Integrand size = 42, antiderivative size = 20 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\frac {-e^{2 x}-3 x}{-625+e+2 x} \] Output:
(-3*x-exp(2*x))/(exp(1)-625+2*x)
Time = 0.42 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\frac {1875-3 e+2 e^{2 x}}{1250-2 e-4 x} \] Input:
Integrate[(1875 - 3*E + E^(2*x)*(1252 - 2*E - 4*x))/(390625 + E^2 - 2500*x + 4*x^2 + E*(-1250 + 4*x)),x]
Output:
(1875 - 3*E + 2*E^(2*x))/(1250 - 2*E - 4*x)
Time = 0.79 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2007, 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.
\(\displaystyle \int \frac {e^{2 x} (-4 x-2 e+1252)-3 e+1875}{4 x^2-2500 x+e (4 x-1250)+e^2+390625} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^{2 x} (-4 x-2 e+1252)-3 e+1875}{(2 x+e-625)^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{2 x} (-4 x-2 e+1252)+1875 \left (1-\frac {e}{625}\right )}{(-2 x-e+625)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2 e^{2 x} (2 x+e-626)}{(2 x+e-625)^2}-\frac {3 (e-625)}{(2 x+e-625)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^{2 x}}{-2 x-e+625}+\frac {3 (625-e)}{2 (-2 x-e+625)}\) |
Input:
Int[(1875 - 3*E + E^(2*x)*(1252 - 2*E - 4*x))/(390625 + E^2 - 2500*x + 4*x ^2 + E*(-1250 + 4*x)),x]
Output:
(3*(625 - E))/(2*(625 - E - 2*x)) + E^(2*x)/(625 - E - 2*x)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 0.60 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15
method | result | size |
norman | \(\frac {-{\mathrm e}^{2 x}-\frac {1875}{2}+\frac {3 \,{\mathrm e}}{2}}{{\mathrm e}-625+2 x}\) | \(23\) |
parallelrisch | \(\frac {3 \,{\mathrm e}-1875-2 \,{\mathrm e}^{2 x}}{-1250+2 \,{\mathrm e}+4 x}\) | \(24\) |
risch | \(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}\) | \(41\) |
parts | \(\frac {-\frac {1875}{2}+\frac {3 \,{\mathrm e}}{2}}{{\mathrm e}-625+2 x}-\frac {{\mathrm e}^{2 x} \left ({\mathrm e}-625\right )}{{\mathrm e}-625+2 x}+\left (-{\mathrm e}+626\right ) {\mathrm e}^{-{\mathrm e}+625} \operatorname {expIntegral}_{1}\left (-{\mathrm e}-2 x +625\right )-{\mathrm e} \left (-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-{\mathrm e}^{-{\mathrm e}+625} \operatorname {expIntegral}_{1}\left (-{\mathrm e}-2 x +625\right )\right )-\frac {626 \,{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-626 \,{\mathrm e}^{-{\mathrm e}+625} \operatorname {expIntegral}_{1}\left (-{\mathrm e}-2 x +625\right )\) | \(137\) |
derivativedivides | \(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {626 \,{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-626 \,{\mathrm e}^{-{\mathrm e}+625} \operatorname {expIntegral}_{1}\left (-{\mathrm e}-2 x +625\right )-\frac {{\mathrm e}^{2 x} \left ({\mathrm e}-625\right )}{{\mathrm e}-625+2 x}+\left (-{\mathrm e}+626\right ) {\mathrm e}^{-{\mathrm e}+625} \operatorname {expIntegral}_{1}\left (-{\mathrm e}-2 x +625\right )-{\mathrm e} \left (-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-{\mathrm e}^{-{\mathrm e}+625} \operatorname {expIntegral}_{1}\left (-{\mathrm e}-2 x +625\right )\right )\) | \(145\) |
default | \(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {626 \,{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-626 \,{\mathrm e}^{-{\mathrm e}+625} \operatorname {expIntegral}_{1}\left (-{\mathrm e}-2 x +625\right )-\frac {{\mathrm e}^{2 x} \left ({\mathrm e}-625\right )}{{\mathrm e}-625+2 x}+\left (-{\mathrm e}+626\right ) {\mathrm e}^{-{\mathrm e}+625} \operatorname {expIntegral}_{1}\left (-{\mathrm e}-2 x +625\right )-{\mathrm e} \left (-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-{\mathrm e}^{-{\mathrm e}+625} \operatorname {expIntegral}_{1}\left (-{\mathrm e}-2 x +625\right )\right )\) | \(145\) |
