Integrand size = 64, antiderivative size = 31 \[ \int \frac {-18-150 x+1875 x^3+1250 e^{2 x} x^3+21250 x^4+e^x \left (-5000 x^3-5000 x^4\right )+\left (150 x+1250 x^2+1250 x^3\right ) \log (x)}{625 x^3} \, dx=-4+x+\left (-e^x+4 x\right )^2+\left (\frac {3}{25 x}-x-\log (x)\right )^2 \] Output:
x-4+(4*x-exp(x))^2+(3/25/x-x-ln(x))^2
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {-18-150 x+1875 x^3+1250 e^{2 x} x^3+21250 x^4+e^x \left (-5000 x^3-5000 x^4\right )+\left (150 x+1250 x^2+1250 x^3\right ) \log (x)}{625 x^3} \, dx=e^{2 x}+\frac {9}{625 x^2}+x-8 e^x x+17 x^2-\frac {6 \log (x)}{25 x}+2 x \log (x)+\log ^2(x) \] Input:
Integrate[(-18 - 150*x + 1875*x^3 + 1250*E^(2*x)*x^3 + 21250*x^4 + E^x*(-5 000*x^3 - 5000*x^4) + (150*x + 1250*x^2 + 1250*x^3)*Log[x])/(625*x^3),x]
Output:
E^(2*x) + 9/(625*x^2) + x - 8*E^x*x + 17*x^2 - (6*Log[x])/(25*x) + 2*x*Log [x] + Log[x]^2
Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {27, 25, 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.
\(\displaystyle \int \frac {21250 x^4+1250 e^{2 x} x^3+1875 x^3+e^x \left (-5000 x^4-5000 x^3\right )+\left (1250 x^3+1250 x^2+150 x\right ) \log (x)-150 x-18}{625 x^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{625} \int -\frac {-21250 x^4-1250 e^{2 x} x^3-1875 x^3+150 x+5000 e^x \left (x^4+x^3\right )-50 \left (25 x^3+25 x^2+3 x\right ) \log (x)+18}{x^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{625} \int \frac {-21250 x^4-1250 e^{2 x} x^3-1875 x^3+150 x+5000 e^x \left (x^4+x^3\right )-50 \left (25 x^3+25 x^2+3 x\right ) \log (x)+18}{x^3}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle -\frac {1}{625} \int \left (5000 e^x (x+1)-1250 e^{2 x}+\frac {-21250 x^4-1250 \log (x) x^3-1875 x^3-1250 \log (x) x^2-150 \log (x) x+150 x+18}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{625} \left (10625 x^2+\frac {9}{x^2}+625 x+5000 e^x+625 e^{2 x}-5000 e^x (x+1)+625 \log ^2(x)+1250 x \log (x)-\frac {150 \log (x)}{x}\right )\) |
Input:
Int[(-18 - 150*x + 1875*x^3 + 1250*E^(2*x)*x^3 + 21250*x^4 + E^x*(-5000*x^ 3 - 5000*x^4) + (150*x + 1250*x^2 + 1250*x^3)*Log[x])/(625*x^3),x]
Output:
(5000*E^x + 625*E^(2*x) + 9/x^2 + 625*x + 10625*x^2 - 5000*E^x*(1 + x) - ( 150*Log[x])/x + 1250*x*Log[x] + 625*Log[x]^2)/625
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 = 1.41 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23
method | result | size |
default | \(-8 \,{\mathrm e}^{x} x +17 x^{2}+x +\frac {9}{625 x^{2}}+{\mathrm e}^{2 x}+2 x \ln \left (x \right )+\ln \left (x \right )^{2}-\frac {6 \ln \left (x \right )}{25 x}\) | \(38\) |
