3.103.30 \(\int \frac {-395641-9 e^2+32708 x-676 x^2+e^{-3+x} (-1965-9 e+78 x)+e (-3774+156 x)}{395641+9 e^2+e (3774-156 x)-32708 x+676 x^2} \, dx\)

Optimal. Leaf size=24 \[ 4-\frac {e^{-3+x}}{-7+e-\frac {26}{3} (-25+x)}-x \]

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Rubi [B]  time = 0.41, antiderivative size = 144, normalized size of antiderivative = 6.00, number of steps used = 11, number of rules used = 5, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {6741, 27, 6742, 2197, 43} \begin {gather*} -x-\frac {3 e^{x-3}}{-26 x+3 e+629}-\frac {395641+9 e^2}{26 (-26 x+3 e+629)}-\frac {(629+3 e)^2}{26 (-26 x+3 e+629)}+\frac {629 (629+3 e)}{13 (-26 x+3 e+629)}+\frac {9 e^2}{13 (-26 x+3 e+629)}-\frac {1}{13} (629+3 e) \log (-26 x+3 e+629)+\frac {3}{13} e \log (-26 x+3 e+629)+\frac {629}{13} \log (-26 x+3 e+629) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-395641 - 9*E^2 + 32708*x - 676*x^2 + E^(-3 + x)*(-1965 - 9*E + 78*x) + E*(-3774 + 156*x))/(395641 + 9*E^
2 + E*(3774 - 156*x) - 32708*x + 676*x^2),x]

[Out]

(9*E^2)/(13*(629 + 3*E - 26*x)) - (3*E^(-3 + x))/(629 + 3*E - 26*x) + (629*(629 + 3*E))/(13*(629 + 3*E - 26*x)
) - (629 + 3*E)^2/(26*(629 + 3*E - 26*x)) - (395641 + 9*E^2)/(26*(629 + 3*E - 26*x)) - x + (629*Log[629 + 3*E
- 26*x])/13 + (3*E*Log[629 + 3*E - 26*x])/13 - ((629 + 3*E)*Log[629 + 3*E - 26*x])/13

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 {-395641 \left (1+\frac {9 e^2}{395641}\right )+32708 x-676 x^2+e^{-3+x} (-1965-9 e+78 x)+e (-3774+156 x)}{(629+3 e)^2-52 (629+3 e) x+676 x^2} \, dx\\ &=\int \frac {-395641 \left (1+\frac {9 e^2}{395641}\right )+32708 x-676 x^2+e^{-3+x} (-1965-9 e+78 x)+e (-3774+156 x)}{(629+3 e-26 x)^2} \, dx\\ &=\int \left (\frac {-395641-9 e^2}{(629+3 e-26 x)^2}-\frac {3 e^{-3+x} (655+3 e-26 x)}{(629+3 e-26 x)^2}+\frac {32708 x}{(629+3 e-26 x)^2}-\frac {676 x^2}{(629+3 e-26 x)^2}+\frac {6 e (-629+26 x)}{(629+3 e-26 x)^2}\right ) \, dx\\ &=-\frac {395641+9 e^2}{26 (629+3 e-26 x)}-3 \int \frac {e^{-3+x} (655+3 e-26 x)}{(629+3 e-26 x)^2} \, dx-676 \int \frac {x^2}{(629+3 e-26 x)^2} \, dx+32708 \int \frac {x}{(629+3 e-26 x)^2} \, dx+(6 e) \int \frac {-629+26 x}{(629+3 e-26 x)^2} \, dx\\ &=-\frac {3 e^{-3+x}}{629+3 e-26 x}-\frac {395641+9 e^2}{26 (629+3 e-26 x)}-676 \int \left (\frac {1}{676}+\frac {(629+3 e)^2}{676 (629+3 e-26 x)^2}+\frac {-629-3 e}{338 (629+3 e-26 x)}\right ) \, dx+32708 \int \left (\frac {629+3 e}{26 (629+3 e-26 x)^2}-\frac {1}{26 (629+3 e-26 x)}\right ) \, dx+(6 e) \int \left (\frac {3 e}{(629+3 e-26 x)^2}+\frac {1}{-629-3 e+26 x}\right ) \, dx\\ &=\frac {9 e^2}{13 (629+3 e-26 x)}-\frac {3 e^{-3+x}}{629+3 e-26 x}+\frac {629 (629+3 e)}{13 (629+3 e-26 x)}-\frac {(629+3 e)^2}{26 (629+3 e-26 x)}-\frac {395641+9 e^2}{26 (629+3 e-26 x)}-x+\frac {629}{13} \log (629+3 e-26 x)+\frac {3}{13} e \log (629+3 e-26 x)-\frac {1}{13} (629+3 e) \log (629+3 e-26 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 26, normalized size = 1.08 \begin {gather*} \frac {-\frac {3 e^x}{629+3 e-26 x}-e^3 x}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-395641 - 9*E^2 + 32708*x - 676*x^2 + E^(-3 + x)*(-1965 - 9*E + 78*x) + E*(-3774 + 156*x))/(395641
+ 9*E^2 + E*(3774 - 156*x) - 32708*x + 676*x^2),x]

