Integrand size = 71, antiderivative size = 25 \[ \int \frac {-102-426 x-585 x^2-301 x^3-65 x^4-5 x^5+\left (10+12 x+7 x^2+x^3\right ) \log (5+x)}{100+420 x+580 x^2+300 x^3+65 x^4+5 x^5} \, dx=5-x+\frac {\log (5+x)}{5 \left (-3-x+\frac {1}{1+x}\right )} \] Output:
5+1/5*ln(5+x)/(1/(1+x)-x-3)-x
Time = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-102-426 x-585 x^2-301 x^3-65 x^4-5 x^5+\left (10+12 x+7 x^2+x^3\right ) \log (5+x)}{100+420 x+580 x^2+300 x^3+65 x^4+5 x^5} \, dx=\frac {1}{5} \left (-5 (5+x)-\frac {(1+x) \log (5+x)}{2+4 x+x^2}\right ) \] Input:
Integrate[(-102 - 426*x - 585*x^2 - 301*x^3 - 65*x^4 - 5*x^5 + (10 + 12*x + 7*x^2 + x^3)*Log[5 + x])/(100 + 420*x + 580*x^2 + 300*x^3 + 65*x^4 + 5*x ^5),x]
Output:
(-5*(5 + x) - ((1 + x)*Log[5 + x])/(2 + 4*x + x^2))/5
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.56 (sec) , antiderivative size = 1206, normalized size of antiderivative = 48.24, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {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.
| \(\displaystyle \int \frac {-5 x^5-65 x^4-301 x^3-585 x^2+\left (x^3+7 x^2+12 x+10\right ) \log (x+5)-426 x-102}{5 x^5+65 x^4+300 x^3+580 x^2+420 x+100} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {-5 x^5-65 x^4-301 x^3-585 x^2+\left (x^3+7 x^2+12 x+10\right ) \log (x+5)-426 x-102}{245 (x+5)}+\frac {(1-x) \left (-5 x^5-65 x^4-301 x^3-585 x^2+\left (x^3+7 x^2+12 x+10\right ) \log (x+5)-426 x-102\right )}{245 \left (x^2+4 x+2\right )}+\frac {(1-x) \left (-5 x^5-65 x^4-301 x^3-585 x^2+\left (x^3+7 x^2+12 x+10\right ) \log (x+5)-426 x-102\right )}{35 \left (x^2+4 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(4 x+3) x^4}{14 \left (x^2+4 x+2\right )}-\frac {13 (4 x+3) x^3}{14 \left (x^2+4 x+2\right )}+\frac {2 x^3}{7}-\frac {43 (4 x+3) x^2}{10 \left (x^2+4 x+2\right )}+\frac {39 x^2}{14}-\frac {117 (4 x+3) x}{14 \left (x^2+4 x+2\right )}+\frac {509 x}{70}-\frac {\left (2-\sqrt {2}\right ) \log (x+5)}{20 \left (x-\sqrt {2}+2\right )}-\frac {\left (2+\sqrt {2}\right ) \log (x+5)}{20 \left (x+\sqrt {2}+2\right )}-\frac {\left (2-\sqrt {2}\right ) \log (x+5)}{20 \left (3+\sqrt {2}\right )}-\frac {\left (2+\sqrt {2}\right ) \log (x+5)}{20 \left (3-\sqrt {2}\right )}+\frac {4}{35} \log (x+5)+\frac {\left (2-\sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )}{20 \left (3+\sqrt {2}\right )}+\frac {117}{56} \left (4-\sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )-\frac {129}{10} \left (3-2 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )+\frac {51}{490} \left (2-3 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )-\frac {213}{245} \left (5-4 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )+\frac {117}{98} \left (18-13 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )-\frac {43}{35} \left (31-22 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )+\frac {13}{28} \left (104-73 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )+\frac {13}{49} \left (106-75 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )-\frac {1}{7} \left (126-89 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )-\frac {2}{49} \left (181-128 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )-\frac {699 \log \left (x-\sqrt {2}+2\right )}{140 \sqrt {2}}+\frac {1}{140} \left (4+3 \sqrt {2}\right ) \log (x+5) \log \left (-\frac {x-\sqrt {2}+2}{3+\sqrt {2}}\right )-\frac {1}{35} \left (1-\sqrt {2}\right ) \log (x+5) \log \left (-\frac {x-\sqrt {2}+2}{3+\sqrt {2}}\right )-\frac {\log (x+5) \log \left (-\frac {x-\sqrt {2}+2}{3+\sqrt {2}}\right )}{10 \sqrt {2}}-\frac {2}{49} \left (181+128 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )-\frac {1}{7} \left (126+89 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )+\frac {13}{49} \left (106+75 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )+\frac {13}{28} \left (104+73 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )-\frac {43}{35} \left (31+22 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )+\frac {117}{98} \left (18+13 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )-\frac {213}{245} \left (5+4 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )+\frac {51}{490} \left (2+3 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )-\frac {129}{10} \left (3+2 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )+\frac {117}{56} \left (4+\sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )+\frac {\left (2+\sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )}{20 \left (3-\sqrt {2}\right )}+\frac {699 \log \left (x+\sqrt {2}+2\right )}{140 \sqrt {2}}-\frac {1}{35} \left (1+\sqrt {2}\right ) \log (x+5) \log \left (-\frac {x+\sqrt {2}+2}{3-\sqrt {2}}\right )+\frac {1}{140} \left (4-3 \sqrt {2}\right ) \log (x+5) \log \left (-\frac {x+\sqrt {2}+2}{3-\sqrt {2}}\right )+\frac {\log (x+5) \log \left (-\frac {x+\sqrt {2}+2}{3-\sqrt {2}}\right )}{10 \sqrt {2}}-\frac {1}{35} \left (1+\sqrt {2}\right ) \operatorname {PolyLog}\left (2,\frac {x+5}{3-\sqrt {2}}\right )+\frac {1}{140} \left (4-3 \sqrt {2}\right ) \operatorname {PolyLog}\left (2,\frac {x+5}{3-\sqrt {2}}\right )+\frac {\operatorname {PolyLog}\left (2,\frac {x+5}{3-\sqrt {2}}\right )}{10 \sqrt {2}}+\frac {1}{140} \left (4+3 \sqrt {2}\right ) \operatorname {PolyLog}\left (2,\frac {x+5}{3+\sqrt {2}}\right )-\frac {1}{35} \left (1-\sqrt {2}\right ) \operatorname {PolyLog}\left (2,\frac {x+5}{3+\sqrt {2}}\right )-\frac {\operatorname {PolyLog}\left (2,\frac {x+5}{3+\sqrt {2}}\right )}{10 \sqrt {2}}+\frac {51 (3 x+4)}{70 \left (x^2+4 x+2\right )}-\frac {213 (4 x+3)}{35 \left (x^2+4 x+2\right )}\) |
Input:
Int[(-102 - 426*x - 585*x^2 - 301*x^3 - 65*x^4 - 5*x^5 + (10 + 12*x + 7*x^ 2 + x^3)*Log[5 + x])/(100 + 420*x + 580*x^2 + 300*x^3 + 65*x^4 + 5*x^5),x]
Output:
