3.101.77 \(\int \frac {e^{-1-x} x (150 x-39 x^2-36 x^3-3 x^4+e (500-500 x-260 x^2-20 x^3))}{(25+12 x+x^2) (25 x+12 x^2+x^3)} \, dx\) [10077]

Optimal. Leaf size=31 \[ \frac {e^{-x} \left (20+\frac {3 x}{e}\right )}{4+\frac {-2 x+(5+x)^2}{x}} \]

[Out]

(3*x/exp(1)+20)/exp(ln(((5+x)^2-2*x)/x+4)+x)

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 1.65, antiderivative size = 510, normalized size of antiderivative = 16.45, number of steps used = 27, number of rules used = 7, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.101, Rules used = {1599, 6873, 6874, 2225, 2208, 2209, 6860} \begin {gather*} -\frac {25 (9-10 e) e^{5+\sqrt {11}} \text {ExpIntegralEi}\left (-x-\sqrt {11}-6\right )}{11 \sqrt {11}}+\frac {25}{11} (9-10 e) e^{5+\sqrt {11}} \text {ExpIntegralEi}\left (-x-\sqrt {11}-6\right )-\frac {3}{22} \left (6+\sqrt {11}\right ) (47-40 e) e^{5+\sqrt {11}} \text {ExpIntegralEi}\left (-x-\sqrt {11}-6\right )+\frac {9 (47-40 e) e^{5+\sqrt {11}} \text {ExpIntegralEi}\left (-x-\sqrt {11}-6\right )}{11 \sqrt {11}}+\frac {1}{22} \left (5 \sqrt {11} (21-20 e)+44 (9-5 e)\right ) e^{5+\sqrt {11}} \text {ExpIntegralEi}\left (-x-\sqrt {11}-6\right )+\frac {25 (9-10 e) e^{5-\sqrt {11}} \text {ExpIntegralEi}\left (-x+\sqrt {11}-6\right )}{11 \sqrt {11}}+\frac {25}{11} (9-10 e) e^{5-\sqrt {11}} \text {ExpIntegralEi}\left (-x+\sqrt {11}-6\right )-\frac {3}{22} \left (6-\sqrt {11}\right ) (47-40 e) e^{5-\sqrt {11}} \text {ExpIntegralEi}\left (-x+\sqrt {11}-6\right )-\frac {9 (47-40 e) e^{5-\sqrt {11}} \text {ExpIntegralEi}\left (-x+\sqrt {11}-6\right )}{11 \sqrt {11}}-\frac {1}{22} \left (5 \sqrt {11} (21-20 e)-44 (9-5 e)\right ) e^{5-\sqrt {11}} \text {ExpIntegralEi}\left (-x+\sqrt {11}-6\right )+3 e^{-x-1}+\frac {25 (9-10 e) e^{-x-1}}{11 \left (x-\sqrt {11}+6\right )}-\frac {3 \left (6-\sqrt {11}\right ) (47-40 e) e^{-x-1}}{22 \left (x-\sqrt {11}+6\right )}+\frac {25 (9-10 e) e^{-x-1}}{11 \left (x+\sqrt {11}+6\right )}-\frac {3 \left (6+\sqrt {11}\right ) (47-40 e) e^{-x-1}}{22 \left (x+\sqrt {11}+6\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-1 - x)*x*(150*x - 39*x^2 - 36*x^3 - 3*x^4 + E*(500 - 500*x - 260*x^2 - 20*x^3)))/((25 + 12*x + x^2)*(
25*x + 12*x^2 + x^3)),x]

[Out]

