Integrand size = 69, antiderivative size = 31 \[ \int \frac {e^{-1-x} x \left (150 x-39 x^2-36 x^3-3 x^4+e \left (500-500 x-260 x^2-20 x^3\right )\right )}{\left (25+12 x+x^2\right ) \left (25 x+12 x^2+x^3\right )} \, dx=\frac {e^{-x} \left (20+\frac {3 x}{e}\right )}{4+\frac {-2 x+(5+x)^2}{x}} \] Output:
(3*x/exp(1)+20)/exp(ln(((5+x)^2-2*x)/x+4)+x)
Time = 2.98 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-1-x} x \left (150 x-39 x^2-36 x^3-3 x^4+e \left (500-500 x-260 x^2-20 x^3\right )\right )}{\left (25+12 x+x^2\right ) \left (25 x+12 x^2+x^3\right )} \, dx=\frac {e^{-1-x} x (20 e+3 x)}{25+12 x+x^2} \] Input:
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]
Output:
(E^(-1 - x)*x*(20*E + 3*x))/(25 + 12*x + x^2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.69 (sec) , antiderivative size = 510, normalized size of antiderivative = 16.45, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {9, 7292, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{-x-1} x \left (-3 x^4-36 x^3-39 x^2+e \left (-20 x^3-260 x^2-500 x+500\right )+150 x\right )}{\left (x^2+12 x+25\right ) \left (x^3+12 x^2+25 x\right )} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {e^{-x-1} \left (-3 x^4-36 x^3-39 x^2+20 e \left (-x^3-13 x^2-25 x+25\right )+150 x\right )}{\left (x^2+12 x+25\right )^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{-x-1} \left (-3 x^4-4 (9+5 e) x^3-13 (3+20 e) x^2+50 (3-10 e) x+500 e\right )}{\left (x^2+12 x+25\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 e^{-x-1} (-3 (47-40 e) x-50 (9-10 e))}{\left (x^2+12 x+25\right )^2}+\frac {e^{-x-1} (4 (9-5 e) x-20 e+111)}{x^2+12 x+25}-3 e^{-x-1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {25 (9-10 e) e^{5+\sqrt {11}} \operatorname {ExpIntegralEi}\left (-x-\sqrt {11}-6\right )}{11 \sqrt {11}}+\frac {25}{11} (9-10 e) e^{5+\sqrt {11}} \operatorname {ExpIntegralEi}\left (-x-\sqrt {11}-6\right )-\frac {3}{22} \left (6+\sqrt {11}\right ) (47-40 e) e^{5+\sqrt {11}} \operatorname {ExpIntegralEi}\left (-x-\sqrt {11}-6\right )+\frac {9 (47-40 e) e^{5+\sqrt {11}} \operatorname {ExpIntegralEi}\left (-x-\sqrt {11}-6\right )}{11 \sqrt {11}}+\frac {1}{22} \left (5 \sqrt {11} (21-20 e)+44 (9-5 e)\right ) e^{5+\sqrt {11}} \operatorname {ExpIntegralEi}\left (-x-\sqrt {11}-6\right )+\frac {25 (9-10 e) e^{5-\sqrt {11}} \operatorname {ExpIntegralEi}\left (-x+\sqrt {11}-6\right )}{11 \sqrt {11}}+\frac {25}{11} (9-10 e) e^{5-\sqrt {11}} \operatorname {ExpIntegralEi}\left (-x+\sqrt {11}-6\right )-\frac {3}{22} \left (6-\sqrt {11}\right ) (47-40 e) e^{5-\sqrt {11}} \operatorname {ExpIntegralEi}\left (-x+\sqrt {11}-6\right )-\frac {9 (47-40 e) e^{5-\sqrt {11}} \operatorname {ExpIntegralEi}\left (-x+\sqrt {11}-6\right )}{11 \sqrt {11}}-\frac {1}{22} \left (5 \sqrt {11} (21-20 e)-44 (9-5 e)\right ) e^{5-\sqrt {11}} \operatorname {ExpIntegralEi}\left (-x+\sqrt {11}-6\right )+3 e^{-x-1}+\frac {25 (9-10 e) e^{-x-1}}{11 \left (x-\sqrt {11}+6\right )}-\frac {3 \left (6-\sqrt {11}\right ) (47-40 e) e^{-x-1}}{22 \left (x-\sqrt {11}+6\right )}+\frac {25 (9-10 e) e^{-x-1}}{11 \left (x+\sqrt {11}+6\right )}-\frac {3 \left (6+\sqrt {11}\right ) (47-40 e) e^{-x-1}}{22 \left (x+\sqrt {11}+6\right )}\) |
