Integrand size = 52, antiderivative size = 20 \[ \int \frac {3600-27 x^3+9 e^5 x^3+e^5 \left (400-3 x^3\right ) \log \left (\frac {1}{100} \left (-400+3 x^3\right )\right )}{-400 x^2+3 x^5} \, dx=\frac {9+e^5 \log \left (-4+\frac {3 x^3}{100}\right )}{x} \] Output:
(9+exp(5)*ln(3/100*x^3-4))/x
Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {3600-27 x^3+9 e^5 x^3+e^5 \left (400-3 x^3\right ) \log \left (\frac {1}{100} \left (-400+3 x^3\right )\right )}{-400 x^2+3 x^5} \, dx=\frac {9+e^5 \log \left (-4+\frac {3 x^3}{100}\right )}{x} \] Input:
Integrate[(3600 - 27*x^3 + 9*E^5*x^3 + E^5*(400 - 3*x^3)*Log[(-400 + 3*x^3 )/100])/(-400*x^2 + 3*x^5),x]
Output:
(9 + E^5*Log[-4 + (3*x^3)/100])/x
Leaf count is larger than twice the leaf count of optimal. \(200\) vs. \(2(20)=40\).
Time = 0.66 (sec) , antiderivative size = 200, normalized size of antiderivative = 10.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6, 2026, 7276, 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.
\(\displaystyle \int \frac {9 e^5 x^3-27 x^3+e^5 \left (400-3 x^3\right ) \log \left (\frac {1}{100} \left (3 x^3-400\right )\right )+3600}{3 x^5-400 x^2} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (9 e^5-27\right ) x^3+e^5 \left (400-3 x^3\right ) \log \left (\frac {1}{100} \left (3 x^3-400\right )\right )+3600}{3 x^5-400 x^2}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (9 e^5-27\right ) x^3+e^5 \left (400-3 x^3\right ) \log \left (\frac {1}{100} \left (3 x^3-400\right )\right )+3600}{x^2 \left (3 x^3-400\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {9 \left (\left (3-e^5\right ) x^3-400\right )}{x^2 \left (400-3 x^3\right )}-\frac {e^5 \log \left (\frac {3 x^3}{100}-4\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^5 \log \left (\frac {3 x^3}{100}-4\right )}{x}-\frac {\sqrt [3]{\frac {3}{2}} e^5 \log \left (3^{2/3} x^2+2\ 5^{2/3} \sqrt [3]{6} x+20\ 2^{2/3} \sqrt [3]{5}\right )}{4\ 5^{2/3}}+\frac {\sqrt [3]{\frac {3}{2}} e^5 \log \left (30^{2/3} x^2+20 \sqrt [3]{15} x+200 \sqrt [3]{2}\right )}{4\ 5^{2/3}}+\frac {9}{x}+\frac {\sqrt [3]{\frac {3}{2}} e^5 \log \left (2 \sqrt [3]{2} 5^{2/3}-\sqrt [3]{3} x\right )}{2\ 5^{2/3}}-\frac {\sqrt [3]{\frac {3}{2}} e^5 \log \left (10\ 2^{2/3}-\sqrt [3]{30} x\right )}{2\ 5^{2/3}}\) |
Input:
Int[(3600 - 27*x^3 + 9*E^5*x^3 + E^5*(400 - 3*x^3)*Log[(-400 + 3*x^3)/100] )/(-400*x^2 + 3*x^5),x]
Output:
9/x + ((3/2)^(1/3)*E^5*Log[2*2^(1/3)*5^(2/3) - 3^(1/3)*x])/(2*5^(2/3)) - ( (3/2)^(1/3)*E^5*Log[10*2^(2/3) - 30^(1/3)*x])/(2*5^(2/3)) - ((3/2)^(1/3)*E ^5*Log[20*2^(2/3)*5^(1/3) + 2*5^(2/3)*6^(1/3)*x + 3^(2/3)*x^2])/(4*5^(2/3) ) + ((3/2)^(1/3)*E^5*Log[200*2^(1/3) + 20*15^(1/3)*x + 30^(2/3)*x^2])/(4*5 ^(2/3)) + (E^5*Log[-4 + (3*x^3)/100])/x
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, 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[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.63 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
method | result | size |
norman | \(\frac {9+{\mathrm e}^{5} \ln \left (\frac {3 x^{3}}{100}-4\right )}{x}\) | \(18\) |
parallelrisch | \(\frac {81+9 \,{\mathrm e}^{5} \ln \left (\frac {3 x^{3}}{100}-4\right )}{9 x}\) | \(20\) |
risch | \(\frac {{\mathrm e}^{5} \ln \left (\frac {3 x^{3}}{100}-4\right )}{x}+\frac {9}{x}\) | \(21\) |