3*E^(-1 - x) - (3*(6 - Sqrt[11])*(47 - 40*E)*E^(-1 - x))/(22*(6 - Sqrt[11] + x)) + (25*(9 - 10*E)*E^(-1 - x))/
(11*(6 - Sqrt[11] + x)) - (3*(6 + Sqrt[11])*(47 - 40*E)*E^(-1 - x))/(22*(6 + Sqrt[11] + x)) + (25*(9 - 10*E)*E
^(-1 - x))/(11*(6 + Sqrt[11] + x)) + ((5*Sqrt[11]*(21 - 20*E) + 44*(9 - 5*E))*E^(5 + Sqrt[11])*ExpIntegralEi[-
6 - Sqrt[11] - x])/22 + (9*(47 - 40*E)*E^(5 + Sqrt[11])*ExpIntegralEi[-6 - Sqrt[11] - x])/(11*Sqrt[11]) - (3*(
6 + Sqrt[11])*(47 - 40*E)*E^(5 + Sqrt[11])*ExpIntegralEi[-6 - Sqrt[11] - x])/22 + (25*(9 - 10*E)*E^(5 + Sqrt[1
1])*ExpIntegralEi[-6 - Sqrt[11] - x])/11 - (25*(9 - 10*E)*E^(5 + Sqrt[11])*ExpIntegralEi[-6 - Sqrt[11] - x])/(
11*Sqrt[11]) - ((5*Sqrt[11]*(21 - 20*E) - 44*(9 - 5*E))*E^(5 - Sqrt[11])*ExpIntegralEi[-6 + Sqrt[11] - x])/22
- (9*(47 - 40*E)*E^(5 - Sqrt[11])*ExpIntegralEi[-6 + Sqrt[11] - x])/(11*Sqrt[11]) - (3*(6 - Sqrt[11])*(47 - 40
*E)*E^(5 - Sqrt[11])*ExpIntegralEi[-6 + Sqrt[11] - x])/22 + (25*(9 - 10*E)*E^(5 - Sqrt[11])*ExpIntegralEi[-6 +
Sqrt[11] - x])/11 + (25*(9 - 10*E)*E^(5 - Sqrt[11])*ExpIntegralEi[-6 + Sqrt[11] - x])/(11*Sqrt[11])

