3.52.13 \(\int \frac {36 x^6-36 x^7+8 x^8+e^e (-64+180 x^6-144 x^7+4 x^8+8 x^9)+e^{2 e} (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10})}{32 x^5+e^e (160 x^5+32 x^6)+e^{2 e} (200 x^5+80 x^6+8 x^7)} \, dx\)

Optimal. Leaf size=29 \[ \frac {1}{16} (-3+x)^2 x^2+\frac {1}{x^4 \left (5+2 e^{-e}+x\right )} \]

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Rubi [B]  time = 0.33, antiderivative size = 123, normalized size of antiderivative = 4.24, number of steps used = 2, number of rules used = 1, integrand size = 124, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2074} \begin {gather*} \frac {x^4}{16}+\frac {e^e}{\left (2+5 e^e\right ) x^4}-\frac {3 x^3}{8}-\frac {e^{2 e}}{\left (2+5 e^e\right )^2 x^3}+\frac {9 x^2}{16}+\frac {e^{3 e}}{\left (2+5 e^e\right )^3 x^2}+\frac {e^{5 e}}{\left (2+5 e^e\right )^4 \left (e^e x+5 e^e+2\right )}-\frac {e^{4 e}}{\left (2+5 e^e\right )^4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36*x^6 - 36*x^7 + 8*x^8 + E^E*(-64 + 180*x^6 - 144*x^7 + 4*x^8 + 8*x^9) + E^(2*E)*(-160 - 40*x + 225*x^6
- 135*x^7 - 31*x^8 + 11*x^9 + 2*x^10))/(32*x^5 + E^E*(160*x^5 + 32*x^6) + E^(2*E)*(200*x^5 + 80*x^6 + 8*x^7)),
x]

[Out]

E^E/((2 + 5*E^E)*x^4) - E^(2*E)/((2 + 5*E^E)^2*x^3) + E^(3*E)/((2 + 5*E^E)^3*x^2) - E^(4*E)/((2 + 5*E^E)^4*x)
+ (9*x^2)/16 - (3*x^3)/8 + x^4/16 + E^(5*E)/((2 + 5*E^E)^4*(2 + 5*E^E + E^E*x))

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4 e^e}{\left (2+5 e^e\right ) x^5}+\frac {3 e^{2 e}}{\left (2+5 e^e\right )^2 x^4}-\frac {2 e^{3 e}}{\left (2+5 e^e\right )^3 x^3}+\frac {e^{4 e}}{\left (2+5 e^e\right )^4 x^2}+\frac {9 x}{8}-\frac {9 x^2}{8}+\frac {x^3}{4}-\frac {e^{6 e}}{\left (2+5 e^e\right )^4 \left (2+5 e^e+e^e x\right )^2}\right ) \, dx\\ &=\frac {e^e}{\left (2+5 e^e\right ) x^4}-\frac {e^{2 e}}{\left (2+5 e^e\right )^2 x^3}+\frac {e^{3 e}}{\left (2+5 e^e\right )^3 x^2}-\frac {e^{4 e}}{\left (2+5 e^e\right )^4 x}+\frac {9 x^2}{16}-\frac {3 x^3}{8}+\frac {x^4}{16}+\frac {e^{5 e}}{\left (2+5 e^e\right )^4 \left (2+5 e^e+e^e x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 53, normalized size = 1.83 \begin {gather*} \frac {2 (-3+x)^2 x^6+e^e \left (16+45 x^6-21 x^7-x^8+x^9\right )}{16 x^4 \left (2+e^e (5+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36*x^6 - 36*x^7 + 8*x^8 + E^E*(-64 + 180*x^6 - 144*x^7 + 4*x^8 + 8*x^9) + E^(2*E)*(-160 - 40*x + 22
5*x^6 - 135*x^7 - 31*x^8 + 11*x^9 + 2*x^10))/(32*x^5 + E^E*(160*x^5 + 32*x^6) + E^(2*E)*(200*x^5 + 80*x^6 + 8*
x^7)),x]

[Out]

(2*(-3 + x)^2*x^6 + E^E*(16 + 45*x^6 - 21*x^7 - x^8 + x^9))/(16*x^4*(2 + E^E*(5 + x)))

