3.101.14 \(\int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x (104-200 x-27 x^2+76 x^3-25 x^4)}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx\)

Optimal. Leaf size=34 \[ -1+x+\frac {e^x x}{\left (-x+25 x^2\right ) \left (-1+\frac {x^2}{2-x}\right )} \]

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Rubi [A]  time = 0.51, antiderivative size = 42, normalized size of antiderivative = 1.24, number of steps used = 13, number of rules used = 4, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {6688, 6742, 2177, 2178} \begin {gather*} x+\frac {1225 e^x}{1224 (1-25 x)}-\frac {e^x}{72 (1-x)}+\frac {4 e^x}{153 (x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 204*x + 2697*x^2 - 2348*x^3 - 1974*x^4 + 1200*x^5 + 625*x^6 + E^x*(104 - 200*x - 27*x^2 + 76*x^3 - 25
*x^4))/(4 - 204*x + 2697*x^2 - 2348*x^3 - 1974*x^4 + 1200*x^5 + 625*x^6),x]

[Out]

(1225*E^x)/(1224*(1 - 25*x)) - E^x/(72*(1 - x)) + x + (4*E^x)/(153*(2 + x))

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 6688

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

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 \left (1+\frac {e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{\left (2-51 x+24 x^2+25 x^3\right )^2}\right ) \, dx\\ &=x+\int \frac {e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{\left (2-51 x+24 x^2+25 x^3\right )^2} \, dx\\ &=x+\int \left (-\frac {e^x}{72 (-1+x)^2}+\frac {e^x}{72 (-1+x)}-\frac {4 e^x}{153 (2+x)^2}+\frac {4 e^x}{153 (2+x)}+\frac {30625 e^x}{1224 (-1+25 x)^2}-\frac {1225 e^x}{1224 (-1+25 x)}\right ) \, dx\\ &=x-\frac {1}{72} \int \frac {e^x}{(-1+x)^2} \, dx+\frac {1}{72} \int \frac {e^x}{-1+x} \, dx-\frac {4}{153} \int \frac {e^x}{(2+x)^2} \, dx+\frac {4}{153} \int \frac {e^x}{2+x} \, dx-\frac {1225 \int \frac {e^x}{-1+25 x} \, dx}{1224}+\frac {30625 \int \frac {e^x}{(-1+25 x)^2} \, dx}{1224}\\ &=\frac {1225 e^x}{1224 (1-25 x)}-\frac {e^x}{72 (1-x)}+x+\frac {4 e^x}{153 (2+x)}+\frac {1}{72} e \text {Ei}(-1+x)+\frac {4 \text {Ei}(2+x)}{153 e^2}-\frac {49 \sqrt [25]{e} \text {Ei}\left (\frac {1}{25} (-1+25 x)\right )}{1224}-\frac {1}{72} \int \frac {e^x}{-1+x} \, dx-\frac {4}{153} \int \frac {e^x}{2+x} \, dx+\frac {1225 \int \frac {e^x}{-1+25 x} \, dx}{1224}\\ &=\frac {1225 e^x}{1224 (1-25 x)}-\frac {e^x}{72 (1-x)}+x+\frac {4 e^x}{153 (2+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 42, normalized size = 1.24 \begin {gather*} \frac {1225 e^x}{1224 (1-25 x)}-\frac {e^x}{72 (1-x)}+x+\frac {4 e^x}{153 (2+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 204*x + 2697*x^2 - 2348*x^3 - 1974*x^4 + 1200*x^5 + 625*x^6 + E^x*(104 - 200*x - 27*x^2 + 76*x^
3 - 25*x^4))/(4 - 204*x + 2697*x^2 - 2348*x^3 - 1974*x^4 + 1200*x^5 + 625*x^6),x]

[Out]

(1225*E^x)/(1224*(1 - 25*x)) - E^x/(72*(1 - x)) + x + (4*E^x)/(153*(2 + x))

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fricas [A]  time = 0.83, size = 44, normalized size = 1.29 \begin {gather*} \frac {25 \, x^{4} + 24 \, x^{3} - 51 \, x^{2} - {\left (x - 2\right )} e^{x} + 2 \, x}{25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^4+76*x^3-27*x^2-200*x+104)*exp(x)+625*x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^2-204*x+4)/(625*
x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^2-204*x+4),x, algorithm="fricas")

[Out]

(25*x^4 + 24*x^3 - 51*x^2 - (x - 2)*e^x + 2*x)/(25*x^3 + 24*x^2 - 51*x + 2)

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giac [A]  time = 0.14, size = 46, normalized size = 1.35 \begin {gather*} \frac {25 \, x^{4} + 24 \, x^{3} - 51 \, x^{2} - x e^{x} + 2 \, x + 2 \, e^{x}}{25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^4+76*x^3-27*x^2-200*x+104)*exp(x)+625*x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^2-204*x+4)/(625*
x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^2-204*x+4),x, algorithm="giac")

