\(\int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} (626-250 x-25 x^2)+e^x (-125-175 x-55 x^2-5 x^3)}{125+50 x+5 x^2} \, dx\) [9684]

Optimal result

Integrand size = 64, antiderivative size = 27 \[ \int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} \left (626-250 x-25 x^2\right )+e^x \left (-125-175 x-55 x^2-5 x^3\right )}{125+50 x+5 x^2} \, dx=e^{\frac {-\frac {1}{5}+5 (5-x) x}{5+x}}-e^x x \]

[Out]

exp((5*x*(5-x)-1/5)/(5+x))-exp(x)*x

Rubi [F]

\[ \int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} \left (626-250 x-25 x^2\right )+e^x \left (-125-175 x-55 x^2-5 x^3\right )}{125+50 x+5 x^2} \, dx=\int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} \left (626-250 x-25 x^2\right )+e^x \left (-125-175 x-55 x^2-5 x^3\right )}{125+50 x+5 x^2} \, dx \]

[In]

Int[(E^((-1 + 125*x - 25*x^2)/(25 + 5*x))*(626 - 250*x - 25*x^2) + E^x*(-125 - 175*x - 55*x^2 - 5*x^3))/(125 +
50*x + 5*x^2),x]

[Out]

E^x - E^x*(1 + x) - 5*Defer[Int][E^((-1 + 125*x - 25*x^2)/(5*(5 + x))), x] + (1251*Defer[Int][E^((-1 + 125*x -
25*x^2)/(5*(5 + x)))/(5 + x)^2, x])/5

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} \left (626-250 x-25 x^2\right )+e^x \left (-125-175 x-55 x^2-5 x^3\right )}{5 (5+x)^2} \, dx \\ & = \frac {1}{5} \int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} \left (626-250 x-25 x^2\right )+e^x \left (-125-175 x-55 x^2-5 x^3\right )}{(5+x)^2} \, dx \\ & = \frac {1}{5} \int \left (-5 e^x (1+x)+\frac {e^{\frac {-1+125 x-25 x^2}{5 (5+x)}} \left (626-250 x-25 x^2\right )}{(5+x)^2}\right ) \, dx \\ & = \frac {1}{5} \int \frac {e^{\frac {-1+125 x-25 x^2}{5 (5+x)}} \left (626-250 x-25 x^2\right )}{(5+x)^2} \, dx-\int e^x (1+x) \, dx \\ & = -e^x (1+x)+\frac {1}{5} \int \left (-25 e^{\frac {-1+125 x-25 x^2}{5 (5+x)}}+\frac {1251 e^{\frac {-1+125 x-25 x^2}{5 (5+x)}}}{(5+x)^2}\right ) \, dx+\int e^x \, dx \\ & = e^x-e^x (1+x)-5 \int e^{\frac {-1+125 x-25 x^2}{5 (5+x)}} \, dx+\frac {1251}{5} \int \frac {e^{\frac {-1+125 x-25 x^2}{5 (5+x)}}}{(5+x)^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} \left (626-250 x-25 x^2\right )+e^x \left (-125-175 x-55 x^2-5 x^3\right )}{125+50 x+5 x^2} \, dx=e^{75-\frac {1251}{5 (5+x)}-5 (5+x)}-e^x x \]

[In]

Integrate[(E^((-1 + 125*x - 25*x^2)/(25 + 5*x))*(626 - 250*x - 25*x^2) + E^x*(-125 - 175*x - 55*x^2 - 5*x^3))/
(125 + 50*x + 5*x^2),x]

[Out]

E^(75 - 1251/(5*(5 + x)) - 5*(5 + x)) - E^x*x

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

method	result	size
risch	\(-{\mathrm e}^{x} x +{\mathrm e}^{-\frac {25 x^{2}-125 x +1}{5 \left (5+x \right )}}\)	\(25\)
parallelrisch	\(-{\mathrm e}^{x} x +{\mathrm e}^{-\frac {25 x^{2}-125 x +1}{5 \left (5+x \right )}}\)	\(25\)
parts	\(-{\mathrm e}^{x} x +\frac {x \,{\mathrm e}^{\frac {-25 x^{2}+125 x -1}{25+5 x}}+5 \,{\mathrm e}^{\frac {-25 x^{2}+125 x -1}{25+5 x}}}{5+x}\)	\(56\)
norman	\(\frac {x \,{\mathrm e}^{\frac {-25 x^{2}+125 x -1}{25+5 x}}-5 \,{\mathrm e}^{x} x -{\mathrm e}^{x} x^{2}+5 \,{\mathrm e}^{\frac {-25 x^{2}+125 x -1}{25+5 x}}}{5+x}\)	\(62\)