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fricas [B]  time = 0.43, size = 63, normalized size = 2.17 \begin {gather*} \frac {2 \, x^{8} - 12 \, x^{7} + 18 \, x^{6} + {\left (x^{9} - x^{8} - 21 \, x^{7} + 45 \, x^{6} + 16\right )} e^{e}}{16 \, {\left (2 \, x^{4} + {\left (x^{5} + 5 \, x^{4}\right )} e^{e}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^10+11*x^9-31*x^8-135*x^7+225*x^6-40*x-160)*exp(exp(1))^2+(8*x^9+4*x^8-144*x^7+180*x^6-64)*exp(
exp(1))+8*x^8-36*x^7+36*x^6)/((8*x^7+80*x^6+200*x^5)*exp(exp(1))^2+(32*x^6+160*x^5)*exp(exp(1))+32*x^5),x, alg
orithm="fricas")

[Out]

1/16*(2*x^8 - 12*x^7 + 18*x^6 + (x^9 - x^8 - 21*x^7 + 45*x^6 + 16)*e^e)/(2*x^4 + (x^5 + 5*x^4)*e^e)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^10+11*x^9-31*x^8-135*x^7+225*x^6-40*x-160)*exp(exp(1))^2+(8*x^9+4*x^8-144*x^7+180*x^6-64)*exp(
exp(1))+8*x^8-36*x^7+36*x^6)/((8*x^7+80*x^6+200*x^5)*exp(exp(1))^2+(32*x^6+160*x^5)*exp(exp(1))+32*x^5),x, alg
orithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 1/8*((1/2*sageVARx^4*exp(2*exp(1))^4-3*s
ageVARx^3*exp(2*exp(1))^4+9/2*sageVARx^2*exp(2*exp(1))^4)/exp(2*exp(1))^4+((-375000*exp(2*exp(1))^5-900000*exp
(2*exp(1))^4*exp(exp(

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maple [A]  time = 0.35, size = 38, normalized size = 1.31

method	result	size
risch	\(\frac {x^{4}}{16}-\frac {3 x^{3}}{8}+\frac {9 x^{2}}{16}+\frac {{\mathrm e}^{{\mathrm e}}}{x^{4} \left (x \,{\mathrm e}^{{\mathrm e}}+5 \,{\mathrm e}^{{\mathrm e}}+2\right )}\)	\(38\)
norman	\(\frac {\left (-\frac {21 \,{\mathrm e}^{{\mathrm e}}}{16}-\frac {3}{4}\right ) x^{7}+\left (-\frac {{\mathrm e}^{{\mathrm e}}}{16}+\frac {1}{8}\right ) x^{8}+\left (\frac {45 \,{\mathrm e}^{{\mathrm e}}}{16}+\frac {9}{8}\right ) x^{6}+\frac {{\mathrm e}^{{\mathrm e}} x^{9}}{16}+{\mathrm e}^{{\mathrm e}}}{x^{4} \left (x \,{\mathrm e}^{{\mathrm e}}+5 \,{\mathrm e}^{{\mathrm e}}+2\right )}\)	\(64\)
gosper	\(\frac {{\mathrm e}^{{\mathrm e}} x^{9}-{\mathrm e}^{{\mathrm e}} x^{8}-21 \,{\mathrm e}^{{\mathrm e}} x^{7}+2 x^{8}+45 \,{\mathrm e}^{{\mathrm e}} x^{6}-12 x^{7}+18 x^{6}+16 \,{\mathrm e}^{{\mathrm e}}}{16 x^{4} \left (x \,{\mathrm e}^{{\mathrm e}}+5 \,{\mathrm e}^{{\mathrm e}}+2\right )}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^10+11*x^9-31*x^8-135*x^7+225*x^6-40*x-160)*exp(exp(1))^2+(8*x^9+4*x^8-144*x^7+180*x^6-64)*exp(exp(1)
)+8*x^8-36*x^7+36*x^6)/((8*x^7+80*x^6+200*x^5)*exp(exp(1))^2+(32*x^6+160*x^5)*exp(exp(1))+32*x^5),x,method=_RE
TURNVERBOSE)

[Out]

1/16*x^4-3/8*x^3+9/16*x^2+exp(exp(1))/x^4/(x*exp(exp(1))+5*exp(exp(1))+2)

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maxima [A]  time = 0.35, size = 41, normalized size = 1.41 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {3}{8} \, x^{3} + \frac {9}{16} \, x^{2} + \frac {e^{e}}{x^{5} e^{e} + x^{4} {\left (5 \, e^{e} + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^10+11*x^9-31*x^8-135*x^7+225*x^6-40*x-160)*exp(exp(1))^2+(8*x^9+4*x^8-144*x^7+180*x^6-64)*exp(
exp(1))+8*x^8-36*x^7+36*x^6)/((8*x^7+80*x^6+200*x^5)*exp(exp(1))^2+(32*x^6+160*x^5)*exp(exp(1))+32*x^5),x, alg
orithm="maxima")