Rule 1599

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rule 2208

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))), x] - Dist[f*g*n*(Log[F]/(d*(m + 1))), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !TrueQ[$UseGamm
a]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-1-x} \left (150 x-39 x^2-36 x^3-3 x^4+e \left (500-500 x-260 x^2-20 x^3\right )\right )}{\left (25+12 x+x^2\right )^2} \, dx\\ &=\int \frac {e^{-1-x} \left (500 e+50 (3-10 e) x-13 (3+20 e) x^2-4 (9+5 e) x^3-3 x^4\right )}{\left (25+12 x+x^2\right )^2} \, dx\\ &=\int \left (-3 e^{-1-x}+\frac {2 e^{-1-x} (-50 (9-10 e)-3 (47-40 e) x)}{\left (25+12 x+x^2\right )^2}+\frac {e^{-1-x} (111-20 e+4 (9-5 e) x)}{25+12 x+x^2}\right ) \, dx\\ &=2 \int \frac {e^{-1-x} (-50 (9-10 e)-3 (47-40 e) x)}{\left (25+12 x+x^2\right )^2} \, dx-3 \int e^{-1-x} \, dx+\int \frac {e^{-1-x} (111-20 e+4 (9-5 e) x)}{25+12 x+x^2} \, dx\\ &=3 e^{-1-x}+2 \int \left (\frac {50 e^{-1-x} (-9+10 e)}{\left (25+12 x+x^2\right )^2}+\frac {3 e^{-1-x} (-47+40 e) x}{\left (25+12 x+x^2\right )^2}\right ) \, dx+\int \left (\frac {e^{-1-x} \left (4 (9-5 e)+\frac {5 (-21+20 e)}{\sqrt {11}}\right )}{12-2 \sqrt {11}+2 x}+\frac {e^{-1-x} \left (4 (9-5 e)-\frac {5 (-21+20 e)}{\sqrt {11}}\right )}{12+2 \sqrt {11}+2 x}\right ) \, dx\\ &=3 e^{-1-x}+\frac {1}{11} \left (-5 \sqrt {11} (21-20 e)+44 (9-5 e)\right ) \int \frac {e^{-1-x}}{12-2 \sqrt {11}+2 x} \, dx+\frac {1}{11} \left (5 \sqrt {11} (21-20 e)+44 (9-5 e)\right ) \int \frac {e^{-1-x}}{12+2 \sqrt {11}+2 x} \, dx-(6 (47-40 e)) \int \frac {e^{-1-x} x}{\left (25+12 x+x^2\right )^2} \, dx-(100 (9-10 e)) \int \frac {e^{-1-x}}{\left (25+12 x+x^2\right )^2} \, dx\\ &=3 e^{-1-x}+\frac {1}{22} \left (5 \sqrt {11} (21-20 e)+44 (9-5 e)\right ) e^{5+\sqrt {11}} \text {Ei}\left (-6-\sqrt {11}-x\right )-\frac {1}{22} \left (5 \sqrt {11} (21-20 e)-44 (9-5 e)\right ) e^{5-\sqrt {11}} \text {Ei}\left (-6+\sqrt {11}-x\right )-(6 (47-40 e)) \int \left (\frac {\left (-12+2 \sqrt {11}\right ) e^{-1-x}}{22 \left (-12+2 \sqrt {11}-2 x\right )^2}-\frac {3 e^{-1-x}}{11 \sqrt {11} \left (-12+2 \sqrt {11}-2 x\right )}+\frac {\left (-12-2 \sqrt {11}\right ) e^{-1-x}}{22 \left (12+2 \sqrt {11}+2 x\right )^2}-\frac {3 e^{-1-x}}{11 \sqrt {11} \left (12+2 \sqrt {11}+2 x\right )}\right ) \, dx-(100 (9-10 e)) \int \left (\frac {e^{-1-x}}{11 \left (-12+2 \sqrt {11}-2 x\right )^2}+\frac {e^{-1-x}}{22 \sqrt {11} \left (-12+2 \sqrt {11}-2 x\right )}+\frac {e^{-1-x}}{11 \left (12+2 \sqrt {11}+2 x\right )^2}+\frac {e^{-1-x}}{22 \sqrt {11} \left (12+2 \sqrt {11}+2 x\right )}\right ) \, dx\\ &=3 e^{-1-x}+\frac {1}{22} \left (5 \sqrt {11} (21-20 e)+44 (9-5 e)\right ) e^{5+\sqrt {11}} \text {Ei}\left (-6-\sqrt {11}-x\right )-\frac {1}{22} \left (5 \sqrt {11} (21-20 e)-44 (9-5 e)\right ) e^{5-\sqrt {11}} \text {Ei}\left (-6+\sqrt {11}-x\right )+\frac {(18 (47-40 e)) \int \frac {e^{-1-x}}{-12+2 \sqrt {11}-2 x} \, dx}{11 \sqrt {11}}+\frac {(18 (47-40 e)) \int \frac {e^{-1-x}}{12+2 \sqrt {11}+2 x} \, dx}{11 \sqrt {11}}+\frac {1}{11} \left (6 \left (6-\sqrt {11}\right ) (47-40 e)\right ) \int \frac {e^{-1-x}}{\left (-12+2 \sqrt {11}-2 x\right )^2} \, dx+\frac {1}{11} \left (6 \left (6+\sqrt {11}\right ) (47-40 e)\right ) \int \frac {e^{-1-x}}{\left (12+2 \sqrt {11}+2 x\right )^2} \, dx-\frac {1}{11} (100 (9-10 