Integrand size = 61, antiderivative size = 32 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {\left (e^{-1-x}+\frac {2 e^x}{x}+\frac {x}{4}\right ) (5+x (5+x))}{x} \] Output:
(5+(5+x)*x)/x*(exp(-1-x)+2*exp(x)/x+1/4*x)
Time = 0.31 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {1}{4} \left (8 e^x \left (1+\frac {5}{x^2}+\frac {5}{x}\right )+5 x+x^2-4 e^{-x} \left (-\frac {5}{e}-\frac {5}{e x}-\frac {x}{e}\right )\right ) \] Input:
Integrate[(5*x^3 + 2*x^4 + E^x*(-80 + 40*x^2 + 8*x^3) + E^(-1 - x)*(-20*x - 20*x^2 - 16*x^3 - 4*x^4))/(4*x^3),x]
Output:
(8*E^x*(1 + 5/x^2 + 5/x) + 5*x + x^2 - (4*(-5/E - 5/(E*x) - x/E))/E^x)/4
Time = 0.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {27, 2010, 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.
\(\displaystyle \int \frac {2 x^4+5 x^3+e^x \left (8 x^3+40 x^2-80\right )+e^{-x-1} \left (-4 x^4-16 x^3-20 x^2-20 x\right )}{4 x^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {2 x^4+5 x^3-8 e^x \left (-x^3-5 x^2+10\right )-4 e^{-x-1} \left (x^4+4 x^3+5 x^2+5 x\right )}{x^3}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {1}{4} \int \left (2 x+5-\frac {4 e^{-x-1} \left (x^3+4 x^2+5 x+5\right )}{x^2}+\frac {8 e^x \left (x^3+5 x^2-10\right )}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (x^2+\frac {40 e^x}{x^2}+4 e^{-x-1} x+5 x+20 e^{-x-1}+8 e^x+\frac {20 e^{-x-1}}{x}+\frac {40 e^x}{x}\right )\) |
Input:
Int[(5*x^3 + 2*x^4 + E^x*(-80 + 40*x^2 + 8*x^3) + E^(-1 - x)*(-20*x - 20*x ^2 - 16*x^3 - 4*x^4))/(4*x^3),x]
Output:
(20*E^(-1 - x) + 8*E^x + (40*E^x)/x^2 + (20*E^(-1 - x))/x + (40*E^x)/x + 5 *x + 4*E^(-1 - x)*x + x^2)/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 0.36 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34
method | result | size |
risch | \(\frac {x^{2}}{4}+\frac {5 x}{4}+\frac {2 \left (x^{2}+5 x +5\right ) {\mathrm e}^{x}}{x^{2}}+\frac {\left (x^{2}+5 x +5\right ) {\mathrm e}^{-1-x}}{x}\) | \(43\) |
parts | \(\frac {5 x}{4}+\frac {x^{2}}{4}+4 \,{\mathrm e}^{-1-x}+\frac {5 \,{\mathrm e}^{-1-x}}{x}-{\mathrm e}^{-1-x} \left (-1-x \right )+\frac {10 \,{\mathrm e}^{x}}{x^{2}}+\frac {10 \,{\mathrm e}^{x}}{x}+2 \,{\mathrm e}^{x}\) | \(60\) |
parallelrisch | \(\frac {4 x^{3} {\mathrm e}^{-1-x}+x^{4}+20 \,{\mathrm e}^{-1-x} x^{2}+8 \,{\mathrm e}^{x} x^{2}+5 x^{3}+20 x \,{\mathrm e}^{-1-x}+40 \,{\mathrm e}^{x} x +40 \,{\mathrm e}^{x}}{4 x^{2}}\) | \(62\) |
norman | \(\frac {\left ({\mathrm e}^{-1} x^{3}+10 \,{\mathrm e}^{2 x}+5 \,{\mathrm e}^{-1} x +10 x \,{\mathrm e}^{2 x}+5 \,{\mathrm e}^{-1} x^{2}+\frac {5 \,{\mathrm e}^{x} x^{3}}{4}+\frac {{\mathrm e}^{x} x^{4}}{4}+2 \,{\mathrm e}^{2 x} x^{2}\right ) {\mathrm e}^{-x}}{x^{2}}\) | \(70\) |