[Out]

1/16*x^4 - 3/8*x^3 + 9/16*x^2 + e^e/(x^5*e^e + x^4*(5*e^e + 2))

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mupad [B]  time = 3.37, size = 314, normalized size = 10.83 \begin {gather*} x^2\,\left (\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (4\,{\mathrm {e}}^{\mathrm {e}}-31\,{\mathrm {e}}^{2\,\mathrm {e}}+8\right )}{16}-\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,{\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}^2}{8}+{\mathrm {e}}^{-\mathrm {e}}\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{2}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{8}\right )\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\right )+\frac {{\mathrm {e}}^{\mathrm {e}}}{{\mathrm {e}}^{\mathrm {e}}\,x^5+\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\,x^4}-x^3\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{6}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{24}\right )+\frac {x^4}{16}-x\,\left (\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (135\,{\mathrm {e}}^{2\,\mathrm {e}}+144\,{\mathrm {e}}^{\mathrm {e}}+36\right )}{8}+2\,{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\,\left (\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (4\,{\mathrm {e}}^{\mathrm {e}}-31\,{\mathrm {e}}^{2\,\mathrm {e}}+8\right )}{8}-\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,{\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}^2}{4}+2\,{\mathrm {e}}^{-\mathrm {e}}\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{2}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{8}\right )\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\right )-{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{2}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{8}\right )\,{\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(1))*(180*x^6 - 144*x^7 + 4*x^8 + 8*x^9 - 64) - exp(2*exp(1))*(40*x - 225*x^6 + 135*x^7 + 31*x^8 -
11*x^9 - 2*x^10 + 160) + 36*x^6 - 36*x^7 + 8*x^8)/(exp(2*exp(1))*(200*x^5 + 80*x^6 + 8*x^7) + 32*x^5 + exp(ex
p(1))*(160*x^5 + 32*x^6)),x)

[Out]

x^2*((exp(-2*exp(1))*(4*exp(exp(1)) - 31*exp(2*exp(1)) + 8))/16 - (exp(-2*exp(1))*(5*exp(exp(1)) + 2)^2)/8 + e
xp(-exp(1))*((exp(-exp(1))*(5*exp(exp(1)) + 2))/2 - (exp(-exp(1))*(11*exp(exp(1)) + 8))/8)*(5*exp(exp(1)) + 2)
) + exp(exp(1))/(x^5*exp(exp(1)) + x^4*(5*exp(exp(1)) + 2)) - x^3*((exp(-exp(1))*(5*exp(exp(1)) + 2))/6 - (exp
(-exp(1))*(11*exp(exp(1)) + 8))/24) + x^4/16 - x*((exp(-2*exp(1))*(135*exp(2*exp(1)) + 144*exp(exp(1)) + 36))/
8 + 2*exp(-exp(1))*(5*exp(exp(1)) + 2)*((exp(-2*exp(1))*(4*exp(exp(1)) - 31*exp(2*exp(1)) + 8))/8 - (exp(-2*ex
p(1))*(5*exp(exp(1)) + 2)^2)/4 + 2*exp(-exp(1))*((exp(-exp(1))*(5*exp(exp(1)) + 2))/2 - (exp(-exp(1))*(11*exp(
exp(1)) + 8))/8)*(5*exp(exp(1)) + 2)) - exp(-2*exp(1))*((exp(-exp(1))*(5*exp(exp(1)) + 2))/2 - (exp(-exp(1))*(
11*exp(exp(1)) + 8))/8)*(5*exp(exp(1)) + 2)^2)

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sympy [A]  time = 0.96, size = 42, normalized size = 1.45 \begin {gather*} \frac {x^{4}}{16} - \frac {3 x^{3}}{8} + \frac {9 x^{2}}{16} + \frac {e^{e}}{x^{5} e^{e} + x^{4} \left (2 + 5 e^{e}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**10+11*x**9-31*x**8-135*x**7+225*x**6-40*x-160)*exp(exp(1))**2+(8*x**9+4*x**8-144*x**7+180*x**
6-64)*exp(exp(1))+8*x**8-36*x**7+36*x**6)/((8*x**7+80*x**6+200*x**5)*exp(exp(1))**2+(32*x**6+160*x**5)*exp(exp
(1))+32*x**5),x)

[Out]

x**4/16 - 3*x**3/8 + 9*x**2/16 + exp(E)/(x**5*exp(E) + x**4*(2 + 5*exp(E)))

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