(509*x)/70 + (39*x^2)/14 + (2*x^3)/7 + (51*(4 + 3*x))/(70*(2 + 4*x + x^2)) - (213*(3 + 4*x))/(35*(2 + 4*x + x^2)) - (117*x*(3 + 4*x))/(14*(2 + 4*x + x^2)) - (43*x^2*(3 + 4*x))/(10*(2 + 4*x + x^2)) - (13*x^3*(3 + 4*x))/(14* (2 + 4*x + x^2)) - (x^4*(3 + 4*x))/(14*(2 + 4*x + x^2)) + (4*Log[5 + x])/3 5 - ((2 + Sqrt[2])*Log[5 + x])/(20*(3 - Sqrt[2])) - ((2 - Sqrt[2])*Log[5 + x])/(20*(3 + Sqrt[2])) - ((2 - Sqrt[2])*Log[5 + x])/(20*(2 - Sqrt[2] + x) ) - ((2 + Sqrt[2])*Log[5 + x])/(20*(2 + Sqrt[2] + x)) - (699*Log[2 - Sqrt[ 2] + x])/(140*Sqrt[2]) - (2*(181 - 128*Sqrt[2])*Log[2 - Sqrt[2] + x])/49 - ((126 - 89*Sqrt[2])*Log[2 - Sqrt[2] + x])/7 + (13*(106 - 75*Sqrt[2])*Log[ 2 - Sqrt[2] + x])/49 + (13*(104 - 73*Sqrt[2])*Log[2 - Sqrt[2] + x])/28 - ( 43*(31 - 22*Sqrt[2])*Log[2 - Sqrt[2] + x])/35 + (117*(18 - 13*Sqrt[2])*Log [2 - Sqrt[2] + x])/98 - (213*(5 - 4*Sqrt[2])*Log[2 - Sqrt[2] + x])/245 + ( 51*(2 - 3*Sqrt[2])*Log[2 - Sqrt[2] + x])/490 - (129*(3 - 2*Sqrt[2])*Log[2 - Sqrt[2] + x])/10 + (117*(4 - Sqrt[2])*Log[2 - Sqrt[2] + x])/56 + ((2 - S qrt[2])*Log[2 - Sqrt[2] + x])/(20*(3 + Sqrt[2])) - (Log[5 + x]*Log[-((2 - Sqrt[2] + x)/(3 + Sqrt[2]))])/(10*Sqrt[2]) - ((1 - Sqrt[2])*Log[5 + x]*Log [-((2 - Sqrt[2] + x)/(3 + Sqrt[2]))])/35 + ((4 + 3*Sqrt[2])*Log[5 + x]*Log [-((2 - Sqrt[2] + x)/(3 + Sqrt[2]))])/140 + (699*Log[2 + Sqrt[2] + x])/(14 0*Sqrt[2]) + ((2 + Sqrt[2])*Log[2 + Sqrt[2] + x])/(20*(3 - Sqrt[2])) + (11 7*(4 + Sqrt[2])*Log[2 + Sqrt[2] + x])/56 - (129*(3 + 2*Sqrt[2])*Log[2 +...
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 = 0.84 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(-\frac {\left (1+x \right ) \ln \left (5+x \right )}{5 \left (x^{2}+4 x +2\right )}-x\) | \(24\) |
| norman | \(\frac {14 x -\frac {\ln \left (5+x \right )}{5}-\frac {x \ln \left (5+x \right )}{5}-x^{3}+8}{x^{2}+4 x +2}\) | \(35\) |
| parallelrisch | \(\frac {-5 x^{3}+140+50 x^{2}-x \ln \left (5+x \right )+270 x -\ln \left (5+x \right )}{5 x^{2}+20 x +10}\) | \(41\) |
| derivativedivides | \(-\frac {\ln \left (5+x \right ) \left (7 \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )^{2}-7 \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )^{2}-42 \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )+42 \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )+49 \sqrt {2}\, \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right )-49 \sqrt {2}\, \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right )+16 \left (5+x \right )^{2}-340-68 x \right )}{140 \left (\left (5+x \right )^{2}-23-6 x \right )}+\frac {\sqrt {2}\, \ln \left (5+x \right ) \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right )}{20}-\frac {\sqrt {2}\, \ln \left (5+x \right ) \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right )}{20}-5-x +\frac {4 \ln \left (5+x \right )}{35}\) | \(234\) |
| default | \(-\frac {\ln \left (5+x \right ) \left (7 \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )^{2}-7 \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )^{2}-42 \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )+42 \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )+49 \sqrt {2}\, \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right )-49 \sqrt {2}\, \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right )+16 \left (5+x \right )^{2}-340-68 x \right )}{140 \left (\left (5+x \right )^{2}-23-6 x \right )}+\frac {\sqrt {2}\, \ln \left (5+x \right ) \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right )}{20}-\frac {\sqrt {2}\, \ln \left (5+x \right ) \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right )}{20}-5-x +\frac {4 \ln \left (5+x \right )}{35}\) | \(234\) |