parts | \(-8 \,{\mathrm e}^{x} x +17 x^{2}+x +\frac {9}{625 x^{2}}+{\mathrm e}^{2 x}+2 x \ln \left (x \right )+\ln \left (x \right )^{2}-\frac {6 \ln \left (x \right )}{25 x}\) | \(38\) |
risch | \(\ln \left (x \right )^{2}+\frac {2 \left (25 x^{2}-3\right ) \ln \left (x \right )}{25 x}+\frac {10625 x^{4}-5000 \,{\mathrm e}^{x} x^{3}+625 \,{\mathrm e}^{2 x} x^{2}+625 x^{3}+9}{625 x^{2}}\) | \(53\) |
parallelrisch | \(-\frac {-10625 x^{4}-1250 x^{3} \ln \left (x \right )+5000 \,{\mathrm e}^{x} x^{3}-625 x^{2} \ln \left (x \right )^{2}-625 \,{\mathrm e}^{2 x} x^{2}-625 x^{3}-9+150 x \ln \left (x \right )}{625 x^{2}}\) | \(55\) |
Input:
int(1/625*((1250*x^3+1250*x^2+150*x)*ln(x)+1250*exp(x)^2*x^3+(-5000*x^4-50 00*x^3)*exp(x)+21250*x^4+1875*x^3-150*x-18)/x^3,x,method=_RETURNVERBOSE)
Output:
-8*exp(x)*x+17*x^2+x+9/625/x^2+exp(x)^2+2*x*ln(x)+ln(x)^2-6/25*ln(x)/x
Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {-18-150 x+1875 x^3+1250 e^{2 x} x^3+21250 x^4+e^x \left (-5000 x^3-5000 x^4\right )+\left (150 x+1250 x^2+1250 x^3\right ) \log (x)}{625 x^3} \, dx=\frac {10625 \, x^{4} - 5000 \, x^{3} e^{x} + 625 \, x^{2} \log \left (x\right )^{2} + 625 \, x^{3} + 625 \, x^{2} e^{\left (2 \, x\right )} + 50 \, {\left (25 \, x^{3} - 3 \, x\right )} \log \left (x\right ) + 9}{625 \, x^{2}} \] Input:
integrate(1/625*((1250*x^3+1250*x^2+150*x)*log(x)+1250*exp(x)^2*x^3+(-5000 *x^4-5000*x^3)*exp(x)+21250*x^4+1875*x^3-150*x-18)/x^3,x, algorithm="frica s")
Output:
1/625*(10625*x^4 - 5000*x^3*e^x + 625*x^2*log(x)^2 + 625*x^3 + 625*x^2*e^( 2*x) + 50*(25*x^3 - 3*x)*log(x) + 9)/x^2
Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {-18-150 x+1875 x^3+1250 e^{2 x} x^3+21250 x^4+e^x \left (-5000 x^3-5000 x^4\right )+\left (150 x+1250 x^2+1250 x^3\right ) \log (x)}{625 x^3} \, dx=17 x^{2} - 8 x e^{x} + x + e^{2 x} + \log {\left (x \right )}^{2} + \frac {\left (50 x^{2} - 6\right ) \log {\left (x \right )}}{25 x} + \frac {9}{625 x^{2}} \] Input:
integrate(1/625*((1250*x**3+1250*x**2+150*x)*ln(x)+1250*exp(x)**2*x**3+(-5 000*x**4-5000*x**3)*exp(x)+21250*x**4+1875*x**3-150*x-18)/x**3,x)
Output:
17*x**2 - 8*x*exp(x) + x + exp(2*x) + log(x)**2 + (50*x**2 - 6)*log(x)/(25 *x) + 9/(625*x**2)
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {-18-150 x+1875 x^3+1250 e^{2 x} x^3+21250 x^4+e^x \left (-5000 x^3-5000 x^4\right )+\left (150 x+1250 x^2+1250 x^3\right ) \log (x)}{625 x^3} \, dx=17 \, x^{2} - 8 \, {\left (x - 1\right )} e^{x} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} + x - \frac {6 \, \log \left (x\right )}{25 \, x} + \frac {9}{625 \, x^{2}} + e^{\left (2 \, x\right )} - 8 \, e^{x} \] Input:
