Integrand size = 58, antiderivative size = 28 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=e^{5-\left (\frac {1}{3}+\frac {3-x}{x}+\frac {47 x}{4}\right )^2}+x \] Output:
exp(5-((3-x)/x+1/3+47/4*x)^2)+x
Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=e^{-\frac {1187}{18}-\frac {9}{x^2}+\frac {4}{x}+\frac {47 x}{3}-\frac {2209 x^2}{16}}+x \] Input:
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 - 6627*x^4))/(24*x^3),x]
Output:
E^(-1187/18 - 9/x^2 + 4/x + (47*x)/3 - (2209*x^2)/16) + x
Time = 0.47 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {27, 2010, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {24 x^3+e^{\frac {-19881 x^4+2256 x^3-9496 x^2+576 x-1296}{144 x^2}} \left (-6627 x^4+376 x^3-96 x+432\right )}{24 x^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{24} \int \frac {24 x^3+e^{-\frac {19881 x^4-2256 x^3+9496 x^2-576 x+1296}{144 x^2}} \left (-6627 x^4+376 x^3-96 x+432\right )}{x^3}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {1}{24} \int \left (\frac {e^{\frac {-19881 x^4+2256 x^3-9496 x^2+576 x-1296}{144 x^2}} \left (-6627 x^4+376 x^3-96 x+432\right )}{x^3}+24\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{24} \left (24 e^{-\frac {19881 x^4-2256 x^3+9496 x^2-576 x+1296}{144 x^2}}+24 x\right )\) |
Input:
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]
Output:
(24/E^((1296 - 576*x + 9496*x^2 - 2256*x^3 + 19881*x^4)/(144*x^2)) + 24*x) /24
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]]
Time = 0.54 (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 | \(\frac {x^{3}+x^{2} {\mathrm e}^{\frac {-19881 x^{4}+2256 x^{3}-9496 x^{2}+576 x -1296}{144 x^{2}}}}{x^{2}}\) | \(39\) |
orering | \(\frac {\left (x +\frac {8}{141}\right ) \left (\left (-6627 x^{4}+376 x^{3}-96 x +432\right ) {\mathrm e}^{\frac {-19881 x^{4}+2256 x^{3}-9496 x^{2}+576 x -1296}{144 x^{2}}}+24 x^{3}\right )}{24 x^{3}}+\frac {8 \left (934407 x^{5}+376 x^{3}+13536 x^{2}-60144 x -3456\right ) x^{3} \left (\frac {\left (-26508 x^{3}+1128 x^{2}-96\right ) {\mathrm e}^{\frac {-19881 x^{4}+2256 x^{3}-9496 x^{2}+576 x -1296}{144 x^{2}}}+\left (-6627 x^{4}+376 x^{3}-96 x +432\right ) \left (\frac {-79524 x^{3}+6768 x^{2}-18992 x +576}{144 x^{2}}-\frac {-19881 x^{4}+2256 x^{3}-9496 x^{2}+576 x -1296}{72 x^{3}}\right ) {\mathrm e}^{\frac {-19881 x^{4}+2256 x^{3}-9496 x^{2}+576 x -1296}{144 x^{2}}}+72 x^{2}}{24 x^{3}}-\frac {\left (-6627 x^{4}+376 x^{3}-96 x +432\right ) {\mathrm e}^{\frac {-19881 x^{4}+2256 x^{3}-9496 x^{2}+576 x -1296}{144 x^{2}}}+24 x^{3}}{8 x^{4}}\right )}{47 \left (43917129 x^{8}-4983504 x^{7}-17672 x^{6}+1272384 x^{5}-5797920 x^{4}+329472 x^{3}-21888 x^{2}-82944 x +186624\right )}\) | \(317\) |
Input:
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)
Output:
x+exp(-1/144*(19881*x^4-2256*x^3+9496*x^2-576*x+1296)/x^2)
Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=x + e^{\left (-\frac {19881 \, x^{4} - 2256 \, x^{3} + 9496 \, x^{2} - 576 \, x + 1296}{144 \, x^{2}}\right )} \] Input:
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")
Output:
x + e^(-1/144*(19881*x^4 - 2256*x^3 + 9496*x^2 - 576*x + 1296)/x^2)
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=x + e^{\frac {- \frac {2209 x^{4}}{16} + \frac {47 x^{3}}{3} - \frac {1187 x^{2}}{18} + 4 x - 9}{x^{2}}} \] Input:
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)
Output:
x + exp((-2209*x**4/16 + 47*x**3/3 - 1187*x**2/18 + 4*x - 9)/x**2)
Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=x + e^{\left (-\frac {2209}{16} \, x^{2} + \frac {47}{3} \, x + \frac {4}{x} - \frac {9}{x^{2}} - \frac {1187}{18}\right )} \] Input:
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")
Output:
x + e^(-2209/16*x^2 + 47/3*x + 4/x - 9/x^2 - 1187/18)
Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=x + e^{\left (-\frac {19881 \, x^{4} - 2256 \, x^{3} + 9496 \, x^{2} - 576 \, x + 1296}{144 \, x^{2}}\right )} \] Input:
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")
Output:
x + e^(-1/144*(19881*x^4 - 2256*x^3 + 9496*x^2 - 576*x + 1296)/x^2)
Time = 2.79 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=x+\frac {{\mathrm {e}}^{-\frac {1187}{18}}\,{\mathrm {e}}^{4/x}\,{\mathrm {e}}^{-\frac {9}{x^2}}\,{\left ({\mathrm {e}}^x\right )}^{47/3}}{{\left ({\mathrm {e}}^{x^2}\right )}^{2209/16}} \] Input:
int(-((exp(-((1187*x^2)/18 - 4*x - (47*x^3)/3 + (2209*x^4)/16 + 9)/x^2)*(9 6*x - 376*x^3 + 6627*x^4 - 432))/24 - x^3)/x^3,x)
Output:
x + (exp(-1187/18)*exp(4/x)*exp(-9/x^2)*exp(x)^(47/3))/exp(x^2)^(2209/16)
\[ \int \frac {24 x^3+e^{\frac {-1296+576 x-9496 x^2+2256 x^3-19881 x^4}{144 x^2}} \left (432-96 x+376 x^3-6627 x^4\right )}{24 x^3} \, dx=\int \frac {\left (-6627 x^{4}+376 x^{3}-96 x +432\right ) {\mathrm e}^{\frac {-19881 x^{4}+2256 x^{3}-9496 x^{2}+576 x -1296}{144 x^{2}}}+24 x^{3}}{24 x^{3}}d x \] Input:
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)
Output:
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)