Integrand size = 58, antiderivative size = 28 \[ \int \frac {2 x^3-4 x^2 \log (24)+2 x \log ^2(24)+\frac {1}{81} e^{20} x^4 (-3 x+4 \log (24))}{x^3-2 x^2 \log (24)+x \log ^2(24)} \, dx=4+2 x+\frac {e^{4 \left (5-\log \left (\frac {3}{x}\right )\right )}}{-x+\log (24)} \]
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {2 x^3-4 x^2 \log (24)+2 x \log ^2(24)+\frac {1}{81} e^{20} x^4 (-3 x+4 \log (24))}{x^3-2 x^2 \log (24)+x \log ^2(24)} \, dx=\frac {162 x^2-e^{20} x^4+162 \log ^2(24)-4 e^{20} \log ^4(24)+4 x \log (24) \left (-81+e^{20} \log ^2(24)\right )}{81 (x-\log (24))} \]
Integrate[(2*x^3 - 4*x^2*Log[24] + 2*x*Log[24]^2 + (E^20*x^4*(-3*x + 4*Log [24]))/81)/(x^3 - 2*x^2*Log[24] + x*Log[24]^2),x]
(162*x^2 - E^20*x^4 + 162*Log[24]^2 - 4*E^20*Log[24]^4 + 4*x*Log[24]*(-81 + E^20*Log[24]^2))/(81*(x - Log[24]))
Time = 0.34 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2026, 9, 27, 2188, 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.
\(\displaystyle \int \frac {\frac {1}{81} e^{20} x^4 (4 \log (24)-3 x)+2 x^3-4 x^2 \log (24)+2 x \log ^2(24)}{x^3-2 x^2 \log (24)+x \log ^2(24)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\frac {1}{81} e^{20} x^4 (4 \log (24)-3 x)+2 x^3-4 x^2 \log (24)+2 x \log ^2(24)}{x \left (x^2-2 x \log (24)+\log ^2(24)\right )}dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {-3 e^{20} x^4+4 e^{20} x^3 \log (24)+162 x^2-324 x \log (24)+162 \log ^2(24)}{81 \left (x^2-2 x \log (24)+\log ^2(24)\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{81} \int \frac {-3 e^{20} x^4+4 e^{20} \log (24) x^3+162 x^2-324 \log (24) x+162 \log ^2(24)}{x^2-2 \log (24) x+\log ^2(24)}dx\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \frac {1}{81} \int \left (-3 e^{20} x^2-2 e^{20} \log (24) x+\frac {e^{20} \log ^4(24)}{x^2-2 \log (24) x+\log ^2(24)}-e^{20} \log ^2(24)+162\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{81} \left (-e^{20} x^3-e^{20} x^2 \log (24)-\frac {e^{20} \log ^4(24)}{x-\log (24)}+x \left (162-e^{20} \log ^2(24)\right )\right )\) |
Int[(2*x^3 - 4*x^2*Log[24] + 2*x*Log[24]^2 + (E^20*x^4*(-3*x + 4*Log[24])) /81)/(x^3 - 2*x^2*Log[24] + x*Log[24]^2),x]
(-(E^20*x^3) - E^20*x^2*Log[24] - (E^20*Log[24]^4)/(x - Log[24]) + x*(162 - E^20*Log[24]^2))/81
3.20.14.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
norman | \(\frac {-2 x^{2}+\frac {{\mathrm e}^{20} x^{4}}{81}+2 \ln \left (24\right )^{2}}{\ln \left (24\right )-x}\) | \(29\) |
parallelrisch | \(\frac {2 \ln \left (24\right )^{2}-2 x^{2}+{\mathrm e}^{-4 \ln \left (\frac {3}{x}\right )+20}}{\ln \left (24\right )-x}\) | \(33\) |
