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

Optimal. Leaf size=19 \[ e^{25+e^2-3 x-x^2} x^2 \]

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Rubi [A]  time = 0.05, antiderivative size = 33, normalized size of antiderivative = 1.74, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2288} \begin {gather*} \frac {e^{-x^2-3 x+e^2+25} x \left (2 x^2+3 x\right )}{2 x+3} \end {gather*}

Antiderivative was successfully verified.

[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 2288

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

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} e^{25+e^2-3 x-x^2} x^2 \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.80, size = 17, normalized size = 0.89 \begin {gather*} e^{\left (-x^{2} - 3 \, x + e^{2} + 2 \, \log \relax (x) + 25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.13, size = 17, normalized size = 0.89 \begin {gather*} e^{\left (-x^{2} - 3 \, x + e^{2} + 2 \, \log \relax (x) + 25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.03, size = 18, normalized size = 0.95

method	result	size
gosper	\({\mathrm e}^{2 \ln \relax (x )+{\mathrm e}^{2}-x^{2}-3 x +25}\)	\(18\)
default	\(x^{2} {\mathrm e}^{25+{\mathrm e}^{2}-x^{2}-3 x}\)	\(18\)
norman	\({\mathrm e}^{2 \ln \relax (x )+{\mathrm e}^{2}-x^{2}-3 x +25}\)	\(18\)
risch	\(x^{2} {\mathrm e}^{25+{\mathrm e}^{2}-x^{2}-3 x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [C]  time = 0.43, size = 231, normalized size = 12.16 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.26, size = 19, normalized size = 1.00 \begin {gather*} x^2\,{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{25}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^{{\mathrm {e}}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.19, size = 15, normalized size = 0.79 \begin {gather*} x^{2} e^{- x^{2} - 3 x + e^{2} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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