[In]

int(((-5*x^3-55*x^2-175*x-125)*exp(x)+(-25*x^2-250*x+626)*exp((-25*x^2+125*x-1)/(25+5*x)))/(5*x^2+50*x+125),x,
method=_RETURNVERBOSE)

[Out]

-exp(x)*x+exp(-1/5*(25*x^2-125*x+1)/(5+x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} \left (626-250 x-25 x^2\right )+e^x \left (-125-175 x-55 x^2-5 x^3\right )}{125+50 x+5 x^2} \, dx=-x e^{x} + e^{\left (-\frac {25 \, x^{2} - 125 \, x + 1}{5 \, {\left (x + 5\right )}}\right )} \]

[In]

integrate(((-5*x^3-55*x^2-175*x-125)*exp(x)+(-25*x^2-250*x+626)*exp((-25*x^2+125*x-1)/(25+5*x)))/(5*x^2+50*x+1
25),x, algorithm="fricas")

[Out]

-x*e^x + e^(-1/5*(25*x^2 - 125*x + 1)/(x + 5))

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} \left (626-250 x-25 x^2\right )+e^x \left (-125-175 x-55 x^2-5 x^3\right )}{125+50 x+5 x^2} \, dx=- x e^{x} + e^{\frac {- 25 x^{2} + 125 x - 1}{5 x + 25}} \]

[In]

integrate(((-5*x**3-55*x**2-175*x-125)*exp(x)+(-25*x**2-250*x+626)*exp((-25*x**2+125*x-1)/(25+5*x)))/(5*x**2+5
0*x+125),x)

[Out]

-x*exp(x) + exp((-25*x**2 + 125*x - 1)/(5*x + 25))

Maxima [F]

\[ \int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} \left (626-250 x-25 x^2\right )+e^x \left (-125-175 x-55 x^2-5 x^3\right )}{125+50 x+5 x^2} \, dx=\int { -\frac {5 \, {\left (x^{3} + 11 \, x^{2} + 35 \, x + 25\right )} e^{x} + {\left (25 \, x^{2} + 250 \, x - 626\right )} e^{\left (-\frac {25 \, x^{2} - 125 \, x + 1}{5 \, {\left (x + 5\right )}}\right )}}{5 \, {\left (x^{2} + 10 \, x + 25\right )}} \,d x } \]

[In]

integrate(((-5*x^3-55*x^2-175*x-125)*exp(x)+(-25*x^2-250*x+626)*exp((-25*x^2+125*x-1)/(25+5*x)))/(5*x^2+50*x+1
25),x, algorithm="maxima")

[Out]

-(x*e^(6*x) - e^(-1251/5/(x + 5) + 50))*e^(-5*x) + 25*e^(-5)*exp_integral_e(2, -x - 5)/(x + 5) + 25*integrate(
e^x/(x^2 + 10*x + 25), x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} \left (626-250 x-25 x^2\right )+e^x \left (-125-175 x-55 x^2-5 x^3\right )}{125+50 x+5 x^2} \, dx=-x e^{x} + e^{\left (-\frac {125 \, x^{2} - 626 \, x}{25 \, {\left (x + 5\right )}} - \frac {1}{25}\right )} \]

[In]

integrate(((-5*x^3-55*x^2-175*x-125)*exp(x)+(-25*x^2-250*x+626)*exp((-25*x^2+125*x-1)/(25+5*x)))/(5*x^2+50*x+1
25),x, algorithm="giac")

[Out]

-x*e^x + e^(-1/25*(125*x^2 - 626*x)/(x + 5) - 1/25)

Mupad [B] (verification not implemented)

Time = 13.94 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^{\frac {-1+125 x-25 x^2}{25+5 x}} \left (626-250 x-25 x^2\right )+e^x \left (-125-175 x-55 x^2-5 x^3\right )}{125+50 x+5 x^2} \, dx={\mathrm {e}}^{-\frac {25\,x^2}{5\,x+25}}\,{\mathrm {e}}^{-\frac {1}{5\,x+25}}\,{\mathrm {e}}^{\frac {125\,x}{5\,x+25}}-x\,{\mathrm {e}}^x \]

[In]

int(-(exp(-(25*x^2 - 125*x + 1)/(5*x + 25))*(250*x + 25*x^2 - 626) + exp(x)*(175*x + 55*x^2 + 5*x^3 + 125))/(5
0*x + 5*x^2 + 125),x)

[Out]

exp(-(25*x^2)/(5*x + 25))*exp(-1/(5*x + 25))*exp((125*x)/(5*x + 25)) - x*exp(x)