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

Optimal. Leaf size=19 \[ e^{\frac {\left ((-5+x)^2+x\right )^2}{x}}+3 x \]

________________________________________________________________________________________

Rubi [A]  time = 0.28, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 2, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {14, 6706} \begin {gather*} e^{\frac {\left (x^2-9 x+25\right )^2}{x}}+3 x \end {gather*}

Antiderivative was successfully verified.

[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 6706

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 24, normalized size = 1.26 \begin {gather*} e^{-450+\frac {625}{x}+131 x-18 x^2+x^3}+3 x \end {gather*}

Antiderivative was successfully verified.

[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

________________________________________________________________________________________

fricas [A]  time = 0.84, size = 27, normalized size = 1.42 \begin {gather*} 3 \, x + e^{\left (\frac {x^{4} - 18 \, x^{3} + 131 \, x^{2} - 450 \, x + 625}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [A]  time = 0.35, size = 27, normalized size = 1.42 \begin {gather*} 3 \, x + e^{\left (\frac {x^{4} - 18 \, x^{3} + 131 \, x^{2} - 450 \, x + 625}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.19, size = 20, normalized size = 1.05

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [A]  time = 0.63, size = 23, normalized size = 1.21 \begin {gather*} 3 \, x + e^{\left (x^{3} - 18 \, x^{2} + 131 \, x + \frac {625}{x} - 450\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 1.33, size = 27, normalized size = 1.42 \begin {gather*} 3\,x+{\mathrm {e}}^{131\,x}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{-450}\,{\mathrm {e}}^{-18\,x^2}\,{\mathrm {e}}^{625/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 24, normalized size = 1.26 \begin {gather*} 3 x + e^{\frac {x^{4} - 18 x^{3} + 131 x^{2} - 450 x + 625}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________