e)) \int \frac {e^{-1-x}}{\left (-12+2 \sqrt {11}-2 x\right )^2} \, dx-\frac {1}{11} (100 (9-10 e)) \int \frac {e^{-1-x}}{\left (12+2 \sqrt {11}+2 x\right )^2} \, dx-\frac {(50 (9-10 e)) \int \frac {e^{-1-x}}{-12+2 \sqrt {11}-2 x} \, dx}{11 \sqrt {11}}-\frac {(50 (9-10 e)) \int \frac {e^{-1-x}}{12+2 \sqrt {11}+2 x} \, dx}{11 \sqrt {11}}\\ &=3 e^{-1-x}-\frac {3 \left (6-\sqrt {11}\right ) (47-40 e) e^{-1-x}}{22 \left (6-\sqrt {11}+x\right )}+\frac {25 (9-10 e) e^{-1-x}}{11 \left (6-\sqrt {11}+x\right )}-\frac {3 \left (6+\sqrt {11}\right ) (47-40 e) e^{-1-x}}{22 \left (6+\sqrt {11}+x\right )}+\frac {25 (9-10 e) e^{-1-x}}{11 \left (6+\sqrt {11}+x\right )}+\frac {1}{22} \left (5 \sqrt {11} (21-20 e)+44 (9-5 e)\right ) e^{5+\sqrt {11}} \text {Ei}\left (-6-\sqrt {11}-x\right )+\frac {9 (47-40 e) e^{5+\sqrt {11}} \text {Ei}\left (-6-\sqrt {11}-x\right )}{11 \sqrt {11}}-\frac {25 (9-10 e) e^{5+\sqrt {11}} \text {Ei}\left (-6-\sqrt {11}-x\right )}{11 \sqrt {11}}-\frac {1}{22} \left (5 \sqrt {11} (21-20 e)-44 (9-5 e)\right ) e^{5-\sqrt {11}} \text {Ei}\left (-6+\sqrt {11}-x\right )-\frac {9 (47-40 e) e^{5-\sqrt {11}} \text {Ei}\left (-6+\sqrt {11}-x\right )}{11 \sqrt {11}}+\frac {25 (9-10 e) e^{5-\sqrt {11}} \text {Ei}\left (-6+\sqrt {11}-x\right )}{11 \sqrt {11}}+\frac {1}{11} \left (3 \left (6-\sqrt {11}\right ) (47-40 e)\right ) \int \frac {e^{-1-x}}{-12+2 \sqrt {11}-2 x} \, dx-\frac {1}{11} \left (3 \left (6+\sqrt {11}\right ) (47-40 e)\right ) \int \frac {e^{-1-x}}{12+2 \sqrt {11}+2 x} \, dx-\frac {1}{11} (50 (9-10 e)) \int \frac {e^{-1-x}}{-12+2 \sqrt {11}-2 x} \, dx+\frac {1}{11} (50 (9-10 e)) \int \frac {e^{-1-x}}{12+2 \sqrt {11}+2 x} \, dx\\ &=3 e^{-1-x}-\frac {3 \left (6-\sqrt {11}\right ) (47-40 e) e^{-1-x}}{22 \left (6-\sqrt {11}+x\right )}+\frac {25 (9-10 e) e^{-1-x}}{11 \left (6-\sqrt {11}+x\right )}-\frac {3 \left (6+\sqrt {11}\right ) (47-40 e) e^{-1-x}}{22 \left (6+\sqrt {11}+x\right )}+\frac {25 (9-10 e) e^{-1-x}}{11 \left (6+\sqrt {11}+x\right )}+\frac {1}{22} \left (5 \sqrt {11} (21-20 e)+44 (9-5 e)\right ) e^{5+\sqrt {11}} \text {Ei}\left (-6-\sqrt {11}-x\right )+\frac {9 (47-40 e) e^{5+\sqrt {11}} \text {Ei}\left (-6-\sqrt {11}-x\right )}{11 \sqrt {11}}-\frac {3}{22} \left (6+\sqrt {11}\right ) (47-40 e) e^{5+\sqrt {11}} \text {Ei}\left (-6-\sqrt {11}-x\right )+\frac {25}{11} (9-10 e) e^{5+\sqrt {11}} \text {Ei}\left (-6-\sqrt {11}-x\right )-\frac {25 (9-10 e) e^{5+\sqrt {11}} \text {Ei}\left (-6-\sqrt {11}-x\right )}{11 \sqrt {11}}-\frac {1}{22} \left (5 \sqrt {11} (21-20 e)-44 (9-5 e)\right ) e^{5-\sqrt {11}} \text {Ei}\left (-6+\sqrt {11}-x\right )-\frac {9 (47-40 e) e^{5-\sqrt {11}} \text {Ei}\left (-6+\sqrt {11}-x\right )}{11 \sqrt {11}}-\frac {3}{22} \left (6-\sqrt {11}\right ) (47-40 e) e^{5-\sqrt {11}} \text {Ei}\left (-6+\sqrt {11}-x\right )+\frac {25}{11} (9-10 e) e^{5-\sqrt {11}} \text {Ei}\left (-6+\sqrt {11}-x\right )+\frac {25 (9-10 e) e^{5-\sqrt {11}} \text {Ei}\left (-6+\sqrt {11}-x\right )}{11 \sqrt {11}}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 2.25, size = 26, normalized size = 0.84 \begin {gather*} \frac {e^{-1-x} x (20 e+3 x)}{25+12 x+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-1 - x)*x*(150*x - 39*x^2 - 36*x^3 - 3*x^4 + E*(500 - 500*x - 260*x^2 - 20*x^3)))/((25 + 12*x +
x^2)*(25*x + 12*x^2 + x^3)),x]