default | \(2 x +{\mathrm e}^{20-4 \ln \left (\frac {3}{x}\right )-4 \ln \left (x \right )} \left (-x \ln \left (3\right )^{2}-6 x \ln \left (2\right ) \ln \left (3\right )-x^{2} \ln \left (3\right )-9 x \ln \left (2\right )^{2}-3 x^{2} \ln \left (2\right )-x^{3}-\frac {\ln \left (3\right )^{4}+12 \ln \left (2\right ) \ln \left (3\right )^{3}+54 \ln \left (3\right )^{2} \ln \left (2\right )^{2}+108 \ln \left (2\right )^{3} \ln \left (3\right )+81 \ln \left (2\right )^{4}}{-3 \ln \left (2\right )-\ln \left (3\right )+x}\right )\) | \(113\) |
parts | \(2 x +{\mathrm e}^{20-4 \ln \left (\frac {3}{x}\right )-4 \ln \left (x \right )} \left (-x \ln \left (3\right )^{2}-6 x \ln \left (2\right ) \ln \left (3\right )-x^{2} \ln \left (3\right )-9 x \ln \left (2\right )^{2}-3 x^{2} \ln \left (2\right )-x^{3}-\frac {\ln \left (3\right )^{4}+12 \ln \left (2\right ) \ln \left (3\right )^{3}+54 \ln \left (3\right )^{2} \ln \left (2\right )^{2}+108 \ln \left (2\right )^{3} \ln \left (3\right )+81 \ln \left (2\right )^{4}}{-3 \ln \left (2\right )-\ln \left (3\right )+x}\right )\) | \(113\) |
risch | \(-\frac {x \ln \left (3\right )^{2} {\mathrm e}^{20}}{81}-\frac {2 x \ln \left (3\right ) \ln \left (2\right ) {\mathrm e}^{20}}{27}-\frac {\ln \left (3\right ) {\mathrm e}^{20} x^{2}}{81}-\frac {x \ln \left (2\right )^{2} {\mathrm e}^{20}}{9}-\frac {\ln \left (2\right ) {\mathrm e}^{20} x^{2}}{27}-\frac {{\mathrm e}^{20} x^{3}}{81}+2 x +\frac {\ln \left (3\right )^{4} {\mathrm e}^{20}}{81 \ln \left (3\right )+243 \ln \left (2\right )-81 x}+\frac {4 \ln \left (3\right )^{3} \ln \left (2\right ) {\mathrm e}^{20}}{27 \left (\ln \left (3\right )+3 \ln \left (2\right )-x \right )}+\frac {2 \ln \left (3\right )^{2} \ln \left (2\right )^{2} {\mathrm e}^{20}}{3 \left (\ln \left (3\right )+3 \ln \left (2\right )-x \right )}+\frac {4 \ln \left (3\right ) \ln \left (2\right )^{3} {\mathrm e}^{20}}{3 \left (\ln \left (3\right )+3 \ln \left (2\right )-x \right )}+\frac {\ln \left (2\right )^{4} {\mathrm e}^{20}}{\ln \left (3\right )+3 \ln \left (2\right )-x}\) | \(164\) |
int(((4*ln(24)-3*x)*exp(-4*ln(3/x)+20)+2*x*ln(24)^2-4*x^2*ln(24)+2*x^3)/(x *ln(24)^2-2*x^2*ln(24)+x^3),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {2 x^3-4 x^2 \log (24)+2 x \log ^2(24)+\frac {1}{81} e^{20} x^4 (-3 x+4 \log (24))}{x^3-2 x^2 \log (24)+x \log ^2(24)} \, dx=-\frac {x^{4} e^{20} - x e^{20} \log \left (24\right )^{3} + e^{20} \log \left (24\right )^{4} - 162 \, x^{2} + 162 \, x \log \left (24\right )}{81 \, {\left (x - \log \left (24\right )\right )}} \]