[Out]

((-3*E^x)/(629 + 3*E - 26*x) - E^3*x)/E^3

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fricas [A]  time = 1.10, size = 33, normalized size = 1.38 \begin {gather*} -\frac {26 \, x^{2} - 3 \, x e - 629 \, x - 3 \, e^{\left (x - 3\right )}}{26 \, x - 3 \, e - 629} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*exp(1)+78*x-1965)*exp(x-3)-9*exp(1)^2+(156*x-3774)*exp(1)-676*x^2+32708*x-395641)/(9*exp(1)^2+(
-156*x+3774)*exp(1)+676*x^2-32708*x+395641),x, algorithm="fricas")

[Out]

-(26*x^2 - 3*x*e - 629*x - 3*e^(x - 3))/(26*x - 3*e - 629)

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giac [A]  time = 0.15, size = 40, normalized size = 1.67 \begin {gather*} -\frac {26 \, x^{2} e^{3} - 3 \, x e^{4} - 629 \, x e^{3} - 3 \, e^{x}}{26 \, x e^{3} - 3 \, e^{4} - 629 \, e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*exp(1)+78*x-1965)*exp(x-3)-9*exp(1)^2+(156*x-3774)*exp(1)-676*x^2+32708*x-395641)/(9*exp(1)^2+(
-156*x+3774)*exp(1)+676*x^2-32708*x+395641),x, algorithm="giac")

[Out]

-(26*x^2*e^3 - 3*x*e^4 - 629*x*e^3 - 3*e^x)/(26*x*e^3 - 3*e^4 - 629*e^3)

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maple [A]  time = 0.27, size = 36, normalized size = 1.50

method	result	size
norman	\(\frac {26 x^{2}-3 \,{\mathrm e}^{x -3}-\frac {395641}{26}-\frac {9 \,{\mathrm e}^{2}}{26}-\frac {1887 \,{\mathrm e}}{13}}{3 \,{\mathrm e}-26 x +629}\)	\(36\)
derivativedivides	\(\frac {303601}{26 \left (3 \,{\mathrm e}-26 x +629\right )}+\frac {26 \left (x -3\right )^{2}-\frac {9 \,{\mathrm e}^{2}}{13}-\frac {3306 \,{\mathrm e}}{13}-\frac {303601}{13}}{3 \,{\mathrm e}-26 x +629}+\frac {9 \,{\mathrm e}^{2}}{26 \left (3 \,{\mathrm e}-26 x +629\right )}-\frac {1731 \,{\mathrm e}^{x -3}}{26 \left (3 \,{\mathrm e}-26 x +629\right )}+\frac {1731 \,{\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \expIntegralEi \left (1, \frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )}{676}+\frac {1653 \,{\mathrm e}}{13 \left (3 \,{\mathrm e}-26 x +629\right )}+\frac {3 \,{\mathrm e}^{x -3} \left (3 \,{\mathrm e}+551\right )}{26 \left (3 \,{\mathrm e}-26 x +629\right )}-78 \left (\frac {3 \,{\mathrm e}}{17576}+\frac {577}{17576}\right ) {\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \expIntegralEi \left (1, \frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )-9 \,{\mathrm e} \left (\frac {{\mathrm e}^{x -3}}{78 \,{\mathrm e}-676 x +16354}-\frac {{\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \expIntegralEi \left (1, \frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )}{676}\right )\)	\(206\)
default	\(\frac {303601}{26 \left (3 \,{\mathrm e}-26 x +629\right )}+\frac {26 \left (x -3\right )^{2}-\frac {9 \,{\mathrm e}^{2}}{13}-\frac {3306 \,{\mathrm e}}{13}-\frac {303601}{13}}{3 \,{\mathrm e}-26 x +629}+\frac {9 \,{\mathrm e}^{2}}{26 \left (3 \,{\mathrm e}-26 x +629\right )}-\frac {1731 \,{\mathrm e}^{x -3}}{26 \left (3 \,{\mathrm e}-26 x +629\right )}+\frac {1731 \,{\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \expIntegralEi \left (1, \frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )}{676}+\frac {1653 \,{\mathrm e}}{13 \left (3 \,{\mathrm e}-26 x +629\right )}+\frac {3 \,{\mathrm e}^{x -3} \left (3 \,{\mathrm e}+551\right )}{26 \left (3 \,{\mathrm e}-26 x +629\right )}-78 \left (\frac {3 \,{\mathrm e}}{17576}+\frac {577}{17576}\right ) {\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \expIntegralEi \left (1, \frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )-9 \,{\mathrm e} \left (\frac {{\mathrm e}^{x -3}}{78 \,{\mathrm e}-676 x +16354}-\frac {{\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \expIntegralEi \left (1, \frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )}{676}\right )\)	\(206\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-9*exp(1)+78*x-1965)*exp(x-3)-9*exp(1)^2+(156*x-3774)*exp(1)-676*x^2+32708*x-395641)/(9*exp(1)^2+(-156*x
+3774)*exp(1)+676*x^2-32708*x+395641),x,method=_RETURNVERBOSE)