parts | \(\frac {9}{x}+9 \left (\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (x -\frac {400^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{3600}-\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (x^{2}+\frac {400^{\frac {1}{3}} 3^{\frac {2}{3}} x}{3}+\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{7200}+\frac {3^{\frac {5}{6}} 400^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} x}{200}+1\right )}{3}\right )}{3600}\right ) {\mathrm e}^{5}-{\mathrm e}^{5} \left (-\frac {\ln \left (3 x^{3}-400\right )}{x}+\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (x -\frac {400^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{400}-\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (x^{2}+\frac {400^{\frac {1}{3}} 3^{\frac {2}{3}} x}{3}+\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{800}+\frac {3^{\frac {5}{6}} 400^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} x}{200}+1\right )}{3}\right )}{400}+\frac {2 \ln \left (10\right )}{x}\right )\) | \(185\) |
default | \(\frac {9}{x}+9 \left (\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (x -\frac {400^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{3600}-\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (x^{2}+\frac {400^{\frac {1}{3}} 3^{\frac {2}{3}} x}{3}+\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{7200}+\frac {3^{\frac {5}{6}} 400^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} x}{200}+1\right )}{3}\right )}{3600}\right ) {\mathrm e}^{5}-\frac {2 \,{\mathrm e}^{5} \ln \left (10\right )}{x}-{\mathrm e}^{5} \left (-\frac {\ln \left (3 x^{3}-400\right )}{x}+\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (x -\frac {400^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{400}-\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (x^{2}+\frac {400^{\frac {1}{3}} 3^{\frac {2}{3}} x}{3}+\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{800}+\frac {3^{\frac {5}{6}} 400^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {400^{\frac {2}{3}} 3^{\frac {1}{3}} x}{200}+1\right )}{3}\right )}{400}\right )\) | \(187\) |
orering | \(-\frac {9 x^{4} \left (\left (-3 x^{3}+400\right ) {\mathrm e}^{5} \ln \left (\frac {3 x^{3}}{100}-4\right )+9 x^{3} {\mathrm e}^{5}-27 x^{3}+3600\right )}{\left (3 x^{3}+800\right ) \left (3 x^{5}-400 x^{2}\right )}-\frac {x^{2} \left (3 x^{3}-400\right ) \left (\frac {-9 x^{2} {\mathrm e}^{5} \ln \left (\frac {3 x^{3}}{100}-4\right )+\frac {9 \left (-3 x^{3}+400\right ) {\mathrm e}^{5} x^{2}}{100 \left (\frac {3 x^{3}}{100}-4\right )}+27 x^{2} {\mathrm e}^{5}-81 x^{2}}{3 x^{5}-400 x^{2}}-\frac {\left (\left (-3 x^{3}+400\right ) {\mathrm e}^{5} \ln \left (\frac {3 x^{3}}{100}-4\right )+9 x^{3} {\mathrm e}^{5}-27 x^{3}+3600\right ) \left (15 x^{4}-800 x \right )}{\left (3 x^{5}-400 x^{2}\right )^{2}}\right )}{3 x^{3}+800}\) | \(204\) |
Input:
int(((-3*x^3+400)*exp(5)*ln(3/100*x^3-4)+9*x^3*exp(5)-27*x^3+3600)/(3*x^5- 400*x^2),x,method=_RETURNVERBOSE)
Output:
(9+exp(5)*ln(3/100*x^3-4))/x
Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {3600-27 x^3+9 e^5 x^3+e^5 \left (400-3 x^3\right ) \log \left (\frac {1}{100} \left (-400+3 x^3\right )\right )}{-400 x^2+3 x^5} \, dx=\frac {e^{5} \log \left (\frac {3}{100} \, x^{3} - 4\right ) + 9}{x} \] Input:
integrate(((-3*x^3+400)*exp(5)*log(3/100*x^3-4)+9*x^3*exp(5)-27*x^3+3600)/ (3*x^5-400*x^2),x, algorithm="fricas")
Output:
(e^5*log(3/100*x^3 - 4) + 9)/x
Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {3600-27 x^3+9 e^5 x^3+e^5 \left (400-3 x^3\right ) \log \left (\frac {1}{100} \left (-400+3 x^3\right )\right )}{-400 x^2+3 x^5} \, dx=\frac {e^{5} \log {\left (\frac {3 x^{3}}{100} - 4 \right )}}{x} + \frac {9}{x} \] Input:
integrate(((-3*x**3+400)*exp(5)*ln(3/100*x**3-4)+9*x**3*exp(5)-27*x**3+360 0)/(3*x**5-400*x**2),x)
Output:
exp(5)*log(3*x**3/100 - 4)/x + 9/x
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (17) = 34\).
