\(\int e^{25+e^2-3 x-x^2} x (2-3 x-2 x^2) \, dx\) [6015]

Optimal result

Integrand size = 27, antiderivative size = 19 \[ \int e^{25+e^2-3 x-x^2} x \left (2-3 x-2 x^2\right ) \, dx=e^{25+e^2-3 x-x^2} x^2 \]

[Out]

exp(2*ln(x)+exp(2)-x^2-3*x+25)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2326} \[ \int e^{25+e^2-3 x-x^2} x \left (2-3 x-2 x^2\right ) \, dx=\frac {e^{-x^2-3 x+e^2+25} x \left (2 x^2+3 x\right )}{2 x+3} \]

[In]

Int[E^(25 + E^2 - 3*x - x^2)*x*(2 - 3*x - 2*x^2),x]

[Out]

(E^(25 + E^2 - 3*x - x^2)*x*(3*x + 2*x^2))/(3 + 2*x)

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{25+e^2-3 x-x^2} x \left (3 x+2 x^2\right )}{3+2 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int e^{25+e^2-3 x-x^2} x \left (2-3 x-2 x^2\right ) \, dx=e^{25+e^2-3 x-x^2} x^2 \]

[In]

Integrate[E^(25 + E^2 - 3*x - x^2)*x*(2 - 3*x - 2*x^2),x]

[Out]

E^(25 + E^2 - 3*x - x^2)*x^2

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
gosper	\({\mathrm e}^{2 \ln \left (x \right )+{\mathrm e}^{2}-x^{2}-3 x +25}\)	\(18\)
norman	\({\mathrm e}^{2 \ln \left (x \right )+{\mathrm e}^{2}-x^{2}-3 x +25}\)	\(18\)
risch	\(x^{2} {\mathrm e}^{25+{\mathrm e}^{2}-x^{2}-3 x}\)	\(18\)
parallelrisch	\({\mathrm e}^{2 \ln \left (x \right )+{\mathrm e}^{2}-x^{2}-3 x +25}\)	\(18\)
default	\(2 \,{\mathrm e}^{25} {\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{-x^{2}-3 x}}{2}-\frac {3 \sqrt {\pi }\, {\mathrm e}^{\frac {9}{4}} \operatorname {erf}\left (x +\frac {3}{2}\right )}{4}\right )-3 \,{\mathrm e}^{25} {\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {x \,{\mathrm e}^{-x^{2}-3 x}}{2}+\frac {3 \,{\mathrm e}^{-x^{2}-3 x}}{4}+\frac {11 \sqrt {\pi }\, {\mathrm e}^{\frac {9}{4}} \operatorname {erf}\left (x +\frac {3}{2}\right )}{8}\right )-2 \,{\mathrm e}^{25} {\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {x^{2} {\mathrm e}^{-x^{2}-3 x}}{2}+\frac {3 x \,{\mathrm e}^{-x^{2}-3 x}}{4}-\frac {13 \,{\mathrm e}^{-x^{2}-3 x}}{8}-\frac {45 \sqrt {\pi }\, {\mathrm e}^{\frac {9}{4}} \operatorname {erf}\left (x +\frac {3}{2}\right )}{16}\right )\)	\(136\)

[In]

int((-2*x^2-3*x+2)*exp(2*ln(x)+exp(2)-x^2-3*x+25)/x,x,method=_RETURNVERBOSE)

[Out]

exp(2*ln(x)+exp(2)-x^2-3*x+25)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int e^{25+e^2-3 x-x^2} x \left (2-3 x-2 x^2\right ) \, dx=e^{\left (-x^{2} - 3 \, x + e^{2} + 2 \, \log \left (x\right ) + 25\right )} \]

[In]

integrate((-2*x^2-3*x+2)*exp(2*log(x)+exp(2)-x^2-3*x+25)/x,x, algorithm="fricas")

[Out]

e^(-x^2 - 3*x + e^2 + 2*log(x) + 25)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int e^{25+e^2-3 x-x^2} x \left (2-3 x-2 x^2\right ) \, dx=x^{2} e^{- x^{2} - 3 x + e^{2} + 25} \]

[In]

integrate((-2*x**2-3*x+2)*exp(2*ln(x)+exp(2)-x**2-3*x+25)/x,x)