[Out]

(26*x^2-3*exp(x-3)-395641/26-9/26*exp(1)^2-1887/13*exp(1))/(3*exp(1)-26*x+629)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {3}{13} \, {\left (\frac {3 \, e + 629}{26 \, x - 3 \, e - 629} - \log \left (26 \, x - 3 \, e - 629\right )\right )} e - \frac {1}{13} \, {\left (3 \, e + 629\right )} \log \left (26 \, x - 3 \, e - 629\right ) - x + \frac {78 \, x e^{x}}{676 \, x^{2} e^{3} - 52 \, x {\left (3 \, e^{4} + 629 \, e^{3}\right )} + 9 \, e^{5} + 3774 \, e^{4} + 395641 \, e^{3}} + \frac {1965 \, e^{\left (\frac {3}{26} \, e + \frac {551}{26}\right )} E_{2}\left (-x + \frac {3}{26} \, e + \frac {629}{26}\right )}{26 \, {\left (26 \, x - 3 \, e - 629\right )}} + \frac {9 \, e^{2} + 3774 \, e + 395641}{26 \, {\left (26 \, x - 3 \, e - 629\right )}} - \frac {629 \, {\left (3 \, e + 629\right )}}{13 \, {\left (26 \, x - 3 \, e - 629\right )}} + \frac {9 \, e^{2}}{26 \, {\left (26 \, x - 3 \, e - 629\right )}} + \frac {1887 \, e}{13 \, {\left (26 \, x - 3 \, e - 629\right )}} + \frac {395641}{26 \, {\left (26 \, x - 3 \, e - 629\right )}} - \int \frac {3 \, {\left (26 \, x {\left (3 \, e - 26\right )} - 9 \, e^{2} - 1965 \, e - 16354\right )} e^{x}}{17576 \, x^{3} e^{3} - 2028 \, x^{2} {\left (3 \, e^{4} + 629 \, e^{3}\right )} + 78 \, x {\left (9 \, e^{5} + 3774 \, e^{4} + 395641 \, e^{3}\right )} - 27 \, e^{6} - 16983 \, e^{5} - 3560769 \, e^{4} - 248858189 \, e^{3}}\,{d x} + \frac {629}{13} \, \log \left (26 \, x - 3 \, e - 629\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*exp(1)+78*x-1965)*exp(x-3)-9*exp(1)^2+(156*x-3774)*exp(1)-676*x^2+32708*x-395641)/(9*exp(1)^2+(
-156*x+3774)*exp(1)+676*x^2-32708*x+395641),x, algorithm="maxima")

[Out]

-3/13*((3*e + 629)/(26*x - 3*e - 629) - log(26*x - 3*e - 629))*e - 1/13*(3*e + 629)*log(26*x - 3*e - 629) - x
+ 78*x*e^x/(676*x^2*e^3 - 52*x*(3*e^4 + 629*e^3) + 9*e^5 + 3774*e^4 + 395641*e^3) + 1965/26*e^(3/26*e + 551/26
)*exp_integral_e(2, -x + 3/26*e + 629/26)/(26*x - 3*e - 629) + 1/26*(9*e^2 + 3774*e + 395641)/(26*x - 3*e - 62
9) - 629/13*(3*e + 629)/(26*x - 3*e - 629) + 9/26*e^2/(26*x - 3*e - 629) + 1887/13*e/(26*x - 3*e - 629) + 3956
41/26/(26*x - 3*e - 629) - integrate(3*(26*x*(3*e - 26) - 9*e^2 - 1965*e - 16354)*e^x/(17576*x^3*e^3 - 2028*x^
2*(3*e^4 + 629*e^3) + 78*x*(9*e^5 + 3774*e^4 + 395641*e^3) - 27*e^6 - 16983*e^5 - 3560769*e^4 - 248858189*e^3)
, x) + 629/13*log(26*x - 3*e - 629)

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mupad [B]  time = 6.01, size = 21, normalized size = 0.88 \begin {gather*} -x-\frac {3\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^x}{3\,\mathrm {e}-26\,x+629} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*exp(2) - 32708*x + exp(x - 3)*(9*exp(1) - 78*x + 1965) + 676*x^2 - exp(1)*(156*x - 3774) + 395641)/(9*
exp(2) - 32708*x + 676*x^2 - exp(1)*(156*x - 3774) + 395641),x)

[Out]

- x - (3*exp(-3)*exp(x))/(3*exp(1) - 26*x + 629)

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sympy [A]  time = 0.16, size = 17, normalized size = 0.71 \begin {gather*} - x + \frac {3 e^{x - 3}}{26 x - 629 - 3 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*exp(1)+78*x-1965)*exp(x-3)-9*exp(1)**2+(156*x-3774)*exp(1)-676*x**2+32708*x-395641)/(9*exp(1)**
2+(-156*x+3774)*exp(1)+676*x**2-32708*x+395641),x)

[Out]

-x + 3*exp(x - 3)/(26*x - 629 - 3*E)

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