\(\int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} (432-96 x+376 x^3-6627 x^4)}{24 x^3} \, dx\) [7063]

Optimal result

Integrand size = 58, antiderivative size = 28 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=e^{5-\left (\frac {1}{3}+\frac {3-x}{x}+\frac {47 x}{4}\right )^2}+x \]

[Out]

exp(5-((-x+3)/x+1/3+47/4*x)^2)+x

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {12, 14, 6838} \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=e^{-\frac {19881 x^4-2256 x^3+9496 x^2-576 x+1296}{144 x^2}}+x \]

[In]

Int[(24*x^3 + E^((-1296 + 576*x - 9496*x^2 + 2256*x^3 - 19881*x^4)/(144*x^2))*(432 - 96*x + 376*x^3 - 6627*x^4
))/(24*x^3),x]

[Out]

E^(-1/144*(1296 - 576*x + 9496*x^2 - 2256*x^3 + 19881*x^4)/x^2) + 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 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}& = \frac {1}{24} \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{x^3} \, dx \\ & = \frac {1}{24} \int \left (24+\frac {e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{x^3}\right ) \, dx \\ & = x+\frac {1}{24} \int \frac {e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{x^3} \, dx \\ & = e^{-\frac {1296-576 x+9496 x^2-2256 x^3+19881 x^4}{144 x^2}}+x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=e^{-\frac {1187}{18}-\frac {9}{x^2}+\frac {4}{x}+\frac {47 x}{3}-\frac {2209 x^2}{16}}+x \]

[In]

Integrate[(24*x^3 + E^((-1296 + 576*x - 9496*x^2 + 2256*x^3 - 19881*x^4)/(144*x^2))*(432 - 96*x + 376*x^3 - 66
27*x^4))/(24*x^3),x]

[Out]

E^(-1187/18 - 9/x^2 + 4/x + (47*x)/3 - (2209*x^2)/16) + x

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04

method	result	size
risch	\(x +{\mathrm e}^{-\frac {19881 x^{4}-2256 x^{3}+9496 x^{2}-576 x +1296}{144 x^{2}}}\)	\(29\)
parallelrisch	\(x +{\mathrm e}^{-\frac {19881 x^{4}-2256 x^{3}+9496 x^{2}-576 x +1296}{144 x^{2}}}\)	\(29\)
parts	\(x +{\mathrm e}^{\frac {-19881 x^{4}+2256 x^{3}-9496 x^{2}+576 x -1296}{144 x^{2}}}\)	\(29\)
norman	\(\frac {x^{3}+x^{2} {\mathrm e}^{\frac {-19881 x^{4}+2256 x^{3}-9496 x^{2}+576 x -1296}{144 x^{2}}}}{x^{2}}\)	\(39\)

[In]

int(1/24*((-6627*x^4+376*x^3-96*x+432)*exp(1/144*(-19881*x^4+2256*x^3-9496*x^2+576*x-1296)/x^2)+24*x^3)/x^3,x,
method=_RETURNVERBOSE)

[Out]

x+exp(-1/144*(19881*x^4-2256*x^3+9496*x^2-576*x+1296)/x^2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=x + e^{\left (-\frac {19881 \, x^{4} - 2256 \, x^{3} + 9496 \, x^{2} - 576 \, x + 1296}{144 \, x^{2}}\right )} \]

[In]

integrate(1/24*((-6627*x^4+376*x^3-96*x+432)*exp(1/144*(-19881*x^4+2256*x^3-9496*x^2+576*x-1296)/x^2)+24*x^3)/
x^3,x, algorithm="fricas")

[Out]

x + e^(-1/144*(19881*x^4 - 2256*x^3 + 9496*x^2 - 576*x + 1296)/x^2)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=x + e^{\frac {- \frac {2209 x^{4}}{16} + \frac {47 x^{3}}{3} - \frac {1187 x^{2}}{18} + 4 x - 9}{x^{2}}} \]

[In]

integrate(1/24*((-6627*x**4+376*x**3-96*x+432)*exp(1/144*(-19881*x**4+2256*x**3-9496*x**2+576*x-1296)/x**2)+24
*x**3)/x**3,x)

[Out]

x + exp((-2209*x**4/16 + 47*x**3/3 - 1187*x**2/18 + 4*x - 9)/x**2)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=x + e^{\left (-\frac {2209}{16} \, x^{2} + \frac {47}{3} \, x + \frac {4}{x} - \frac {9}{x^{2}} - \frac {1187}{18}\right )} \]

[In]

integrate(1/24*((-6627*x^4+376*x^3-96*x+432)*exp(1/144*(-19881*x^4+2256*x^3-9496*x^2+576*x-1296)/x^2)+24*x^3)/
x^3,x, algorithm="maxima")

[Out]

x + e^(-2209/16*x^2 + 47/3*x + 4/x - 9/x^2 - 1187/18)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=x + e^{\left (-\frac {19881 \, x^{4} - 2256 \, x^{3} + 9496 \, x^{2} - 576 \, x + 1296}{144 \, x^{2}}\right )} \]

[In]

integrate(1/24*((-6627*x^4+376*x^3-96*x+432)*exp(1/144*(-19881*x^4+2256*x^3-9496*x^2+576*x-1296)/x^2)+24*x^3)/
x^3,x, algorithm="giac")

[Out]

x + e^(-1/144*(19881*x^4 - 2256*x^3 + 9496*x^2 - 576*x + 1296)/x^2)

Mupad [B] (verification not implemented)

Time = 12.98 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=x+\frac {{\mathrm {e}}^{-\frac {1187}{18}}\,{\mathrm {e}}^{4/x}\,{\mathrm {e}}^{-\frac {9}{x^2}}\,{\left ({\mathrm {e}}^x\right )}^{47/3}}{{\left ({\mathrm {e}}^{x^2}\right )}^{2209/16}} \]

[In]

int(-((exp(-((1187*x^2)/18 - 4*x - (47*x^3)/3 + (2209*x^4)/16 + 9)/x^2)*(96*x - 376*x^3 + 6627*x^4 - 432))/24
- x^3)/x^3,x)

[Out]

x + (exp(-1187/18)*exp(4/x)*exp(-9/x^2)*exp(x)^(47/3))/exp(x^2)^(2209/16)