default | \(\frac {x^{2}}{4}+\frac {5 x}{4}+\frac {10 \,{\mathrm e}^{x}}{x^{2}}+\frac {10 \,{\mathrm e}^{x}}{x}+4 \,{\mathrm e}^{-1} {\mathrm e}^{-x}-5 \,{\mathrm e}^{-1} \left (-\frac {{\mathrm e}^{-x}}{x}+\operatorname {expIntegral}_{1}\left (x \right )\right )+5 \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (x \right )-{\mathrm e}^{-1} \left (-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}\right )+2 \,{\mathrm e}^{x}\) | \(78\) |
orering | \(\frac {\left (4 x^{11}+50 x^{10}-68 x^{9}-1693 x^{8}-2090 x^{7}-1375 x^{6}-19630 x^{5}-18965 x^{4}+20000 x^{3}+1200 x^{2}-3000 x +37500\right ) \left (\left (8 x^{3}+40 x^{2}-80\right ) {\mathrm e}^{x}+\left (-4 x^{4}-16 x^{3}-20 x^{2}-20 x \right ) {\mathrm e}^{-1-x}+2 x^{4}+5 x^{3}\right )}{8 \left (4 x^{10}+40 x^{9}+119 x^{8}+150 x^{7}+365 x^{6}+950 x^{5}+185 x^{4}-1160 x^{3}-1200 x -1500\right ) x^{3}}+\frac {3 x \left (2 x^{9}+18 x^{8}+865 x^{5}+1405 x^{4}+1675 x^{3}+3350 x^{2}-1250 x +6250\right ) \left (\frac {\left (24 x^{2}+80 x \right ) {\mathrm e}^{x}+\left (8 x^{3}+40 x^{2}-80\right ) {\mathrm e}^{x}+\left (-16 x^{3}-48 x^{2}-40 x -20\right ) {\mathrm e}^{-1-x}-\left (-4 x^{4}-16 x^{3}-20 x^{2}-20 x \right ) {\mathrm e}^{-1-x}+8 x^{3}+15 x^{2}}{4 x^{3}}-\frac {3 \left (\left (8 x^{3}+40 x^{2}-80\right ) {\mathrm e}^{x}+\left (-4 x^{4}-16 x^{3}-20 x^{2}-20 x \right ) {\mathrm e}^{-1-x}+2 x^{4}+5 x^{3}\right )}{4 x^{4}}\right )}{4 x^{10}+40 x^{9}+119 x^{8}+150 x^{7}+365 x^{6}+950 x^{5}+185 x^{4}-1160 x^{3}-1200 x -1500}-\frac {\left (4 x^{9}+54 x^{8}-1731 x^{6}-5190 x^{5}-4215 x^{4}-3350 x^{3}-5025 x^{2}+1500 x -6250\right ) x^{2} \left (\frac {\left (48 x +80\right ) {\mathrm e}^{x}+2 \left (24 x^{2}+80 x \right ) {\mathrm e}^{x}+\left (8 x^{3}+40 x^{2}-80\right ) {\mathrm e}^{x}+\left (-48 x^{2}-96 x -40\right ) {\mathrm e}^{-1-x}-2 \left (-16 x^{3}-48 x^{2}-40 x -20\right ) {\mathrm e}^{-1-x}+\left (-4 x^{4}-16 x^{3}-20 x^{2}-20 x \right ) {\mathrm e}^{-1-x}+24 x^{2}+30 x}{4 x^{3}}-\frac {3 \left (\left (24 x^{2}+80 x \right ) {\mathrm e}^{x}+\left (8 x^{3}+40 x^{2}-80\right ) {\mathrm e}^{x}+\left (-16 x^{3}-48 x^{2}-40 x -20\right ) {\mathrm e}^{-1-x}-\left (-4 x^{4}-16 x^{3}-20 x^{2}-20 x \right ) {\mathrm e}^{-1-x}+8 x^{3}+15 x^{2}\right )}{2 x^{4}}+\frac {3 \left (8 x^{3}+40 x^{2}-80\right ) {\mathrm e}^{x}+3 \left (-4 x^{4}-16 x^{3}-20 x^{2}-20 x \right ) {\mathrm e}^{-1-x}+6 x^{4}+15 x^{3}}{x^{5}}\right )}{2 \left (4 x^{10}+40 x^{9}+119 x^{8}+150 x^{7}+365 x^{6}+950 x^{5}+185 x^{4}-1160 x^{3}-1200 x -1500\right )}\) | \(754\) |
Input:
int(1/4*((8*x^3+40*x^2-80)*exp(x)+(-4*x^4-16*x^3-20*x^2-20*x)*exp(-1-x)+2* x^4+5*x^3)/x^3,x,method=_RETURNVERBOSE)
Output:
1/4*x^2+5/4*x+2*(x^2+5*x+5)/x^2*exp(x)+1/x*(x^2+5*x+5)*exp(-1-x)
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {{\left (4 \, x^{3} + 20 \, x^{2} + 8 \, {\left (x^{2} + 5 \, x + 5\right )} e^{\left (2 \, x + 1\right )} + {\left (x^{4} + 5 \, x^{3}\right )} e^{\left (x + 1\right )} + 20 \, x\right )} e^{\left (-x - 1\right )}}{4 \, x^{2}} \] Input:
