3.14.90 \(\int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx\)

Optimal. Leaf size=20 \[ \frac {-e^{2 x}-3 x}{-625+e+2 x} \]

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Rubi [A]  time = 0.27, antiderivative size = 36, normalized size of antiderivative = 1.80, number of steps used = 5, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6741, 27, 6742, 2197} \begin {gather*} \frac {e^{2 x}}{-2 x-e+625}+\frac {3 (625-e)}{2 (-2 x-e+625)} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

(3*(625 - E))/(2*(625 - E - 2*x)) + E^(2*x)/(625 - E - 2*x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1875 \left (1-\frac {e}{625}\right )+e^{2 x} (1252-2 e-4 x)}{(-625+e)^2-4 (625-e) x+4 x^2} \, dx\\ &=\int \frac {1875 \left (1-\frac {e}{625}\right )+e^{2 x} (1252-2 e-4 x)}{(-625+e+2 x)^2} \, dx\\ &=\int \left (-\frac {3 (-625+e)}{(-625+e+2 x)^2}-\frac {2 e^{2 x} (-626+e+2 x)}{(-625+e+2 x)^2}\right ) \, dx\\ &=\frac {3 (625-e)}{2 (625-e-2 x)}-2 \int \frac {e^{2 x} (-626+e+2 x)}{(-625+e+2 x)^2} \, dx\\ &=\frac {3 (625-e)}{2 (625-e-2 x)}+\frac {e^{2 x}}{625-e-2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 23, normalized size = 1.15 \begin {gather*} \frac {1875-3 e+2 e^{2 x}}{1250-2 e-4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

(1875 - 3*E + 2*E^(2*x))/(1250 - 2*E - 4*x)

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fricas [A]  time = 1.25, size = 23, normalized size = 1.15 \begin {gather*} \frac {3 \, e - 2 \, e^{\left (2 \, x\right )} - 1875}{2 \, {\left (2 \, x + e - 625\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-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, al
gorithm="fricas")

[Out]

1/2*(3*e - 2*e^(2*x) - 1875)/(2*x + e - 625)

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giac [A]  time = 0.58, size = 23, normalized size = 1.15 \begin {gather*} \frac {3 \, e - 2 \, e^{\left (2 \, x\right )} - 1875}{2 \, {\left (2 \, x + e - 625\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-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, al
gorithm="giac")

[Out]

1/2*(3*e - 2*e^(2*x) - 1875)/(2*x + e - 625)

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maple [A]  time = 0.32, size = 23, normalized size = 1.15

method	result	size
norman	\(\frac {-{\mathrm e}^{2 x}-\frac {1875}{2}+\frac {3 \,{\mathrm e}}{2}}{{\mathrm e}-625+2 x}\)	\(23\)
risch	\(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}\)	\(41\)
derivativedivides	\(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {626 \,{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-626 \,{\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )-\frac {{\mathrm e}^{2 x} \left ({\mathrm e}-625\right )}{{\mathrm e}-625+2 x}+\left (-{\mathrm e}+626\right ) {\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )-{\mathrm e} \left (-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-{\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )\right )\)	\(145\)
default	\(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {626 \,{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-626 \,{\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )-\frac {{\mathrm e}^{2 x} \left ({\mathrm e}-625\right )}{{\mathrm e}-625+2 x}+\left (-{\mathrm e}+626\right ) {\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )-{\mathrm e} \left (-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-{\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )\right )\)	\(145\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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=_R
ETURNVERBOSE)

[Out]

(-exp(2*x)-1875/2+3/2*exp(1))/(exp(1)-625+2*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {2 \, x e^{\left (2 \, x\right )}}{4 \, x^{2} + 4 \, x {\left (e - 625\right )} + e^{2} - 1250 \, e + 390625} - \frac {626 \, e^{\left (-e + 625\right )} E_{2}\left (-2 \, x - e + 625\right )}{2 \, x + e - 625} + \frac {3 \, e}{2 \, {\left (2 \, x + e - 625\right )}} - \frac {1875}{2 \, {\left (2 \, x + e - 625\right )}} - \int \frac {2 \, {\left (2 \, x {\left (e + 1\right )} + e^{2} - 626 \, e + 625\right )} e^{\left (2 \, x\right )}}{8 \, x^{3} + 12 \, x^{2} {\left (e - 625\right )} + 6 \, x {\left (e^{2} - 1250 \, e + 390625\right )} + e^{3} - 1875 \, e^{2} + 1171875 \, e - 244140625}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-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, al
gorithm="maxima")

[Out]

-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 + 6
25)/(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 - 24414062
5), x)

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mupad [B]  time = 1.04, size = 19, normalized size = 0.95 \begin {gather*} -\frac {3\,x+{\mathrm {e}}^{2\,x}}{2\,x+\mathrm {e}-625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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) + 390
625),x)

[Out]

-(3*x + exp(2*x))/(2*x + exp(1) - 625)

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sympy [A]  time = 0.23, size = 31, normalized size = 1.55 \begin {gather*} - \frac {1875 - 3 e}{4 x - 1250 + 2 e} - \frac {e^{2 x}}{2 x - 625 + e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-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)

[Out]

-(1875 - 3*E)/(4*x - 1250 + 2*E) - exp(2*x)/(2*x - 625 + E)

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