Integrand size = 77, antiderivative size = 29 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=5+64 \left (-(-2+e)^2-e^{x/3}+\frac {5 e}{x}\right )^2 \]
Time = 2.61 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=64 \left (e^{2 x/3}+2 e^{x/3} \left ((-2+e)^2-\frac {5 e}{x}\right )+\frac {25 e^2}{x^2}-\frac {10 (-2+e)^2 e}{x}\right ) \]
Integrate[(E^2*(-9600 - 7680*x) + 7680*E*x + 1920*E^3*x + 128*E^((2*x)/3)* x^3 + E^(x/3)*(512*x^3 + 128*E^2*x^3 + E*(1920*x - 640*x^2 - 512*x^3)))/(3 *x^3),x]
Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(29)=58\).
Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.38, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6, 27, 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.
\(\displaystyle \int \frac {128 e^{2 x/3} x^3+e^{x/3} \left (128 e^2 x^3+512 x^3+e \left (-512 x^3-640 x^2+1920 x\right )\right )+1920 e^3 x+7680 e x+e^2 (-7680 x-9600)}{3 x^3} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {128 e^{2 x/3} x^3+e^{x/3} \left (128 e^2 x^3+512 x^3+e \left (-512 x^3-640 x^2+1920 x\right )\right )+\left (7680 e+1920 e^3\right ) x+e^2 (-7680 x-9600)}{3 x^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {128 \left (e^{2 x/3} x^3+15 e \left (4+e^2\right ) x-15 e^2 (4 x+5)+e^{x/3} \left (e^2 x^3+4 x^3+e \left (-4 x^3-5 x^2+15 x\right )\right )\right )}{x^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {128}{3} \int \frac {e^{2 x/3} x^3+15 e \left (4+e^2\right ) x-15 e^2 (4 x+5)+e^{x/3} \left (e^2 x^3+4 x^3+e \left (-4 x^3-5 x^2+15 x\right )\right )}{x^3}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {128}{3} \int \left (\frac {15 e \left ((2-e)^2 x-5 e\right )}{x^3}+e^{2 x/3}+\frac {e^{x/3} \left ((2-e)^2 x^2-5 e x+15 e\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {128}{3} \left (\frac {3 \left (5 e-(2-e)^2 x\right )^2}{2 x^2}+3 (2-e)^2 e^{x/3}+\frac {3}{2} e^{2 x/3}-\frac {15 e^{\frac {x}{3}+1}}{x}\right )\) |
Int[(E^2*(-9600 - 7680*x) + 7680*E*x + 1920*E^3*x + 128*E^((2*x)/3)*x^3 + E^(x/3)*(512*x^3 + 128*E^2*x^3 + E*(1920*x - 640*x^2 - 512*x^3)))/(3*x^3), x]
(128*(3*(2 - E)^2*E^(x/3) + (3*E^((2*x)/3))/2 - (15*E^(1 + x/3))/x + (3*(5 *E - (2 - E)^2*x)^2)/(2*x^2)))/3
3.8.51.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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 = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03
method | result | size |
risch | \(\frac {\left (-1920 \,{\mathrm e}^{3}+7680 \,{\mathrm e}^{2}-7680 \,{\mathrm e}\right ) x +4800 \,{\mathrm e}^{2}}{3 x^{2}}+64 \,{\mathrm e}^{\frac {2 x}{3}}+\frac {128 \left ({\mathrm e}^{2} x -4 x \,{\mathrm e}-5 \,{\mathrm e}+4 x \right ) {\mathrm e}^{\frac {x}{3}}}{x}\) | \(59\) |
norman | \(\frac {\left (-640 \,{\mathrm e}^{3}+2560 \,{\mathrm e}^{2}-2560 \,{\mathrm e}\right ) x +\left (128 \,{\mathrm e}^{2}-512 \,{\mathrm e}+512\right ) x^{2} {\mathrm e}^{\frac {x}{3}}+1600 \,{\mathrm e}^{2}+64 \,{\mathrm e}^{\frac {2 x}{3}} x^{2}-640 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}} x}{x^{2}}\) | \(71\) |
