Integrand size = 76, antiderivative size = 32 \[ \int \frac {3+7 x+11 x^2+15 x^3+7 x^4+x^5+e^8 \left (-3-5 x-2 x^2\right )+\left (-3-5 x-2 x^2\right ) \log \left (x+x^2\right )}{9 x^2+15 x^3+7 x^4+x^5} \, dx=x+\frac {e^8}{x (3+x)}-\frac {\log \left (x+x^2\right )}{x-x (4+x)} \] Output:
x-ln(x^2+x)/(x-(4+x)*x)+exp(4)^2/(3+x)/x
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {3+7 x+11 x^2+15 x^3+7 x^4+x^5+e^8 \left (-3-5 x-2 x^2\right )+\left (-3-5 x-2 x^2\right ) \log \left (x+x^2\right )}{9 x^2+15 x^3+7 x^4+x^5} \, dx=\frac {e^8+x^2 (3+x)+\log (x (1+x))}{x (3+x)} \] Input:
Integrate[(3 + 7*x + 11*x^2 + 15*x^3 + 7*x^4 + x^5 + E^8*(-3 - 5*x - 2*x^2 ) + (-3 - 5*x - 2*x^2)*Log[x + x^2])/(9*x^2 + 15*x^3 + 7*x^4 + x^5),x]
Output:
(E^8 + x^2*(3 + x) + Log[x*(1 + x)])/(x*(3 + x))
Leaf count is larger than twice the leaf count of optimal. \(192\) vs. \(2(32)=64\).
Time = 1.76 (sec) , antiderivative size = 192, normalized size of antiderivative = 6.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2026, 2463, 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.
\(\displaystyle \int \frac {x^5+7 x^4+15 x^3+11 x^2+e^8 \left (-2 x^2-5 x-3\right )+\left (-2 x^2-5 x-3\right ) \log \left (x^2+x\right )+7 x+3}{x^5+7 x^4+15 x^3+9 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {x^5+7 x^4+15 x^3+11 x^2+e^8 \left (-2 x^2-5 x-3\right )+\left (-2 x^2-5 x-3\right ) \log \left (x^2+x\right )+7 x+3}{x^2 \left (x^3+7 x^2+15 x+9\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {x^5+7 x^4+15 x^3+11 x^2+e^8 \left (-2 x^2-5 x-3\right )+\left (-2 x^2-5 x-3\right ) \log \left (x^2+x\right )+7 x+3}{4 x^2 (x+1)}-\frac {x^5+7 x^4+15 x^3+11 x^2+e^8 \left (-2 x^2-5 x-3\right )+\left (-2 x^2-5 x-3\right ) \log \left (x^2+x\right )+7 x+3}{4 x^2 (x+3)}-\frac {x^5+7 x^4+15 x^3+11 x^2+e^8 \left (-2 x^2-5 x-3\right )+\left (-2 x^2-5 x-3\right ) \log \left (x^2+x\right )+7 x+3}{2 x^2 (x+3)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x-\frac {3 \left (1-e^8\right )}{4 x}-\frac {5 e^8}{12 x}+\frac {3}{4 x}-\frac {e^8}{3 (x+3)}-\frac {1}{2} \log ^2(x)-\frac {1}{2} \log (x+1) \log (x)+\frac {1}{2} (\log (x)+\log (x+1)-\log (x (x+1))) \log (x)+\frac {1}{2} \log (x (x+1)) \log (x)+\frac {3 \log (x)}{4 x}+\frac {1}{2} \left (2-e^8\right ) \log (x)+\frac {1}{2} e^8 \log (x)-\log (x)+\frac {3 \log (x+1)}{4 x}-\frac {3 (\log (x)+\log (x+1)-\log (x (x+1)))}{4 x}-\frac {5 \log (x (x+1))}{12 x}-\frac {\log (x (x+1))}{3 (x+3)}\) |