[Out]

(E^(-1 - x)*x*(20*E + 3*x))/(25 + 12*x + x^2)

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.27, size = 513, normalized size = 16.55

method	result	size
risch	\(\frac {x \left (20 \,{\mathrm e}+3 x \right ) {\mathrm e}^{-x -1}}{x^{2}+12 x +25}\)	\(27\)
norman	\(\frac {\left (3 \,{\mathrm e}^{-1} x +20\right ) x \,{\mathrm e}^{-x}}{x^{2}+12 x +25}\)	\(29\)
gosper	\(\frac {\left (20 \,{\mathrm e}+3 x \right ) x \,{\mathrm e}^{-x} {\mathrm e}^{-1}}{x^{2}+12 x +25}\)	\(32\)
default	\({\mathrm e}^{-1} \left (-\frac {75 \,{\mathrm e}^{-x} \left (-6 x -25\right )}{11 \left (x^{2}+12 x +25\right )}-\frac {39 \,{\mathrm e}^{-x} \left (-47 x -150\right )}{22 \left (x^{2}+12 x +25\right )}+\frac {18 \,{\mathrm e}^{-x} \left (-414 x -1175\right )}{11 \left (x^{2}+12 x +25\right )}+3 \,{\mathrm e}^{-x}-\frac {3 \,{\mathrm e}^{-x} \left (-3793 x -10350\right )}{22 \left (x^{2}+12 x +25\right )}+500 \,{\mathrm e} \left (\frac {{\mathrm e}^{-x} \left (-x -6\right )}{22 x^{2}+264 x +550}+\frac {{\mathrm e}^{6+\sqrt {11}} \expIntegral \left (1, x +6+\sqrt {11}\right )}{44}-\frac {\sqrt {11}\, {\mathrm e}^{6+\sqrt {11}} \expIntegral \left (1, x +6+\sqrt {11}\right )}{484}+\frac {{\mathrm e}^{6-\sqrt {11}} \expIntegral \left (1, x +6-\sqrt {11}\right )}{44}+\frac {\sqrt {11}\, {\mathrm e}^{6-\sqrt {11}} \expIntegral \left (1, x +6-\sqrt {11}\right )}{484}\right )+500 \,{\mathrm e} \left (\frac {{\mathrm e}^{-x} \left (-6 x -25\right )}{22 x^{2}+264 x +550}+\frac {3 \,{\mathrm e}^{6+\sqrt {11}} \expIntegral \left (1, x +6+\sqrt {11}\right )}{22}+\frac {5 \sqrt {11}\, {\mathrm e}^{6+\sqrt {11}} \expIntegral \left (1, x +6+\sqrt {11}\right )}{484}+\frac {3 \,{\mathrm e}^{6-\sqrt {11}} \expIntegral \left (1, x +6-\sqrt {11}\right )}{22}-\frac {5 \sqrt {11}\, {\mathrm e}^{6-\sqrt {11}} \expIntegral \left (1, x +6-\sqrt {11}\right )}{484}\right )-260 \,{\mathrm e} \left (\frac {{\mathrm e}^{-x} \left (-47 x -150\right )}{22 x^{2}+264 x +550}+\frac {47 \,{\mathrm e}^{6+\sqrt {11}} \expIntegral \left (1, x +6+\sqrt {11}\right )}{44}+\frac {107 \sqrt {11}\, {\mathrm e}^{6+\sqrt {11}} \expIntegral \left (1, x +6+\sqrt {11}\right )}{484}+\frac {47 \,{\mathrm e}^{6-\sqrt {11}} \expIntegral \left (1, x +6-\sqrt {11}\right )}{44}-\frac {107 \sqrt {11}\, {\mathrm e}^{6-\sqrt {11}} \expIntegral \left (1, x +6-\sqrt {11}\right )}{484}\right )+20 \,{\mathrm e} \left (\frac {{\mathrm e}^{-x} \left (-414 x -1175\right )}{22 x^{2}+264 x +550}+\frac {109 \,{\mathrm e}^{6+\sqrt {11}} \expIntegral \left (1, x +6+\sqrt {11}\right )}{11}+\frac {1291 \sqrt {11}\, {\mathrm e}^{6+\sqrt {11}} \expIntegral \left (1, x +6+\sqrt {11}\right )}{484}+\frac {109 \,{\mathrm e}^{6-\sqrt {11}} \expIntegral \left (1, x +6-\sqrt {11}\right )}{11}-\frac {1291 \sqrt {11}\, {\mathrm e}^{6-\sqrt {11}} \expIntegral \left (1, x +6-\sqrt {11}\right )}{484}\right )\right )\)	\(513\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-20*x^3-260*x^2-500*x+500)*exp(1)-3*x^4-36*x^3-39*x^2+150*x)/(x^3+12*x^2+25*x)/exp(1)/exp(ln((x^2+12*x+2
5)/x)+x),x,method=_RETURNVERBOSE)