parallelrisch | \(-\frac {-384 \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{3}} x^{2}+1920 x \,{\mathrm e}^{3}+1536 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}} x^{2}-192 \,{\mathrm e}^{\frac {2 x}{3}} x^{2}-7680 \,{\mathrm e}^{2} x +1920 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}} x -1536 \,{\mathrm e}^{\frac {x}{3}} x^{2}-4800 \,{\mathrm e}^{2}+7680 x \,{\mathrm e}}{3 x^{2}}\) | \(85\) |
parts | \(64 \,{\mathrm e}^{\frac {2 x}{3}}-512 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}}+128 \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{3}}+\frac {640 \,{\mathrm e} \left (-\frac {3 \,{\mathrm e}^{\frac {x}{3}}}{x}-\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )\right )}{3}+\frac {640 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )}{3}+512 \,{\mathrm e}^{\frac {x}{3}}+640 \,{\mathrm e} \left (-\frac {{\mathrm e}^{2}-4 \,{\mathrm e}+4}{x}+\frac {5 \,{\mathrm e}}{2 x^{2}}\right )\) | \(89\) |
derivativedivides | \(64 \,{\mathrm e}^{\frac {2 x}{3}}+\frac {1600 \,{\mathrm e}^{2}}{x^{2}}-\frac {2560 \,{\mathrm e}}{x}+\frac {2560 \,{\mathrm e}^{2}}{x}-\frac {640 \,{\mathrm e}^{3}}{x}-512 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}}+128 \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{3}}+\frac {640 \,{\mathrm e} \left (-\frac {3 \,{\mathrm e}^{\frac {x}{3}}}{x}-\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )\right )}{3}+\frac {640 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )}{3}+512 \,{\mathrm e}^{\frac {x}{3}}\) | \(98\) |
default | \(64 \,{\mathrm e}^{\frac {2 x}{3}}+\frac {1600 \,{\mathrm e}^{2}}{x^{2}}-\frac {2560 \,{\mathrm e}}{x}+\frac {2560 \,{\mathrm e}^{2}}{x}-\frac {640 \,{\mathrm e}^{3}}{x}-512 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}}+128 \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{3}}+\frac {640 \,{\mathrm e} \left (-\frac {3 \,{\mathrm e}^{\frac {x}{3}}}{x}-\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )\right )}{3}+\frac {640 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )}{3}+512 \,{\mathrm e}^{\frac {x}{3}}\) | \(98\) |
int(1/3*(128*x^3*exp(1/3*x)^2+(128*x^3*exp(1)^2+(-512*x^3-640*x^2+1920*x)* exp(1)+512*x^3)*exp(1/3*x)+1920*x*exp(1)^3+(-7680*x-9600)*exp(1)^2+7680*x* exp(1))/x^3,x,method=_RETURNVERBOSE)
1/3*((-1920*exp(3)+7680*exp(2)-7680*exp(1))*x+4800*exp(2))/x^2+64*exp(2/3* x)+128*(exp(2)*x-4*x*exp(1)-5*exp(1)+4*x)/x*exp(1/3*x)
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (24) = 48\).
Time = 0.23 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=\frac {64 \, {\left (x^{2} e^{\left (\frac {2}{3} \, x\right )} - 10 \, x e^{3} + 5 \, {\left (8 \, x + 5\right )} e^{2} - 40 \, x e + 2 \, {\left (x^{2} e^{2} + 4 \, x^{2} - {\left (4 \, x^{2} + 5 \, x\right )} e\right )} e^{\left (\frac {1}{3} \, x\right )}\right )}}{x^{2}} \]
integrate(1/3*(128*x^3*exp(1/3*x)^2+(128*x^3*exp(1)^2+(-512*x^3-640*x^2+19 20*x)*exp(1)+512*x^3)*exp(1/3*x)+1920*x*exp(1)^3+(-7680*x-9600)*exp(1)^2+7 680*x*exp(1))/x^3,x, algorithm=\