integrate(1/4*((8*x^3+40*x^2-80)*exp(x)+(-4*x^4-16*x^3-20*x^2-20*x)*exp(-1 -x)+2*x^4+5*x^3)/x^3,x, algorithm="fricas")
Output:
1/4*(4*x^3 + 20*x^2 + 8*(x^2 + 5*x + 5)*e^(2*x + 1) + (x^4 + 5*x^3)*e^(x + 1) + 20*x)*e^(-x - 1)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {x^{2}}{4} + \frac {5 x}{4} + \frac {\left (x^{4} + 5 x^{3} + 5 x^{2}\right ) e^{- x} + \left (2 e x^{3} + 10 e x^{2} + 10 e x\right ) e^{x}}{e x^{3}} \] Input:
integrate(1/4*((8*x**3+40*x**2-80)*exp(x)+(-4*x**4-16*x**3-20*x**2-20*x)*e xp(-1-x)+2*x**4+5*x**3)/x**3,x)
Output:
x**2/4 + 5*x/4 + ((x**4 + 5*x**3 + 5*x**2)*exp(-x) + (2*E*x**3 + 10*E*x**2 + 10*E*x)*exp(x))*exp(-1)/x**3
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {1}{4} \, x^{2} - 5 \, {\rm Ei}\left (-x\right ) e^{\left (-1\right )} + {\left (x + 1\right )} e^{\left (-x - 1\right )} + 5 \, e^{\left (-1\right )} \Gamma \left (-1, x\right ) + \frac {5}{4} \, x + 10 \, {\rm Ei}\left (x\right ) + 2 \, e^{x} + 4 \, e^{\left (-x - 1\right )} + 20 \, \Gamma \left (-2, -x\right ) \] Input:
integrate(1/4*((8*x^3+40*x^2-80)*exp(x)+(-4*x^4-16*x^3-20*x^2-20*x)*exp(-1 -x)+2*x^4+5*x^3)/x^3,x, algorithm="maxima")
Output:
1/4*x^2 - 5*Ei(-x)*e^(-1) + (x + 1)*e^(-x - 1) + 5*e^(-1)*gamma(-1, x) + 5 /4*x + 10*Ei(x) + 2*e^x + 4*e^(-x - 1) + 20*gamma(-2, -x)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (29) = 58\).
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {{\left (x^{4} e + 5 \, x^{3} e + 4 \, x^{3} e^{\left (-x\right )} + 20 \, x^{2} e^{\left (-x\right )} + 8 \, x^{2} e^{\left (x + 1\right )} + 20 \, x e^{\left (-x\right )} + 40 \, x e^{\left (x + 1\right )} + 40 \, e^{\left (x + 1\right )}\right )} e^{\left (-1\right )}}{4 \, x^{2}} \] Input:
integrate(1/4*((8*x^3+40*x^2-80)*exp(x)+(-4*x^4-16*x^3-20*x^2-20*x)*exp(-1 -x)+2*x^4+5*x^3)/x^3,x, algorithm="giac")
Output:
1/4*(x^4*e + 5*x^3*e + 4*x^3*e^(-x) + 20*x^2*e^(-x) + 8*x^2*e^(x + 1) + 20 *x*e^(-x) + 40*x*e^(x + 1) + 40*e^(x + 1))*e^(-1)/x^2
Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {5\,x}{4}+5\,{\mathrm {e}}^{-x-1}+2\,{\mathrm {e}}^x+\frac {10\,{\mathrm {e}}^x}{x}+\frac {10\,{\mathrm {e}}^x}{x^2}+x\,{\mathrm {e}}^{-x-1}+\frac {5\,{\mathrm {e}}^{-x-1}}{x}+\frac {x^2}{4} \] Input:
int(((exp(x)*(40*x^2 + 8*x^3 - 80))/4 - (exp(- x - 1)*(20*x + 20*x^2 + 16* x^3 + 4*x^4))/4 + (5*x^3)/4 + x^4/2)/x^3,x)
Output:
(5*x)/4 + 5*exp(- x - 1) + 2*exp(x) + (10*exp(x))/x + (10*exp(x))/x^2 + x* exp(- x - 1) + (5*exp(- x - 1))/x + x^2/4
Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.25 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {8 e^{2 x} e \,x^{2}+40 e^{2 x} e x +40 e^{2 x} e +e^{x} e \,x^{4}+5 e^{x} e \,x^{3}+4 x^{3}+20 x^{2}+20 x}{4 e^{x} e \,x^{2}} \] Input:
int(1/4*((8*x^3+40*x^2-80)*exp(x)+(-4*x^4-16*x^3-20*x^2-20*x)*exp(-1-x)+2* x^4+5*x^3)/x^3,x)
Output:
(8*e**(2*x)*e*x**2 + 40*e**(2*x)*e*x + 40*e**(2*x)*e + e**x*e*x**4 + 5*e** x*e*x**3 + 4*x**3 + 20*x**2 + 20*x)/(4*e**x*e*x**2)