Time = 0.51 (sec) , antiderivative size = 216, normalized size of antiderivative = 10.80 \[ \int \frac {3600-27 x^3+9 e^5 x^3+e^5 \left (400-3 x^3\right ) \log \left (\frac {1}{100} \left (-400+3 x^3\right )\right )}{-400 x^2+3 x^5} \, dx=-\frac {1}{100} \cdot 50^{\frac {2}{3}} 3^{\frac {5}{6}} \arctan \left (\frac {1}{150} \cdot 50^{\frac {2}{3}} 3^{\frac {1}{6}} {\left (3^{\frac {2}{3}} x + 50^{\frac {1}{3}} 3^{\frac {1}{3}}\right )}\right ) e^{5} + \frac {1}{200} \cdot 50^{\frac {2}{3}} 3^{\frac {1}{3}} e^{5} \log \left (3^{\frac {2}{3}} x^{2} + 2 \cdot 50^{\frac {1}{3}} 3^{\frac {1}{3}} x + 4 \cdot 50^{\frac {2}{3}}\right ) - \frac {1}{100} \cdot 50^{\frac {2}{3}} 3^{\frac {1}{3}} e^{5} \log \left (\frac {1}{3} \cdot 3^{\frac {2}{3}} {\left (3^{\frac {1}{3}} x - 2 \cdot 50^{\frac {1}{3}}\right )}\right ) + \frac {1}{200} \, {\left (2 \cdot 50^{\frac {2}{3}} 3^{\frac {5}{6}} \arctan \left (\frac {1}{150} \cdot 50^{\frac {2}{3}} 3^{\frac {1}{6}} {\left (3^{\frac {2}{3}} x + 50^{\frac {1}{3}} 3^{\frac {1}{3}}\right )}\right ) - 50^{\frac {2}{3}} 3^{\frac {1}{3}} \log \left (3^{\frac {2}{3}} x^{2} + 2 \cdot 50^{\frac {1}{3}} 3^{\frac {1}{3}} x + 4 \cdot 50^{\frac {2}{3}}\right ) + 2 \cdot 50^{\frac {2}{3}} 3^{\frac {1}{3}} \log \left (\frac {1}{3} \cdot 3^{\frac {2}{3}} {\left (3^{\frac {1}{3}} x - 2 \cdot 50^{\frac {1}{3}}\right )}\right )\right )} e^{5} - \frac {2 \, {\left (\log \left (5\right ) + \log \left (2\right )\right )} e^{5} - e^{5} \log \left (3 \, x^{3} - 400\right )}{x} + \frac {9}{x} \] Input:
integrate(((-3*x^3+400)*exp(5)*log(3/100*x^3-4)+9*x^3*exp(5)-27*x^3+3600)/ (3*x^5-400*x^2),x, algorithm="maxima")
Output:
-1/100*50^(2/3)*3^(5/6)*arctan(1/150*50^(2/3)*3^(1/6)*(3^(2/3)*x + 50^(1/3 )*3^(1/3)))*e^5 + 1/200*50^(2/3)*3^(1/3)*e^5*log(3^(2/3)*x^2 + 2*50^(1/3)* 3^(1/3)*x + 4*50^(2/3)) - 1/100*50^(2/3)*3^(1/3)*e^5*log(1/3*3^(2/3)*(3^(1 /3)*x - 2*50^(1/3))) + 1/200*(2*50^(2/3)*3^(5/6)*arctan(1/150*50^(2/3)*3^( 1/6)*(3^(2/3)*x + 50^(1/3)*3^(1/3))) - 50^(2/3)*3^(1/3)*log(3^(2/3)*x^2 + 2*50^(1/3)*3^(1/3)*x + 4*50^(2/3)) + 2*50^(2/3)*3^(1/3)*log(1/3*3^(2/3)*(3 ^(1/3)*x - 2*50^(1/3))))*e^5 - (2*(log(5) + log(2))*e^5 - e^5*log(3*x^3 - 400))/x + 9/x
Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {3600-27 x^3+9 e^5 x^3+e^5 \left (400-3 x^3\right ) \log \left (\frac {1}{100} \left (-400+3 x^3\right )\right )}{-400 x^2+3 x^5} \, dx=\frac {e^{5} \log \left (\frac {3}{100} \, x^{3} - 4\right ) + 9}{x} \] Input:
integrate(((-3*x^3+400)*exp(5)*log(3/100*x^3-4)+9*x^3*exp(5)-27*x^3+3600)/ (3*x^5-400*x^2),x, algorithm="giac")
Output:
(e^5*log(3/100*x^3 - 4) + 9)/x
Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {3600-27 x^3+9 e^5 x^3+e^5 \left (400-3 x^3\right ) \log \left (\frac {1}{100} \left (-400+3 x^3\right )\right )}{-400 x^2+3 x^5} \, dx=\frac {\ln \left (\frac {3\,x^3}{100}-4\right )\,{\mathrm {e}}^5+9}{x} \] Input:
int(-(9*x^3*exp(5) - 27*x^3 - log((3*x^3)/100 - 4)*exp(5)*(3*x^3 - 400) + 3600)/(400*x^2 - 3*x^5),x)
Output:
(log((3*x^3)/100 - 4)*exp(5) + 9)/x
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 5.30 \[ \int \frac {3600-27 x^3+9 e^5 x^3+e^5 \left (400-3 x^3\right ) \log \left (\frac {1}{100} \left (-400+3 x^3\right )\right )}{-400 x^2+3 x^5} \, dx=\frac {\left (4 \,2^{\frac {1}{3}} 15^{\frac {2}{3}} \mathrm {log}\left (\frac {3 x^{3}}{100}-4\right ) e^{5}+36 \,2^{\frac {1}{3}} 15^{\frac {2}{3}}-3 \,\mathrm {log}\left (-2 \,2^{\frac {1}{3}} 5^{\frac {2}{3}}+3^{\frac {1}{3}} x \right ) e^{5} x -3 \,\mathrm {log}\left (2 \,6^{\frac {1}{3}} 5^{\frac {2}{3}} x +20 \,2^{\frac {2}{3}} 5^{\frac {1}{3}}+3^{\frac {2}{3}} x^{2}\right ) e^{5} x +3 \,\mathrm {log}\left (\frac {3 x^{3}}{100}-4\right ) e^{5} x \right ) 2^{\frac {2}{3}} 15^{\frac {1}{3}}}{120 x} \] Input:
int(((-3*x^3+400)*exp(5)*log(3/100*x^3-4)+9*x^3*exp(5)-27*x^3+3600)/(3*x^5 -400*x^2),x)
Output:
(4*2**(1/3)*15**(2/3)*log((3*x**3 - 400)/100)*e**5 + 36*2**(1/3)*15**(2/3) - 3*log( - 2*2**(1/3)*5**(2/3) + 3**(1/3)*x)*e**5*x - 3*log(2*6**(1/3)*5* *(2/3)*x + 20*2**(2/3)*5**(1/3) + 3**(2/3)*x**2)*e**5*x + 3*log((3*x**3 - 400)/100)*e**5*x)/(4*2**(1/3)*15**(2/3)*x)