[Out]

1/exp(1)*(-75/11*exp(-x)*(-6*x-25)/(x^2+12*x+25)-39/22*exp(-x)*(-47*x-150)/(x^2+12*x+25)+18/11*exp(-x)*(-414*x
-1175)/(x^2+12*x+25)+3*exp(-x)-3/22*exp(-x)*(-3793*x-10350)/(x^2+12*x+25)+500*exp(1)*(1/22*exp(-x)*(-x-6)/(x^2
+12*x+25)+1/44*exp(6+11^(1/2))*Ei(1,x+6+11^(1/2))-1/484*11^(1/2)*exp(6+11^(1/2))*Ei(1,x+6+11^(1/2))+1/44*exp(6
-11^(1/2))*Ei(1,x+6-11^(1/2))+1/484*11^(1/2)*exp(6-11^(1/2))*Ei(1,x+6-11^(1/2)))+500*exp(1)*(1/22*exp(-x)*(-6*
x-25)/(x^2+12*x+25)+3/22*exp(6+11^(1/2))*Ei(1,x+6+11^(1/2))+5/484*11^(1/2)*exp(6+11^(1/2))*Ei(1,x+6+11^(1/2))+
3/22*exp(6-11^(1/2))*Ei(1,x+6-11^(1/2))-5/484*11^(1/2)*exp(6-11^(1/2))*Ei(1,x+6-11^(1/2)))-260*exp(1)*(1/22*ex
p(-x)*(-47*x-150)/(x^2+12*x+25)+47/44*exp(6+11^(1/2))*Ei(1,x+6+11^(1/2))+107/484*11^(1/2)*exp(6+11^(1/2))*Ei(1
,x+6+11^(1/2))+47/44*exp(6-11^(1/2))*Ei(1,x+6-11^(1/2))-107/484*11^(1/2)*exp(6-11^(1/2))*Ei(1,x+6-11^(1/2)))+2
0*exp(1)*(1/22*exp(-x)*(-414*x-1175)/(x^2+12*x+25)+109/11*exp(6+11^(1/2))*Ei(1,x+6+11^(1/2))+1291/484*11^(1/2)
*exp(6+11^(1/2))*Ei(1,x+6+11^(1/2))+109/11*exp(6-11^(1/2))*Ei(1,x+6-11^(1/2))-1291/484*11^(1/2)*exp(6-11^(1/2)
)*Ei(1,x+6-11^(1/2))))

