3.87.16 \(\int \frac {-450 x-60 e^2 x-2 e^4 x+6 e^{\frac {x^2}{225+30 e^2+e^4}} x}{225+30 e^2+e^4} \, dx\)

Optimal. Leaf size=22 \[ 4+3 e^{\frac {x^2}{\left (15+e^2\right )^2}}-x^2 \]

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Rubi [A]  time = 0.03, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 3, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {6, 12, 2209} \begin {gather*} 3 e^{\frac {x^2}{\left (15+e^2\right )^2}}-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-450*x - 60*E^2*x - 2*E^4*x + 6*E^(x^2/(225 + 30*E^2 + E^4))*x)/(225 + 30*E^2 + E^4),x]

[Out]

3*E^(x^2/(15 + E^2)^2) - x^2

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 e^4 x+6 e^{\frac {x^2}{225+30 e^2+e^4}} x+\left (-450-60 e^2\right ) x}{225+30 e^2+e^4} \, dx\\ &=\int \frac {6 e^{\frac {x^2}{225+30 e^2+e^4}} x+\left (-450-60 e^2-2 e^4\right ) x}{225+30 e^2+e^4} \, dx\\ &=\frac {\int \left (6 e^{\frac {x^2}{225+30 e^2+e^4}} x+\left (-450-60 e^2-2 e^4\right ) x\right ) \, dx}{\left (15+e^2\right )^2}\\ &=-x^2+\frac {6 \int e^{\frac {x^2}{225+30 e^2+e^4}} x \, dx}{\left (15+e^2\right )^2}\\ &=3 e^{\frac {x^2}{\left (15+e^2\right )^2}}-x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 21, normalized size = 0.95 \begin {gather*} 3 e^{\frac {x^2}{\left (15+e^2\right )^2}}-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-450*x - 60*E^2*x - 2*E^4*x + 6*E^(x^2/(225 + 30*E^2 + E^4))*x)/(225 + 30*E^2 + E^4),x]

[Out]

3*E^(x^2/(15 + E^2)^2) - x^2

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fricas [A]  time = 0.62, size = 23, normalized size = 1.05 \begin {gather*} -x^{2} + 3 \, e^{\left (\frac {x^{2}}{e^{4} + 30 \, e^{2} + 225}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(x^2/(exp(2)^2+30*exp(2)+225))-2*x*exp(2)^2-60*exp(2)*x-450*x)/(exp(2)^2+30*exp(2)+225),x, a
lgorithm="fricas")

[Out]

-x^2 + 3*e^(x^2/(e^4 + 30*e^2 + 225))

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giac [B]  time = 0.14, size = 56, normalized size = 2.55 \begin {gather*} -\frac {x^{2} e^{4} + 30 \, x^{2} e^{2} + 225 \, x^{2} - 3 \, {\left (e^{4} + 30 \, e^{2} + 225\right )} e^{\left (\frac {x^{2}}{e^{4} + 30 \, e^{2} + 225}\right )}}{e^{4} + 30 \, e^{2} + 225} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(x^2/(exp(2)^2+30*exp(2)+225))-2*x*exp(2)^2-60*exp(2)*x-450*x)/(exp(2)^2+30*exp(2)+225),x, a
lgorithm="giac")

[Out]

-(x^2*e^4 + 30*x^2*e^2 + 225*x^2 - 3*(e^4 + 30*e^2 + 225)*e^(x^2/(e^4 + 30*e^2 + 225)))/(e^4 + 30*e^2 + 225)

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maple [B]  time = 0.06, size = 43, normalized size = 1.95

method	result	size
norman	\(\frac {\left (-{\mathrm e}^{2}-15\right ) x^{2}+\left (3 \,{\mathrm e}^{2}+45\right ) {\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{4}+30 \,{\mathrm e}^{2}+225}}}{{\mathrm e}^{2}+15}\)	\(43\)
default	\(\frac {-225 x^{2}-x^{2} {\mathrm e}^{4}+6 \left (\frac {{\mathrm e}^{4}}{2}+15 \,{\mathrm e}^{2}+\frac {225}{2}\right ) {\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{4}+30 \,{\mathrm e}^{2}+225}}-30 x^{2} {\mathrm e}^{2}}{{\mathrm e}^{4}+30 \,{\mathrm e}^{2}+225}\)	\(68\)
risch	\(-x^{2}+\frac {3 \,{\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{4}+30 \,{\mathrm e}^{2}+225}} {\mathrm e}^{4}}{{\mathrm e}^{4}+30 \,{\mathrm e}^{2}+225}+\frac {90 \,{\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{4}+30 \,{\mathrm e}^{2}+225}} {\mathrm e}^{2}}{{\mathrm e}^{4}+30 \,{\mathrm e}^{2}+225}+\frac {675 \,{\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{4}+30 \,{\mathrm e}^{2}+225}}}{{\mathrm e}^{4}+30 \,{\mathrm e}^{2}+225}\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x*exp(x^2/(exp(2)^2+30*exp(2)+225))-2*x*exp(2)^2-60*exp(2)*x-450*x)/(exp(2)^2+30*exp(2)+225),x,method=_
RETURNVERBOSE)

[Out]

((-exp(2)-15)*x^2+(3*exp(2)+45)*exp(x^2/(exp(2)^2+30*exp(2)+225)))/(exp(2)+15)

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maxima [B]  time = 0.35, size = 56, normalized size = 2.55 \begin {gather*} -\frac {x^{2} e^{4} + 30 \, x^{2} e^{2} + 225 \, x^{2} - 3 \, {\left (e^{4} + 30 \, e^{2} + 225\right )} e^{\left (\frac {x^{2}}{e^{4} + 30 \, e^{2} + 225}\right )}}{e^{4} + 30 \, e^{2} + 225} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(x^2/(exp(2)^2+30*exp(2)+225))-2*x*exp(2)^2-60*exp(2)*x-450*x)/(exp(2)^2+30*exp(2)+225),x, a
lgorithm="maxima")

[Out]

-(x^2*e^4 + 30*x^2*e^2 + 225*x^2 - 3*(e^4 + 30*e^2 + 225)*e^(x^2/(e^4 + 30*e^2 + 225)))/(e^4 + 30*e^2 + 225)

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mupad [B]  time = 0.16, size = 23, normalized size = 1.05 \begin {gather*} 3\,{\mathrm {e}}^{\frac {x^2}{30\,{\mathrm {e}}^2+{\mathrm {e}}^4+225}}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(450*x + 60*x*exp(2) + 2*x*exp(4) - 6*x*exp(x^2/(30*exp(2) + exp(4) + 225)))/(30*exp(2) + exp(4) + 225),x
)

[Out]

3*exp(x^2/(30*exp(2) + exp(4) + 225)) - x^2

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sympy [A]  time = 0.13, size = 19, normalized size = 0.86 \begin {gather*} - x^{2} + 3 e^{\frac {x^{2}}{e^{4} + 30 e^{2} + 225}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(x**2/(exp(2)**2+30*exp(2)+225))-2*x*exp(2)**2-60*exp(2)*x-450*x)/(exp(2)**2+30*exp(2)+225),
x)

[Out]

-x**2 + 3*exp(x**2/(exp(4) + 30*exp(2) + 225))

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