Input:
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]
Output:
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])/2 2 + (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*Sqr t[11]) - ((5*Sqrt[11]*(21 - 20*E) - 44*(9 - 5*E))*E^(5 - Sqrt[11])*ExpInte gralEi[-6 + Sqrt[11] - x])/22 - (9*(47 - 40*E)*E^(5 - Sqrt[11])*ExpIntegra lEi[-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 - Sqr t[11])*ExpIntegralEi[-6 + Sqrt[11] - x])/(11*Sqrt[11])
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Time = 0.60 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87
method | result | size |
risch | \(\frac {x \left (20 \,{\mathrm e}+3 x \right ) {\mathrm e}^{-1-x}}{x^{2}+12 x +25}\) | \(27\) |
norman | \(\frac {\left (3 \,{\mathrm e}^{-1} x +20\right ) x \,{\mathrm e}^{-x}}{x^{2}+12 x +25}\) | \(29\) |
gosper | \(\frac {\left (20 \,{\mathrm e}+3 x \right ) x \,{\mathrm e}^{-x} {\mathrm e}^{-1}}{x^{2}+12 x +25}\) | \(32\) |
parallelrisch | \(\frac {\left (20 \,{\mathrm e}+3 x \right ) x \,{\mathrm e}^{-x} {\mathrm e}^{-1}}{x^{2}+12 x +25}\) | \(32\) |
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 (-6-x \right )}{22 x^{2}+264 x +550}+\frac {{\mathrm e}^{6+\sqrt {11}} \operatorname {expIntegral}_{1}\left (x +6+\sqrt {11}\right )}{44}-\frac {\sqrt {11}\, {\mathrm e}^{6+\sqrt {11}} \operatorname {expIntegral}_{1}\left (x +6+\sqrt {11}\right )}{484}+\frac {{\mathrm e}^{6-\sqrt {11}} \operatorname {expIntegral}_{1}\left (x +6-\sqrt {11}\right )}{44}+\frac {\sqrt {11}\, {\mathrm e}^{6-\sqrt {11}} \operatorname {expIntegral}_{1}\left (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}} \operatorname {expIntegral}_{1}\left (x +6+\sqrt {11}\right )}{22}+\frac {5 \sqrt {11}\, {\mathrm e}^{6+\sqrt {11}} \operatorname {expIntegral}_{1}\left (x +6+\sqrt {11}\right )}{484}+\frac {3 \,{\mathrm e}^{6-\sqrt {11}} \operatorname {expIntegral}_{1}\left (x +6-\sqrt {11}\right )}{22}-\frac {5 \sqrt {11}\, {\mathrm e}^{6-\sqrt {11}} \operatorname {expIntegral}_{1}\left (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}} \operatorname {expIntegral}_{1}\left (x +6+\sqrt {11}\right )}{44}+\frac {107 \sqrt {11}\, {\mathrm e}^{6+\sqrt {11}} \operatorname {expIntegral}_{1}\left (x +6+\sqrt {11}\right )}{484}+\frac {47 \,{\mathrm e}^{6-\sqrt {11}} \operatorname {expIntegral}_{1}\left (x +6-\sqrt {11}\right )}{44}-\frac {107 \sqrt {11}\, {\mathrm e}^{6-\sqrt {11}} \operatorname {expIntegral}_{1}\left (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}} \operatorname {expIntegral}_{1}\left (x +6+\sqrt {11}\right )}{11}+\frac {1291 \sqrt {11}\, {\mathrm e}^{6+\sqrt {11}} \operatorname {expIntegral}_{1}\left (x +6+\sqrt {11}\right )}{484}+\frac {109 \,{\mathrm e}^{6-\sqrt {11}} \operatorname {expIntegral}_{1}\left (x +6-\sqrt {11}\right )}{11}-\frac {1291 \sqrt {11}\, {\mathrm e}^{6-\sqrt {11}} \operatorname {expIntegral}_{1}\left (x +6-\sqrt {11}\right )}{484}\right )\right )\) | \(513\) |