integrate(1/625*((1250*x^3+1250*x^2+150*x)*log(x)+1250*exp(x)^2*x^3+(-5000 *x^4-5000*x^3)*exp(x)+21250*x^4+1875*x^3-150*x-18)/x^3,x, algorithm="maxim a")
Output:
17*x^2 - 8*(x - 1)*e^x + 2*x*log(x) + log(x)^2 + x - 6/25*log(x)/x + 9/625 /x^2 + e^(2*x) - 8*e^x
\[ \int \frac {-18-150 x+1875 x^3+1250 e^{2 x} x^3+21250 x^4+e^x \left (-5000 x^3-5000 x^4\right )+\left (150 x+1250 x^2+1250 x^3\right ) \log (x)}{625 x^3} \, dx=\int { \frac {21250 \, x^{4} + 1250 \, x^{3} e^{\left (2 \, x\right )} + 1875 \, x^{3} - 5000 \, {\left (x^{4} + x^{3}\right )} e^{x} + 50 \, {\left (25 \, x^{3} + 25 \, x^{2} + 3 \, x\right )} \log \left (x\right ) - 150 \, x - 18}{625 \, x^{3}} \,d x } \] Input:
integrate(1/625*((1250*x^3+1250*x^2+150*x)*log(x)+1250*exp(x)^2*x^3+(-5000 *x^4-5000*x^3)*exp(x)+21250*x^4+1875*x^3-150*x-18)/x^3,x, algorithm="giac" )
Output:
integrate(1/625*(21250*x^4 + 1250*x^3*e^(2*x) + 1875*x^3 - 5000*(x^4 + x^3 )*e^x + 50*(25*x^3 + 25*x^2 + 3*x)*log(x) - 150*x - 18)/x^3, x)
Time = 3.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {-18-150 x+1875 x^3+1250 e^{2 x} x^3+21250 x^4+e^x \left (-5000 x^3-5000 x^4\right )+\left (150 x+1250 x^2+1250 x^3\right ) \log (x)}{625 x^3} \, dx={\mathrm {e}}^{2\,x}+{\ln \left (x\right )}^2-\frac {\frac {6\,x\,\ln \left (x\right )}{25}-\frac {9}{625}}{x^2}+x\,\left (2\,\ln \left (x\right )-8\,{\mathrm {e}}^x+1\right )+17\,x^2 \] Input:
int((2*x^3*exp(2*x) - (exp(x)*(5000*x^3 + 5000*x^4))/625 - (6*x)/25 + 3*x^ 3 + 34*x^4 + (log(x)*(150*x + 1250*x^2 + 1250*x^3))/625 - 18/625)/x^3,x)
Output:
exp(2*x) + log(x)^2 - ((6*x*log(x))/25 - 9/625)/x^2 + x*(2*log(x) - 8*exp( x) + 1) + 17*x^2
Time = 0.18 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {-18-150 x+1875 x^3+1250 e^{2 x} x^3+21250 x^4+e^x \left (-5000 x^3-5000 x^4\right )+\left (150 x+1250 x^2+1250 x^3\right ) \log (x)}{625 x^3} \, dx=\frac {625 e^{2 x} x^{2}-5000 e^{x} x^{3}+625 \mathrm {log}\left (x \right )^{2} x^{2}+1250 \,\mathrm {log}\left (x \right ) x^{3}-150 \,\mathrm {log}\left (x \right ) x +10625 x^{4}+625 x^{3}+9}{625 x^{2}} \] Input:
int(1/625*((1250*x^3+1250*x^2+150*x)*log(x)+1250*exp(x)^2*x^3+(-5000*x^4-5 000*x^3)*exp(x)+21250*x^4+1875*x^3-150*x-18)/x^3,x)
Output:
(625*e**(2*x)*x**2 - 5000*e**x*x**3 + 625*log(x)**2*x**2 + 1250*log(x)*x** 3 - 150*log(x)*x + 10625*x**4 + 625*x**3 + 9)/(625*x**2)