[Out]

(25*x^4 + 24*x^3 - 51*x^2 - x*e^x + 2*x + 2*e^x)/(25*x^3 + 24*x^2 - 51*x + 2)

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maple [A]  time = 0.15, size = 27, normalized size = 0.79

method	result	size
risch	\(x -\frac {\left (x -2\right ) {\mathrm e}^{x}}{25 x^{3}+24 x^{2}-51 x +2}\)	\(27\)
default	\(x +\frac {{\mathrm e}^{x}}{72 x -72}+\frac {4 \,{\mathrm e}^{x}}{153 \left (2+x \right )}-\frac {49 \,{\mathrm e}^{x}}{1224 \left (x -\frac {1}{25}\right )}\)	\(30\)
norman	\(\frac {-51 x^{2}+2 x +24 x^{3}+25 x^{4}-{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}}{25 x^{3}+24 x^{2}-51 x +2}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-25*x^4+76*x^3-27*x^2-200*x+104)*exp(x)+625*x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^2-204*x+4)/(625*x^6+12
00*x^5-1974*x^4-2348*x^3+2697*x^2-204*x+4),x,method=_RETURNVERBOSE)

[Out]

x-(x-2)/(25*x^3+24*x^2-51*x+2)*exp(x)

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maxima [B]  time = 0.41, size = 229, normalized size = 6.74 \begin {gather*} x - \frac {{\left (x - 2\right )} e^{x}}{25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2} - \frac {285481771 \, x^{2} - 240455729 \, x + 9161458}{6242400 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} + \frac {4580729 \, x^{2} - 7021771 \, x + 273542}{130050 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} + \frac {329 \, {\left (136771 \, x^{2} - 51929 \, x + 1858\right )}}{1040400 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} - \frac {4075 \, x^{2} + 4687 \, x - 5294}{62424 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} - \frac {899 \, {\left (2275 \, x^{2} + 1255 \, x - 62\right )}}{83232 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} - \frac {587 \, {\left (929 \, x^{2} - 4579 \, x + 182\right )}}{62424 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} + \frac {775 \, x^{2} + 3019 \, x - 326}{1224 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^4+76*x^3-27*x^2-200*x+104)*exp(x)+625*x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^2-204*x+4)/(625*
x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^2-204*x+4),x, algorithm="maxima")

[Out]

x - (x - 2)*e^x/(25*x^3 + 24*x^2 - 51*x + 2) - 1/6242400*(285481771*x^2 - 240455729*x + 9161458)/(25*x^3 + 24*
x^2 - 51*x + 2) + 1/130050*(4580729*x^2 - 7021771*x + 273542)/(25*x^3 + 24*x^2 - 51*x + 2) + 329/1040400*(1367
71*x^2 - 51929*x + 1858)/(25*x^3 + 24*x^2 - 51*x + 2) - 1/62424*(4075*x^2 + 4687*x - 5294)/(25*x^3 + 24*x^2 -
51*x + 2) - 899/83232*(2275*x^2 + 1255*x - 62)/(25*x^3 + 24*x^2 - 51*x + 2) - 587/62424*(929*x^2 - 4579*x + 18
2)/(25*x^3 + 24*x^2 - 51*x + 2) + 1/1224*(775*x^2 + 3019*x - 326)/(25*x^3 + 24*x^2 - 51*x + 2)

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mupad [B]  time = 7.11, size = 30, normalized size = 0.88 \begin {gather*} x+\frac {2\,{\mathrm {e}}^x-x\,{\mathrm {e}}^x}{\left (25\,x-1\right )\,\left (x-1\right )\,\left (x+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(204*x + exp(x)*(200*x + 27*x^2 - 76*x^3 + 25*x^4 - 104) - 2697*x^2 + 2348*x^3 + 1974*x^4 - 1200*x^5 - 62
5*x^6 - 4)/(2697*x^2 - 204*x - 2348*x^3 - 1974*x^4 + 1200*x^5 + 625*x^6 + 4),x)

[Out]

x + (2*exp(x) - x*exp(x))/((25*x - 1)*(x - 1)*(x + 2))

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sympy [A]  time = 0.13, size = 22, normalized size = 0.65 \begin {gather*} x + \frac {\left (2 - x\right ) e^{x}}{25 x^{3} + 24 x^{2} - 51 x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x**4+76*x**3-27*x**2-200*x+104)*exp(x)+625*x**6+1200*x**5-1974*x**4-2348*x**3+2697*x**2-204*x+
4)/(625*x**6+1200*x**5-1974*x**4-2348*x**3+2697*x**2-204*x+4),x)

[Out]

x + (2 - x)*exp(x)/(25*x**3 + 24*x**2 - 51*x + 2)

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