integrate(((4*log(24)-3*x)*exp(-4*log(3/x)+20)+2*x*log(24)^2-4*x^2*log(24) +2*x^3)/(x*log(24)^2-2*x^2*log(24)+x^3),x, algorithm=\
-1/81*(x^4*e^20 - x*e^20*log(24)^3 + e^20*log(24)^4 - 162*x^2 + 162*x*log( 24))/(x - log(24))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {2 x^3-4 x^2 \log (24)+2 x \log ^2(24)+\frac {1}{81} e^{20} x^4 (-3 x+4 \log (24))}{x^3-2 x^2 \log (24)+x \log ^2(24)} \, dx=- \frac {x^{3} e^{20}}{81} - \frac {x^{2} e^{20} \log {\left (24 \right )}}{81} - x \left (-2 + \frac {e^{20} \log {\left (24 \right )}^{2}}{81}\right ) - \frac {e^{20} \log {\left (24 \right )}^{4}}{81 x - 81 \log {\left (24 \right )}} \]
integrate(((4*ln(24)-3*x)*exp(-4*ln(3/x)+20)+2*x*ln(24)**2-4*x**2*ln(24)+2 *x**3)/(x*ln(24)**2-2*x**2*ln(24)+x**3),x)
-x**3*exp(20)/81 - x**2*exp(20)*log(24)/81 - x*(-2 + exp(20)*log(24)**2/81 ) - exp(20)*log(24)**4/(81*x - 81*log(24))
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {2 x^3-4 x^2 \log (24)+2 x \log ^2(24)+\frac {1}{81} e^{20} x^4 (-3 x+4 \log (24))}{x^3-2 x^2 \log (24)+x \log ^2(24)} \, dx=-\frac {1}{81} \, x^{3} e^{20} - \frac {1}{81} \, x^{2} e^{20} \log \left (24\right ) - \frac {e^{20} \log \left (24\right )^{4}}{81 \, {\left (x - \log \left (24\right )\right )}} - \frac {1}{81} \, {\left (e^{20} \log \left (24\right )^{2} - 162\right )} x \]
integrate(((4*log(24)-3*x)*exp(-4*log(3/x)+20)+2*x*log(24)^2-4*x^2*log(24) +2*x^3)/(x*log(24)^2-2*x^2*log(24)+x^3),x, algorithm=\
-1/81*x^3*e^20 - 1/81*x^2*e^20*log(24) - 1/81*e^20*log(24)^4/(x - log(24)) - 1/81*(e^20*log(24)^2 - 162)*x
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {2 x^3-4 x^2 \log (24)+2 x \log ^2(24)+\frac {1}{81} e^{20} x^4 (-3 x+4 \log (24))}{x^3-2 x^2 \log (24)+x \log ^2(24)} \, dx=-\frac {1}{81} \, x^{3} e^{20} - \frac {1}{81} \, x^{2} e^{20} \log \left (24\right ) - \frac {1}{81} \, x e^{20} \log \left (24\right )^{2} - \frac {e^{20} \log \left (24\right )^{4}}{81 \, {\left (x - \log \left (24\right )\right )}} + 2 \, x \]
integrate(((4*log(24)-3*x)*exp(-4*log(3/x)+20)+2*x*log(24)^2-4*x^2*log(24) +2*x^3)/(x*log(24)^2-2*x^2*log(24)+x^3),x, algorithm=\
-1/81*x^3*e^20 - 1/81*x^2*e^20*log(24) - 1/81*x*e^20*log(24)^2 - 1/81*e^20 *log(24)^4/(x - log(24)) + 2*x
Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {2 x^3-4 x^2 \log (24)+2 x \log ^2(24)+\frac {1}{81} e^{20} x^4 (-3 x+4 \log (24))}{x^3-2 x^2 \log (24)+x \log ^2(24)} \, dx=-\frac {{\mathrm {e}}^{20}\,x^4-162\,x^2+162\,\ln \left (24\right )\,x}{81\,x-81\,\ln \left (24\right )} \]