\(\int \frac {3 x^2+e^{\frac {625-450 x+131 x^2-18 x^3+x^4}{x}} (-625+131 x^2-36 x^3+3 x^4)}{x^2} \, dx\) [2223]

Optimal result

Integrand size = 52, antiderivative size = 19 \[ \int \frac {3 x^2+e^{\frac {625-450 x+131 x^2-18 x^3+x^4}{x}} \left (-625+131 x^2-36 x^3+3 x^4\right )}{x^2} \, dx=e^{\frac {\left ((-5+x)^2+x\right )^2}{x}}+3 x \]

[Out]

3*x+exp((x+(-5+x)^2)^2/x)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {14, 6838} \[ \int \frac {3 x^2+e^{\frac {625-450 x+131 x^2-18 x^3+x^4}{x}} \left (-625+131 x^2-36 x^3+3 x^4\right )}{x^2} \, dx=e^{\frac {\left (x^2-9 x+25\right )^2}{x}}+3 x \]

[In]

Int[(3*x^2 + E^((625 - 450*x + 131*x^2 - 18*x^3 + x^4)/x)*(-625 + 131*x^2 - 36*x^3 + 3*x^4))/x^2,x]

[Out]

E^((25 - 9*x + x^2)^2/x) + 3*x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 6838

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

Rubi steps \begin{align*} \text {integral}& = \int \left (3+\frac {e^{\frac {\left (25-9 x+x^2\right )^2}{x}} \left (25-9 x+x^2\right ) \left (-25-9 x+3 x^2\right )}{x^2}\right ) \, dx \\ & = 3 x+\int \frac {e^{\frac {\left (25-9 x+x^2\right )^2}{x}} \left (25-9 x+x^2\right ) \left (-25-9 x+3 x^2\right )}{x^2} \, dx \\ & = e^{\frac {\left (25-9 x+x^2\right )^2}{x}}+3 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {3 x^2+e^{\frac {625-450 x+131 x^2-18 x^3+x^4}{x}} \left (-625+131 x^2-36 x^3+3 x^4\right )}{x^2} \, dx=e^{-450+\frac {625}{x}+131 x-18 x^2+x^3}+3 x \]

[In]

Integrate[(3*x^2 + E^((625 - 450*x + 131*x^2 - 18*x^3 + x^4)/x)*(-625 + 131*x^2 - 36*x^3 + 3*x^4))/x^2,x]

[Out]

E^(-450 + 625/x + 131*x - 18*x^2 + x^3) + 3*x

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05

method	result	size
risch	\(3 x +{\mathrm e}^{\frac {\left (x^{2}-9 x +25\right )^{2}}{x}}\)	\(20\)
parallelrisch	\(3 x +{\mathrm e}^{\frac {x^{4}-18 x^{3}+131 x^{2}-450 x +625}{x}}\)	\(28\)
parts	\(3 x +{\mathrm e}^{\frac {x^{4}-18 x^{3}+131 x^{2}-450 x +625}{x}}\)	\(28\)
norman	\(\frac {x \,{\mathrm e}^{\frac {x^{4}-18 x^{3}+131 x^{2}-450 x +625}{x}}+3 x^{2}}{x}\)	\(36\)

[In]

int(((3*x^4-36*x^3+131*x^2-625)*exp((x^4-18*x^3+131*x^2-450*x+625)/x)+3*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

3*x+exp((x^2-9*x+25)^2/x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {3 x^2+e^{\frac {625-450 x+131 x^2-18 x^3+x^4}{x}} \left (-625+131 x^2-36 x^3+3 x^4\right )}{x^2} \, dx=3 \, x + e^{\left (\frac {x^{4} - 18 \, x^{3} + 131 \, x^{2} - 450 \, x + 625}{x}\right )} \]

[In]

integrate(((3*x^4-36*x^3+131*x^2-625)*exp((x^4-18*x^3+131*x^2-450*x+625)/x)+3*x^2)/x^2,x, algorithm="fricas")

[Out]

3*x + e^((x^4 - 18*x^3 + 131*x^2 - 450*x + 625)/x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {3 x^2+e^{\frac {625-450 x+131 x^2-18 x^3+x^4}{x}} \left (-625+131 x^2-36 x^3+3 x^4\right )}{x^2} \, dx=3 x + e^{\frac {x^{4} - 18 x^{3} + 131 x^{2} - 450 x + 625}{x}} \]

[In]

integrate(((3*x**4-36*x**3+131*x**2-625)*exp((x**4-18*x**3+131*x**2-450*x+625)/x)+3*x**2)/x**2,x)

[Out]

3*x + exp((x**4 - 18*x**3 + 131*x**2 - 450*x + 625)/x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {3 x^2+e^{\frac {625-450 x+131 x^2-18 x^3+x^4}{x}} \left (-625+131 x^2-36 x^3+3 x^4\right )}{x^2} \, dx=3 \, x + e^{\left (x^{3} - 18 \, x^{2} + 131 \, x + \frac {625}{x} - 450\right )} \]

[In]

integrate(((3*x^4-36*x^3+131*x^2-625)*exp((x^4-18*x^3+131*x^2-450*x+625)/x)+3*x^2)/x^2,x, algorithm="maxima")

[Out]

3*x + e^(x^3 - 18*x^2 + 131*x + 625/x - 450)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {3 x^2+e^{\frac {625-450 x+131 x^2-18 x^3+x^4}{x}} \left (-625+131 x^2-36 x^3+3 x^4\right )}{x^2} \, dx=3 \, x + e^{\left (\frac {x^{4} - 18 \, x^{3} + 131 \, x^{2} - 450 \, x + 625}{x}\right )} \]

[In]

integrate(((3*x^4-36*x^3+131*x^2-625)*exp((x^4-18*x^3+131*x^2-450*x+625)/x)+3*x^2)/x^2,x, algorithm="giac")

[Out]

3*x + e^((x^4 - 18*x^3 + 131*x^2 - 450*x + 625)/x)

Mupad [B] (verification not implemented)

Time = 9.57 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {3 x^2+e^{\frac {625-450 x+131 x^2-18 x^3+x^4}{x}} \left (-625+131 x^2-36 x^3+3 x^4\right )}{x^2} \, dx=3\,x+{\mathrm {e}}^{131\,x}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{-450}\,{\mathrm {e}}^{-18\,x^2}\,{\mathrm {e}}^{625/x} \]

[In]

int((exp((131*x^2 - 450*x - 18*x^3 + x^4 + 625)/x)*(131*x^2 - 36*x^3 + 3*x^4 - 625) + 3*x^2)/x^2,x)

[Out]

3*x + exp(131*x)*exp(x^3)*exp(-450)*exp(-18*x^2)*exp(625/x)