Input:
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+25)/x)+x),x,method=_RETURNVERBOSE)
Output:
x/(x^2+12*x+25)*(20*exp(1)+3*x)*exp(-1-x)
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-1-x} x \left (150 x-39 x^2-36 x^3-3 x^4+e \left (500-500 x-260 x^2-20 x^3\right )\right )}{\left (25+12 x+x^2\right ) \left (25 x+12 x^2+x^3\right )} \, dx={\left (3 \, x + 20 \, e\right )} e^{\left (-x - \log \left (\frac {x^{2} + 12 \, x + 25}{x}\right ) - 1\right )} \] Input:
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")
Output:
(3*x + 20*e)*e^(-x - log((x^2 + 12*x + 25)/x) - 1)
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-1-x} x \left (150 x-39 x^2-36 x^3-3 x^4+e \left (500-500 x-260 x^2-20 x^3\right )\right )}{\left (25+12 x+x^2\right ) \left (25 x+12 x^2+x^3\right )} \, dx=\frac {\left (3 x^{2} + 20 e x\right ) e^{- x}}{e x^{2} + 12 e x + 25 e} \] Input:
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(ln((x**2+12*x+25)/x)+x),x)
Output:
(3*x**2 + 20*E*x)*exp(-x)/(E*x**2 + 12*E*x + 25*E)
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{-1-x} x \left (150 x-39 x^2-36 x^3-3 x^4+e \left (500-500 x-260 x^2-20 x^3\right )\right )}{\left (25+12 x+x^2\right ) \left (25 x+12 x^2+x^3\right )} \, dx=\frac {{\left (3 \, x^{2} + 20 \, x e\right )} e^{\left (-x\right )}}{x^{2} e + 12 \, x e + 25 \, e} \] Input:
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")
Output:
(3*x^2 + 20*x*e)*e^(-x)/(x^2*e + 12*x*e + 25*e)
Time = 0.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-1-x} x \left (150 x-39 x^2-36 x^3-3 x^4+e \left (500-500 x-260 x^2-20 x^3\right )\right )}{\left (25+12 x+x^2\right ) \left (25 x+12 x^2+x^3\right )} \, dx=\frac {3 \, x^{2} e^{\left (-x\right )} + 20 \, x e^{\left (-x + 1\right )}}{x^{2} e + 12 \, x e + 25 \, e} \] Input:
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")
Output:
(3*x^2*e^(-x) + 20*x*e^(-x + 1))/(x^2*e + 12*x*e + 25*e)
Time = 8.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-1-x} x \left (150 x-39 x^2-36 x^3-3 x^4+e \left (500-500 x-260 x^2-20 x^3\right )\right )}{\left (25+12 x+x^2\right ) \left (25 x+12 x^2+x^3\right )} \, dx=\frac {x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-1}\,\left (3\,x+20\,\mathrm {e}\right )}{x^2+12\,x+25} \] Input:
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)
Output:
(x*exp(-x)*exp(-1)*(3*x + 20*exp(1)))/(12*x + x^2 + 25)
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {e^{-1-x} x \left (150 x-39 x^2-36 x^3-3 x^4+e \left (500-500 x-260 x^2-20 x^3\right )\right )}{\left (25+12 x+x^2\right ) \left (25 x+12 x^2+x^3\right )} \, dx=\frac {x \left (20 e +3 x \right )}{e^{x} e \left (x^{2}+12 x +25\right )} \] Input:
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(log((x^2+12*x+25)/x)+x),x)
Output:
(x*(20*e + 3*x))/(e**x*e*(x**2 + 12*x + 25))