64*(x^2*e^(2/3*x) - 10*x*e^3 + 5*(8*x + 5)*e^2 - 40*x*e + 2*(x^2*e^2 + 4*x ^2 - (4*x^2 + 5*x)*e)*e^(1/3*x))/x^2
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=\frac {64 x e^{\frac {2 x}{3}} + \left (- 512 e x + 512 x + 128 x e^{2} - 640 e\right ) e^{\frac {x}{3}}}{x} + \frac {x \left (- 640 e^{3} - 2560 e + 2560 e^{2}\right ) + 1600 e^{2}}{x^{2}} \]
integrate(1/3*(128*x**3*exp(1/3*x)**2+(128*x**3*exp(1)**2+(-512*x**3-640*x **2+1920*x)*exp(1)+512*x**3)*exp(1/3*x)+1920*x*exp(1)**3+(-7680*x-9600)*ex p(1)**2+7680*x*exp(1))/x**3,x)
(64*x*exp(2*x/3) + (-512*E*x + 512*x + 128*x*exp(2) - 640*E)*exp(x/3))/x + (x*(-640*exp(3) - 2560*E + 2560*exp(2)) + 1600*exp(2))/x**2
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.55 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=-\frac {640}{3} \, {\rm Ei}\left (\frac {1}{3} \, x\right ) e + \frac {640}{3} \, e \Gamma \left (-1, -\frac {1}{3} \, x\right ) - \frac {640 \, e^{3}}{x} + \frac {2560 \, e^{2}}{x} - \frac {2560 \, e}{x} + \frac {1600 \, e^{2}}{x^{2}} + 64 \, e^{\left (\frac {2}{3} \, x\right )} + 512 \, e^{\left (\frac {1}{3} \, x\right )} + 128 \, e^{\left (\frac {1}{3} \, x + 2\right )} - 512 \, e^{\left (\frac {1}{3} \, x + 1\right )} \]
integrate(1/3*(128*x^3*exp(1/3*x)^2+(128*x^3*exp(1)^2+(-512*x^3-640*x^2+19 20*x)*exp(1)+512*x^3)*exp(1/3*x)+1920*x*exp(1)^3+(-7680*x-9600)*exp(1)^2+7 680*x*exp(1))/x^3,x, algorithm=\
-640/3*Ei(1/3*x)*e + 640/3*e*gamma(-1, -1/3*x) - 640*e^3/x + 2560*e^2/x - 2560*e/x + 1600*e^2/x^2 + 64*e^(2/3*x) + 512*e^(1/3*x) + 128*e^(1/3*x + 2) - 512*e^(1/3*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (24) = 48\).
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.52 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=\frac {64 \, {\left (x^{2} e^{\left (\frac {2}{3} \, x\right )} + 8 \, x^{2} e^{\left (\frac {1}{3} \, x\right )} + 2 \, x^{2} e^{\left (\frac {1}{3} \, x + 2\right )} - 8 \, x^{2} e^{\left (\frac {1}{3} \, x + 1\right )} - 10 \, x e^{3} + 40 \, x e^{2} - 40 \, x e - 10 \, x e^{\left (\frac {1}{3} \, x + 1\right )} + 25 \, e^{2}\right )}}{x^{2}} \]
integrate(1/3*(128*x^3*exp(1/3*x)^2+(128*x^3*exp(1)^2+(-512*x^3-640*x^2+19 20*x)*exp(1)+512*x^3)*exp(1/3*x)+1920*x*exp(1)^3+(-7680*x-9600)*exp(1)^2+7 680*x*exp(1))/x^3,x, algorithm=\
64*(x^2*e^(2/3*x) + 8*x^2*e^(1/3*x) + 2*x^2*e^(1/3*x + 2) - 8*x^2*e^(1/3*x + 1) - 10*x*e^3 + 40*x*e^2 - 40*x*e - 10*x*e^(1/3*x + 1) + 25*e^2)/x^2
Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=64\,{\mathrm {e}}^{\frac {2\,x}{3}}+\frac {1600\,{\mathrm {e}}^2-x\,\left (640\,{\mathrm {e}}^{\frac {x}{3}+1}+640\,\mathrm {e}\,{\left (\mathrm {e}-2\right )}^2\right )}{x^2}+128\,{\mathrm {e}}^{x/3}\,{\left (\mathrm {e}-2\right )}^2 \]
int(((exp(x/3)*(128*x^3*exp(2) - exp(1)*(640*x^2 - 1920*x + 512*x^3) + 512 *x^3))/3 + 2560*x*exp(1) + 640*x*exp(3) + (128*x^3*exp((2*x)/3))/3 - (exp( 2)*(7680*x + 9600))/3)/x^3,x)