orering | \(\frac {\left (-\frac {{\mathrm e}}{2}-x +\frac {627}{2}\right ) \left (\left (-2 \,{\mathrm e}-4 x +1252\right ) {\mathrm e}^{2 x}-3 \,{\mathrm e}+1875\right )}{{\mathrm e}^{2}+\left (4 x -1250\right ) {\mathrm e}+4 x^{2}-2500 x +390625}+\left (\frac {x}{2}+\frac {{\mathrm e}}{4}-\frac {625}{4}\right ) \left (\frac {-4 \,{\mathrm e}^{2 x}+2 \left (-2 \,{\mathrm e}-4 x +1252\right ) {\mathrm e}^{2 x}}{{\mathrm e}^{2}+\left (4 x -1250\right ) {\mathrm e}+4 x^{2}-2500 x +390625}-\frac {\left (\left (-2 \,{\mathrm e}-4 x +1252\right ) {\mathrm e}^{2 x}-3 \,{\mathrm e}+1875\right ) \left (4 \,{\mathrm e}+8 x -2500\right )}{\left ({\mathrm e}^{2}+\left (4 x -1250\right ) {\mathrm e}+4 x^{2}-2500 x +390625\right )^{2}}\right )\) | \(169\) |
Input:
int(((-2*exp(1)-4*x+1252)*exp(2*x)-3*exp(1)+1875)/(exp(1)^2+(4*x-1250)*exp (1)+4*x^2-2500*x+390625),x,method=_RETURNVERBOSE)
Output:
(-exp(2*x)-1875/2+3/2*exp(1))/(exp(1)-625+2*x)
Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\frac {3 \, e - 2 \, e^{\left (2 \, x\right )} - 1875}{2 \, {\left (2 \, x + e - 625\right )}} \] Input:
integrate(((-2*exp(1)-4*x+1252)*exp(2*x)-3*exp(1)+1875)/(exp(1)^2+(4*x-125 0)*exp(1)+4*x^2-2500*x+390625),x, algorithm="fricas")
Output:
1/2*(3*e - 2*e^(2*x) - 1875)/(2*x + e - 625)
Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=- \frac {1875 - 3 e}{4 x - 1250 + 2 e} - \frac {e^{2 x}}{2 x - 625 + e} \] Input:
integrate(((-2*exp(1)-4*x+1252)*exp(2*x)-3*exp(1)+1875)/(exp(1)**2+(4*x-12 50)*exp(1)+4*x**2-2500*x+390625),x)
Output:
-(1875 - 3*E)/(4*x - 1250 + 2*E) - exp(2*x)/(2*x - 625 + E)
\[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\int { -\frac {2 \, {\left (2 \, x + e - 626\right )} e^{\left (2 \, x\right )} + 3 \, e - 1875}{4 \, x^{2} + 2 \, {\left (2 \, x - 625\right )} e - 2500 \, x + e^{2} + 390625} \,d x } \] Input:
integrate(((-2*exp(1)-4*x+1252)*exp(2*x)-3*exp(1)+1875)/(exp(1)^2+(4*x-125 0)*exp(1)+4*x^2-2500*x+390625),x, algorithm="maxima")
Output:
-2*x*e^(2*x)/(4*x^2 + 4*x*(e - 625) + e^2 - 1250*e + 390625) - 626*e^(-e + 625)*exp_integral_e(2, -2*x - e + 625)/(2*x + e - 625) + 3/2*e/(2*x + e - 625) - 1875/2/(2*x + e - 625) - integrate(2*(2*x*(e + 1) + e^2 - 626*e + 625)*e^(2*x)/(8*x^3 + 12*x^2*(e - 625) + 6*x*(e^2 - 1250*e + 390625) + e^3 - 1875*e^2 + 1171875*e - 244140625), x)
Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\frac {3 \, e - 2 \, e^{\left (2 \, x\right )} - 1875}{2 \, {\left (2 \, x + e - 625\right )}} \] Input:
integrate(((-2*exp(1)-4*x+1252)*exp(2*x)-3*exp(1)+1875)/(exp(1)^2+(4*x-125 0)*exp(1)+4*x^2-2500*x+390625),x, algorithm="giac")
Output:
1/2*(3*e - 2*e^(2*x) - 1875)/(2*x + e - 625)
Time = 1.82 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=-\frac {3\,x+{\mathrm {e}}^{2\,x}}{2\,x+\mathrm {e}-625} \] Input:
int(-(3*exp(1) + exp(2*x)*(4*x + 2*exp(1) - 1252) - 1875)/(exp(2) - 2500*x + 4*x^2 + exp(1)*(4*x - 1250) + 390625),x)
Output:
-(3*x + exp(2*x))/(2*x + exp(1) - 625)
Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\frac {-e^{2 x}-3 x}{e +2 x -625} \] Input:
int(((-2*exp(1)-4*x+1252)*exp(2*x)-3*exp(1)+1875)/(exp(1)^2+(4*x-1250)*exp (1)+4*x^2-2500*x+390625),x)
Output:
( - e**(2*x) - 3*x)/(e + 2*x - 625)