[Out]

x**2*exp(-x**2 - 3*x + exp(2) + 25)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.27 (sec) , antiderivative size = 231, normalized size of antiderivative = 12.16 \[ \int e^{25+e^2-3 x-x^2} x \left (2-3 x-2 x^2\right ) \, dx=\frac {1}{8} i \, {\left (\frac {36 i \, {\left (2 \, x + 3\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{4} \, {\left (2 \, x + 3\right )}^{2}\right )}{{\left ({\left (2 \, x + 3\right )}^{2}\right )}^{\frac {3}{2}}} - \frac {27 i \, \sqrt {\pi } {\left (2 \, x + 3\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {{\left (2 \, x + 3\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (2 \, x + 3\right )}^{2}}} - 54 i \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 3\right )}^{2}\right )} - 8 i \, \Gamma \left (2, \frac {1}{4} \, {\left (2 \, x + 3\right )}^{2}\right )\right )} e^{\left (e^{2} + \frac {109}{4}\right )} + \frac {3}{8} i \, {\left (-\frac {4 i \, {\left (2 \, x + 3\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{4} \, {\left (2 \, x + 3\right )}^{2}\right )}{{\left ({\left (2 \, x + 3\right )}^{2}\right )}^{\frac {3}{2}}} + \frac {9 i \, \sqrt {\pi } {\left (2 \, x + 3\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {{\left (2 \, x + 3\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (2 \, x + 3\right )}^{2}}} + 12 i \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 3\right )}^{2}\right )}\right )} e^{\left (e^{2} + \frac {109}{4}\right )} - \frac {1}{2} i \, {\left (-\frac {3 i \, \sqrt {\pi } {\left (2 \, x + 3\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {{\left (2 \, x + 3\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (2 \, x + 3\right )}^{2}}} - 2 i \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 3\right )}^{2}\right )}\right )} e^{\left (e^{2} + \frac {109}{4}\right )} \]

[In]

integrate((-2*x^2-3*x+2)*exp(2*log(x)+exp(2)-x^2-3*x+25)/x,x, algorithm="maxima")

[Out]

1/8*I*(36*I*(2*x + 3)^3*gamma(3/2, 1/4*(2*x + 3)^2)/((2*x + 3)^2)^(3/2) - 27*I*sqrt(pi)*(2*x + 3)*(erf(1/2*sqr
t((2*x + 3)^2)) - 1)/sqrt((2*x + 3)^2) - 54*I*e^(-1/4*(2*x + 3)^2) - 8*I*gamma(2, 1/4*(2*x + 3)^2))*e^(e^2 + 1
09/4) + 3/8*I*(-4*I*(2*x + 3)^3*gamma(3/2, 1/4*(2*x + 3)^2)/((2*x + 3)^2)^(3/2) + 9*I*sqrt(pi)*(2*x + 3)*(erf(
1/2*sqrt((2*x + 3)^2)) - 1)/sqrt((2*x + 3)^2) + 12*I*e^(-1/4*(2*x + 3)^2))*e^(e^2 + 109/4) - 1/2*I*(-3*I*sqrt(
pi)*(2*x + 3)*(erf(1/2*sqrt((2*x + 3)^2)) - 1)/sqrt((2*x + 3)^2) - 2*I*e^(-1/4*(2*x + 3)^2))*e^(e^2 + 109/4)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int e^{25+e^2-3 x-x^2} x \left (2-3 x-2 x^2\right ) \, dx=e^{\left (-x^{2} - 3 \, x + e^{2} + 2 \, \log \left (x\right ) + 25\right )} \]

[In]

integrate((-2*x^2-3*x+2)*exp(2*log(x)+exp(2)-x^2-3*x+25)/x,x, algorithm="giac")

[Out]

e^(-x^2 - 3*x + e^2 + 2*log(x) + 25)

Mupad [B] (verification not implemented)

Time = 11.83 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int e^{25+e^2-3 x-x^2} x \left (2-3 x-2 x^2\right ) \, dx=x^2\,{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{25}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^{{\mathrm {e}}^2} \]

[In]

int(-(exp(exp(2) - 3*x + 2*log(x) - x^2 + 25)*(3*x + 2*x^2 - 2))/x,x)

[Out]

x^2*exp(-3*x)*exp(25)*exp(-x^2)*exp(exp(2))