 e (-2+x)^2 x} x} \, dx+(1215 e) \int \frac {x^4}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx+(1620 e) \int \frac {x^2}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx-(3240 e) \int \frac {x^3}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx+\int \frac {x}{81+e^{5 e (-2+x)^2 x} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.50, size = 21, normalized size = 0.91 \begin {gather*} \frac {x^2}{81+e^{5 e (-2+x)^2 x} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(162*x + E^(E*(20*x - 20*x^2 + 5*x^3))*(x^2 + E*(-20*x^3 + 40*x^4 - 15*x^5)))/(6561 + 162*E^(E*(20*x
- 20*x^2 + 5*x^3))*x + E^(2*E*(20*x - 20*x^2 + 5*x^3))*x^2),x]

[Out]

x^2/(81 + E^(5*E*(-2 + x)^2*x)*x)

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fricas [A]  time = 1.92, size = 27, normalized size = 1.17 \begin {gather*} \frac {x^{2}}{x e^{\left (5 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e\right )} + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x^5+40*x^4-20*x^3)*exp(1)+x^2)*exp((5*x^3-20*x^2+20*x)*exp(1))+162*x)/(x^2*exp((5*x^3-20*x^2+
20*x)*exp(1))^2+162*x*exp((5*x^3-20*x^2+20*x)*exp(1))+6561),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.84, size = 31, normalized size = 1.35 \begin {gather*} \frac {x^{2}}{x e^{\left (5 \, x^{3} e - 20 \, x^{2} e + 20 \, x e\right )} + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x^5+40*x^4-20*x^3)*exp(1)+x^2)*exp((5*x^3-20*x^2+20*x)*exp(1))+162*x)/(x^2*exp((5*x^3-20*x^2+
20*x)*exp(1))^2+162*x*exp((5*x^3-20*x^2+20*x)*exp(1))+6561),x, algorithm="giac")

[Out]

x^2/(x*e^(5*x^3*e - 20*x^2*e + 20*x*e) + 81)

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maple [A]  time = 0.12, size = 22, normalized size = 0.96

method	result	size
risch	\(\frac {x^{2}}{x \,{\mathrm e}^{5 x \left (x -2\right )^{2} {\mathrm e}}+81}\)	\(22\)
norman	\(\frac {x^{2}}{x \,{\mathrm e}^{\left (5 x^{3}-20 x^{2}+20 x \right ) {\mathrm e}}+81}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-15*x^5+40*x^4-20*x^3)*exp(1)+x^2)*exp((5*x^3-20*x^2+20*x)*exp(1))+162*x)/(x^2*exp((5*x^3-20*x^2+20*x)*
exp(1))^2+162*x*exp((5*x^3-20*x^2+20*x)*exp(1))+6561),x,method=_RETURNVERBOSE)

[Out]

x^2/(x*exp(5*x*(x-2)^2*exp(1))+81)

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maxima [A]  time = 0.70, size = 41, normalized size = 1.78 \begin {gather*} \frac {x^{2} e^{\left (20 \, x^{2} e\right )}}{x e^{\left (5 \, x^{3} e + 20 \, x e\right )} + 81 \, e^{\left (20 \, x^{2} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x^5+40*x^4-20*x^3)*exp(1)+x^2)*exp((5*x^3-20*x^2+20*x)*exp(1))+162*x)/(x^2*exp((5*x^3-20*x^2+
20*x)*exp(1))^2+162*x*exp((5*x^3-20*x^2+20*x)*exp(1))+6561),x, algorithm="maxima")

[Out]

x^2*e^(20*x^2*e)/(x*e^(5*x^3*e + 20*x*e) + 81*e^(20*x^2*e))

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mupad [B]  time = 0.26, size = 32, normalized size = 1.39 \begin {gather*} \frac {x^2}{x\,{\mathrm {e}}^{5\,x^3\,\mathrm {e}}\,{\mathrm {e}}^{-20\,x^2\,\mathrm {e}}\,{\mathrm {e}}^{20\,x\,\mathrm {e}}+81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((162*x - exp(exp(1)*(20*x - 20*x^2 + 5*x^3))*(exp(1)*(20*x^3 - 40*x^4 + 15*x^5) - x^2))/(x^2*exp(2*exp(1)*
(20*x - 20*x^2 + 5*x^3)) + 162*x*exp(exp(1)*(20*x - 20*x^2 + 5*x^3)) + 6561),x)

[Out]

x^2/(x*exp(5*x^3*exp(1))*exp(-20*x^2*exp(1))*exp(20*x*exp(1)) + 81)

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sympy [A]  time = 0.24, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^{2}}{x e^{e \left (5 x^{3} - 20 x^{2} + 20 x\right )} + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x**5+40*x**4-20*x**3)*exp(1)+x**2)*exp((5*x**3-20*x**2+20*x)*exp(1))+162*x)/(x**2*exp((5*x**3
-20*x**2+20*x)*exp(1))**2+162*x*exp((5*x**3-20*x**2+20*x)*exp(1))+6561),x)

[Out]

x**2/(x*exp(E*(5*x**3 - 20*x**2 + 20*x)) + 81)

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