Input:
Int[(3 + 7*x + 11*x^2 + 15*x^3 + 7*x^4 + x^5 + E^8*(-3 - 5*x - 2*x^2) + (- 3 - 5*x - 2*x^2)*Log[x + x^2])/(9*x^2 + 15*x^3 + 7*x^4 + x^5),x]
Output:
3/(4*x) - (5*E^8)/(12*x) - (3*(1 - E^8))/(4*x) + x - E^8/(3*(3 + x)) - Log [x] + (E^8*Log[x])/2 + ((2 - E^8)*Log[x])/2 + (3*Log[x])/(4*x) - Log[x]^2/ 2 + (3*Log[1 + x])/(4*x) - (Log[x]*Log[1 + x])/2 - (3*(Log[x] + Log[1 + x] - Log[x*(1 + x)]))/(4*x) + (Log[x]*(Log[x] + Log[1 + x] - Log[x*(1 + x)]) )/2 - (5*Log[x*(1 + x)])/(12*x) - Log[x*(1 + x)]/(3*(3 + x)) + (Log[x]*Log [x*(1 + x)])/2
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_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.81 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
norman | \(\frac {x^{3}+{\mathrm e}^{8}-9 x +\ln \left (x^{2}+x \right )}{\left (3+x \right ) x}\) | \(27\) |
parallelrisch | \(\frac {x^{3}+{\mathrm e}^{8}-2 x^{2}-15 x +\ln \left (x^{2}+x \right )}{x \left (3+x \right )}\) | \(32\) |
risch | \(\frac {\ln \left (x^{2}+x \right )}{\left (3+x \right ) x}+\frac {x^{3}+{\mathrm e}^{8}+3 x^{2}}{\left (3+x \right ) x}\) | \(37\) |
default | \(x -\frac {\ln \left (1+x \right )}{2}-\frac {{\mathrm e}^{8}}{3 \left (3+x \right )}-\frac {\ln \left (x \right )}{2}+\frac {\frac {{\mathrm e}^{8}}{3}-\frac {1}{3}}{x}+\frac {1+\frac {x}{3}+\frac {3 \ln \left (x^{2}+x \right ) x}{2}+\frac {\ln \left (x^{2}+x \right ) x^{2}}{2}+\ln \left (x^{2}+x \right )}{\left (3+x \right ) x}\) | \(72\) |
parts | \(x -\frac {\ln \left (1+x \right )}{2}-\frac {{\mathrm e}^{8}}{3 \left (3+x \right )}-\frac {\ln \left (x \right )}{2}+\frac {\frac {{\mathrm e}^{8}}{3}-\frac {1}{3}}{x}+\frac {1+\frac {x}{3}+\frac {3 \ln \left (x^{2}+x \right ) x}{2}+\frac {\ln \left (x^{2}+x \right ) x^{2}}{2}+\ln \left (x^{2}+x \right )}{\left (3+x \right ) x}\) | \(72\) |
