3.19.46 \(\int \frac {e^{\frac {-26 x+15 x^2}{-26+13 x}} (52-60 x+15 x^2)}{52-52 x+13 x^2} \, dx\)

Optimal. Leaf size=21 \[ e^{\frac {\left (2-\frac {15 x}{13}\right ) x}{2-x}}+\log (5) \]

________________________________________________________________________________________

Rubi [A]  time = 0.37, antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {27, 12, 6741, 6706} \begin {gather*} e^{\frac {(26-15 x) x}{13 (2-x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-26*x + 15*x^2)/(-26 + 13*x))*(52 - 60*x + 15*x^2))/(52 - 52*x + 13*x^2),x]

[Out]

E^(((26 - 15*x)*x)/(13*(2 - x)))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {-26 x+15 x^2}{-26+13 x}} \left (52-60 x+15 x^2\right )}{13 (-2+x)^2} \, dx\\ &=\frac {1}{13} \int \frac {e^{\frac {-26 x+15 x^2}{-26+13 x}} \left (52-60 x+15 x^2\right )}{(-2+x)^2} \, dx\\ &=\frac {1}{13} \int \frac {e^{\frac {x (-26+15 x)}{-26+13 x}} \left (52-60 x+15 x^2\right )}{(2-x)^2} \, dx\\ &=e^{\frac {(26-15 x) x}{13 (2-x)}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 17, normalized size = 0.81 \begin {gather*} e^{\frac {x (-26+15 x)}{13 (-2+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-26*x + 15*x^2)/(-26 + 13*x))*(52 - 60*x + 15*x^2))/(52 - 52*x + 13*x^2),x]

[Out]

E^((x*(-26 + 15*x))/(13*(-2 + x)))

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 17, normalized size = 0.81 \begin {gather*} e^{\left (\frac {15 \, x^{2} - 26 \, x}{13 \, {\left (x - 2\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x^2-60*x+52)*exp((15*x^2-26*x)/(13*x-26))/(13*x^2-52*x+52),x, algorithm="fricas")

[Out]

e^(1/13*(15*x^2 - 26*x)/(x - 2))

________________________________________________________________________________________

giac [A]  time = 0.17, size = 20, normalized size = 0.95 \begin {gather*} e^{\left (\frac {15 \, x^{2}}{13 \, {\left (x - 2\right )}} - \frac {2 \, x}{x - 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x^2-60*x+52)*exp((15*x^2-26*x)/(13*x-26))/(13*x^2-52*x+52),x, algorithm="giac")

[Out]

e^(15/13*x^2/(x - 2) - 2*x/(x - 2))

________________________________________________________________________________________

maple [A]  time = 0.38, size = 15, normalized size = 0.71

method	result	size
gosper	\({\mathrm e}^{\frac {x \left (15 x -26\right )}{13 x -26}}\)	\(15\)
risch	\({\mathrm e}^{\frac {x \left (15 x -26\right )}{13 x -26}}\)	\(15\)
norman	\(\frac {x \,{\mathrm e}^{\frac {15 x^{2}-26 x}{13 x -26}}-2 \,{\mathrm e}^{\frac {15 x^{2}-26 x}{13 x -26}}}{x -2}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((15*x^2-60*x+52)*exp((15*x^2-26*x)/(13*x-26))/(13*x^2-52*x+52),x,method=_RETURNVERBOSE)

[Out]

exp(1/13*x*(15*x-26)/(x-2))

________________________________________________________________________________________

maxima [A]  time = 0.53, size = 13, normalized size = 0.62 \begin {gather*} e^{\left (\frac {15}{13} \, x + \frac {8}{13 \, {\left (x - 2\right )}} + \frac {4}{13}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x^2-60*x+52)*exp((15*x^2-26*x)/(13*x-26))/(13*x^2-52*x+52),x, algorithm="maxima")

[Out]

e^(15/13*x + 8/13/(x - 2) + 4/13)

________________________________________________________________________________________

mupad [B]  time = 1.22, size = 23, normalized size = 1.10 \begin {gather*} {\mathrm {e}}^{\frac {15\,x^2}{13\,x-26}}\,{\mathrm {e}}^{-\frac {2\,x}{x-2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(26*x - 15*x^2)/(13*x - 26))*(15*x^2 - 60*x + 52))/(13*x^2 - 52*x + 52),x)

[Out]

exp((15*x^2)/(13*x - 26))*exp(-(2*x)/(x - 2))

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 14, normalized size = 0.67 \begin {gather*} e^{\frac {15 x^{2} - 26 x}{13 x - 26}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x**2-60*x+52)*exp((15*x**2-26*x)/(13*x-26))/(13*x**2-52*x+52),x)

[Out]

exp((15*x**2 - 26*x)/(13*x - 26))

________________________________________________________________________________________