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Maxima [A]
time = 0.30, size = 34, normalized size = 1.10 \begin {gather*} \frac {{\left (3 \, x^{2} + 20 \, x e\right )} e^{\left (-x\right )}}{x^{2} e + 12 \, x e + 25 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^3-260*x^2-500*x+500)*exp(1)-3*x^4-36*x^3-39*x^2+150*x)/(x^3+12*x^2+25*x)/exp(1)/exp(log((x^2
+12*x+25)/x)+x),x, algorithm="maxima")

[Out]

(3*x^2 + 20*x*e)*e^(-x)/(x^2*e + 12*x*e + 25*e)

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Fricas [A]
time = 0.56, size = 30, normalized size = 0.97 \begin {gather*} {\left (3 \, x + 20 \, e\right )} e^{\left (-x - \log \left (\frac {x^{2} + 12 \, x + 25}{x}\right ) - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^3-260*x^2-500*x+500)*exp(1)-3*x^4-36*x^3-39*x^2+150*x)/(x^3+12*x^2+25*x)/exp(1)/exp(log((x^2
+12*x+25)/x)+x),x, algorithm="fricas")

[Out]

(3*x + 20*e)*e^(-x - log((x^2 + 12*x + 25)/x) - 1)

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Sympy [A]
time = 0.09, size = 32, normalized size = 1.03 \begin {gather*} \frac {\left (3 x^{2} + 20 e x\right ) e^{- x}}{e x^{2} + 12 e x + 25 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x**3-260*x**2-500*x+500)*exp(1)-3*x**4-36*x**3-39*x**2+150*x)/(x**3+12*x**2+25*x)/exp(1)/exp(l
n((x**2+12*x+25)/x)+x),x)

[Out]

(3*x**2 + 20*E*x)*exp(-x)/(E*x**2 + 12*E*x + 25*E)

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Giac [A]
time = 0.42, size = 38, normalized size = 1.23 \begin {gather*} \frac {3 \, x^{2} e^{\left (-x\right )} + 20 \, x e^{\left (-x + 1\right )}}{x^{2} e + 12 \, x e + 25 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^3-260*x^2-500*x+500)*exp(1)-3*x^4-36*x^3-39*x^2+150*x)/(x^3+12*x^2+25*x)/exp(1)/exp(log((x^2
+12*x+25)/x)+x),x, algorithm="giac")

[Out]

(3*x^2*e^(-x) + 20*x*e^(-x + 1))/(x^2*e + 12*x*e + 25*e)

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Mupad [B]
time = 7.61, size = 26, normalized size = 0.84 \begin {gather*} \frac {x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-1}\,\left (3\,x+20\,\mathrm {e}\right )}{x^2+12\,x+25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-1)*exp(- x - log((12*x + x^2 + 25)/x))*(exp(1)*(500*x + 260*x^2 + 20*x^3 - 500) - 150*x + 39*x^2 +
36*x^3 + 3*x^4))/(25*x + 12*x^2 + x^3),x)

[Out]

(x*exp(-x)*exp(-1)*(3*x + 20*exp(1)))/(12*x + x^2 + 25)

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