orering | \(\frac {x \left (6 x^{6}+3 x^{5}-75 x^{4}-206 x^{3}-193 x^{2}-52 x +9\right ) \left (\left (-2 x^{2}-5 x -3\right ) \ln \left (x^{2}+x \right )+\left (-2 x^{2}-5 x -3\right ) {\mathrm e}^{8}+x^{5}+7 x^{4}+15 x^{3}+11 x^{2}+7 x +3\right )}{\left (6 x^{6}+30 x^{5}+60 x^{4}+46 x^{3}+2 x^{2}-10 x -3\right ) \left (x^{5}+7 x^{4}+15 x^{3}+9 x^{2}\right )}+\frac {\left (6 x^{4}-33 x^{2}-23 x +2\right ) \left (3+x \right ) x^{2} \left (1+x \right ) \left (\frac {\left (-4 x -5\right ) \ln \left (x^{2}+x \right )+\frac {\left (-2 x^{2}-5 x -3\right ) \left (1+2 x \right )}{x^{2}+x}+\left (-4 x -5\right ) {\mathrm e}^{8}+5 x^{4}+28 x^{3}+45 x^{2}+22 x +7}{x^{5}+7 x^{4}+15 x^{3}+9 x^{2}}-\frac {\left (\left (-2 x^{2}-5 x -3\right ) \ln \left (x^{2}+x \right )+\left (-2 x^{2}-5 x -3\right ) {\mathrm e}^{8}+x^{5}+7 x^{4}+15 x^{3}+11 x^{2}+7 x +3\right ) \left (5 x^{4}+28 x^{3}+45 x^{2}+18 x \right )}{\left (x^{5}+7 x^{4}+15 x^{3}+9 x^{2}\right )^{2}}\right )}{12 x^{6}+60 x^{5}+120 x^{4}+92 x^{3}+4 x^{2}-20 x -6}\) | \(385\) |
Input:
int(((-2*x^2-5*x-3)*ln(x^2+x)+(-2*x^2-5*x-3)*exp(4)^2+x^5+7*x^4+15*x^3+11* x^2+7*x+3)/(x^5+7*x^4+15*x^3+9*x^2),x,method=_RETURNVERBOSE)
Output:
(x^3+exp(4)^2-9*x+ln(x^2+x))/(3+x)/x
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {3+7 x+11 x^2+15 x^3+7 x^4+x^5+e^8 \left (-3-5 x-2 x^2\right )+\left (-3-5 x-2 x^2\right ) \log \left (x+x^2\right )}{9 x^2+15 x^3+7 x^4+x^5} \, dx=\frac {x^{3} + 3 \, x^{2} + e^{8} + \log \left (x^{2} + x\right )}{x^{2} + 3 \, x} \] Input:
integrate(((-2*x^2-5*x-3)*log(x^2+x)+(-2*x^2-5*x-3)*exp(4)^2+x^5+7*x^4+15* x^3+11*x^2+7*x+3)/(x^5+7*x^4+15*x^3+9*x^2),x, algorithm="fricas")
Output:
(x^3 + 3*x^2 + e^8 + log(x^2 + x))/(x^2 + 3*x)
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {3+7 x+11 x^2+15 x^3+7 x^4+x^5+e^8 \left (-3-5 x-2 x^2\right )+\left (-3-5 x-2 x^2\right ) \log \left (x+x^2\right )}{9 x^2+15 x^3+7 x^4+x^5} \, dx=x + \frac {\log {\left (x^{2} + x \right )}}{x^{2} + 3 x} + \frac {e^{8}}{x^{2} + 3 x} \] Input:
integrate(((-2*x**2-5*x-3)*ln(x**2+x)+(-2*x**2-5*x-3)*exp(4)**2+x**5+7*x** 4+15*x**3+11*x**2+7*x+3)/(x**5+7*x**4+15*x**3+9*x**2),x)
Output:
x + log(x**2 + x)/(x**2 + 3*x) + exp(8)/(x**2 + 3*x)
Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (31) = 62\).
Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 5.06 \[ \int \frac {3+7 x+11 x^2+15 x^3+7 x^4+x^5+e^8 \left (-3-5 x-2 x^2\right )+\left (-3-5 x-2 x^2\right ) \log \left (x+x^2\right )}{9 x^2+15 x^3+7 x^4+x^5} \, dx=\frac {1}{36} \, {\left (\frac {6 \, {\left (x + 6\right )}}{x^{2} + 3 \, x} + 7 \, \log \left (x + 3\right ) - 27 \, \log \left (x + 1\right ) + 20 \, \log \left (x\right )\right )} e^{8} + \frac {5}{36} \, {\left (\frac {6}{x + 3} - 5 \, \log \left (x + 3\right ) + 9 \, \log \left (x + 1\right ) - 4 \, \log \left (x\right )\right )} e^{8} - \frac {1}{2} \, {\left (\frac {2}{x + 3} - \log \left (x + 3\right ) + \log \left (x + 1\right )\right )} e^{8} + x + \frac {9 \, {\left (x^{2} + 3 \, x + 2\right )} \log \left (x + 1\right ) - 2 \, {\left (2 \, x^{2} + 6 \, x - 9\right )} \log \left (x\right ) + 6 \, x + 18}{18 \, {\left (x^{2} + 3 \, x\right )}} - \frac {x + 6}{6 \, {\left (x^{2} + 3 \, x\right )}} - \frac {1}{6 \, {\left (x + 3\right )}} - \frac {1}{2} \, \log \left (x + 1\right ) + \frac {2}{9} \, \log \left (x\right ) \] Input:
integrate(((-2*x^2-5*x-3)*log(x^2+x)+(-2*x^2-5*x-3)*exp(4)^2+x^5+7*x^4+15* x^3+11*x^2+7*x+3)/(x^5+7*x^4+15*x^3+9*x^2),x, algorithm="maxima")
Output:
1/36*(6*(x + 6)/(x^2 + 3*x) + 7*log(x + 3) - 27*log(x + 1) + 20*log(x))*e^ 8 + 5/36*(6/(x + 3) - 5*log(x + 3) + 9*log(x + 1) - 4*log(x))*e^8 - 1/2*(2 /(x + 3) - log(x + 3) + log(x + 1))*e^8 + x + 1/18*(9*(x^2 + 3*x + 2)*log( x + 1) - 2*(2*x^2 + 6*x - 9)*log(x) + 6*x + 18)/(x^2 + 3*x) - 1/6*(x + 6)/ (x^2 + 3*x) - 1/6/(x + 3) - 1/2*log(x + 1) + 2/9*log(x)
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {3+7 x+11 x^2+15 x^3+7 x^4+x^5+e^8 \left (-3-5 x-2 x^2\right )+\left (-3-5 x-2 x^2\right ) \log \left (x+x^2\right )}{9 x^2+15 x^3+7 x^4+x^5} \, dx=\frac {x^{3} + 3 \, x^{2} + e^{8} + \log \left (x^{2} + x\right )}{x^{2} + 3 \, x} \] Input:
integrate(((-2*x^2-5*x-3)*log(x^2+x)+(-2*x^2-5*x-3)*exp(4)^2+x^5+7*x^4+15* x^3+11*x^2+7*x+3)/(x^5+7*x^4+15*x^3+9*x^2),x, algorithm="giac")
Output:
(x^3 + 3*x^2 + e^8 + log(x^2 + x))/(x^2 + 3*x)
Time = 3.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {3+7 x+11 x^2+15 x^3+7 x^4+x^5+e^8 \left (-3-5 x-2 x^2\right )+\left (-3-5 x-2 x^2\right ) \log \left (x+x^2\right )}{9 x^2+15 x^3+7 x^4+x^5} \, dx=x+\frac {\ln \left (x^2+x\right )+{\mathrm {e}}^8}{x\,\left (x+3\right )} \] Input:
int((7*x - exp(8)*(5*x + 2*x^2 + 3) + 11*x^2 + 15*x^3 + 7*x^4 + x^5 - log( x + x^2)*(5*x + 2*x^2 + 3) + 3)/(9*x^2 + 15*x^3 + 7*x^4 + x^5),x)
Output:
x + (log(x + x^2) + exp(8))/(x*(x + 3))
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {3+7 x+11 x^2+15 x^3+7 x^4+x^5+e^8 \left (-3-5 x-2 x^2\right )+\left (-3-5 x-2 x^2\right ) \log \left (x+x^2\right )}{9 x^2+15 x^3+7 x^4+x^5} \, dx=\frac {\mathrm {log}\left (x^{2}+x \right )+e^{8}+x^{3}+3 x^{2}}{x \left (x +3\right )} \] Input:
int(((-2*x^2-5*x-3)*log(x^2+x)+(-2*x^2-5*x-3)*exp(4)^2+x^5+7*x^4+15*x^3+11 *x^2+7*x+3)/(x^5+7*x^4+15*x^3+9*x^2),x)
Output:
(log(x**2 + x) + e**8 + x**3 + 3*x**2)/(x*(x + 3))