| parts | \(-x -\frac {2 \ln \left (x^{2}+4 x +2\right )}{35}+\frac {4 \ln \left (5+x \right )}{35}+\frac {\sqrt {2}\, \ln \left (5+x \right ) \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right )}{20}-\frac {\sqrt {2}\, \ln \left (5+x \right ) \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right )}{20}+\frac {2 \ln \left (\left (5+x \right )^{2}-23-6 x \right )}{35}-\frac {\ln \left (5+x \right ) \left (7 \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )^{2}-7 \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )^{2}-42 \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )+42 \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right ) \sqrt {2}\, \left (5+x \right )+49 \sqrt {2}\, \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right )-49 \sqrt {2}\, \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right )+16 \left (5+x \right )^{2}-340-68 x \right )}{140 \left (\left (5+x \right )^{2}-23-6 x \right )}\) | \(257\) |
| orering | \(\frac {\left (10 x^{6}-40 x^{5}-1597 x^{4}-8938 x^{3}-21821 x^{2}-34148 x -25478\right ) \left (\left (x^{3}+7 x^{2}+12 x +10\right ) \ln \left (5+x \right )-5 x^{5}-65 x^{4}-301 x^{3}-585 x^{2}-426 x -102\right )}{\left (10 x^{5}+130 x^{4}+609 x^{3}+1407 x^{2}+2098 x +1508\right ) \left (5 x^{5}+65 x^{4}+300 x^{3}+580 x^{2}+420 x +100\right )}+\frac {\left (10 x^{4}-379 x^{2}-808 x -799\right ) \left (x^{2}+4 x +2\right ) \left (5+x \right ) \left (\frac {\left (3 x^{2}+14 x +12\right ) \ln \left (5+x \right )+\frac {x^{3}+7 x^{2}+12 x +10}{5+x}-25 x^{4}-260 x^{3}-903 x^{2}-1170 x -426}{5 x^{5}+65 x^{4}+300 x^{3}+580 x^{2}+420 x +100}-\frac {\left (\left (x^{3}+7 x^{2}+12 x +10\right ) \ln \left (5+x \right )-5 x^{5}-65 x^{4}-301 x^{3}-585 x^{2}-426 x -102\right ) \left (25 x^{4}+260 x^{3}+900 x^{2}+1160 x +420\right )}{\left (5 x^{5}+65 x^{4}+300 x^{3}+580 x^{2}+420 x +100\right )^{2}}\right )}{10 x^{5}+130 x^{4}+609 x^{3}+1407 x^{2}+2098 x +1508}\) | \(359\) |
Input:
int(((x^3+7*x^2+12*x+10)*ln(5+x)-5*x^5-65*x^4-301*x^3-585*x^2-426*x-102)/( 5*x^5+65*x^4+300*x^3+580*x^2+420*x+100),x,method=_RETURNVERBOSE)
Output:
-1/5*(1+x)/(x^2+4*x+2)*ln(5+x)-x
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-102-426 x-585 x^2-301 x^3-65 x^4-5 x^5+\left (10+12 x+7 x^2+x^3\right ) \log (5+x)}{100+420 x+580 x^2+300 x^3+65 x^4+5 x^5} \, dx=-\frac {5 \, x^{3} + 20 \, x^{2} + {\left (x + 1\right )} \log \left (x + 5\right ) + 10 \, x}{5 \, {\left (x^{2} + 4 \, x + 2\right )}} \] Input:
integrate(((x^3+7*x^2+12*x+10)*log(5+x)-5*x^5-65*x^4-301*x^3-585*x^2-426*x -102)/(5*x^5+65*x^4+300*x^3+580*x^2+420*x+100),x, algorithm="fricas")
Output:
-1/5*(5*x^3 + 20*x^2 + (x + 1)*log(x + 5) + 10*x)/(x^2 + 4*x + 2)
Time = 0.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-102-426 x-585 x^2-301 x^3-65 x^4-5 x^5+\left (10+12 x+7 x^2+x^3\right ) \log (5+x)}{100+420 x+580 x^2+300 x^3+65 x^4+5 x^5} \, dx=- x + \frac {\left (- x - 1\right ) \log {\left (x + 5 \right )}}{5 x^{2} + 20 x + 10} \] Input:
integrate(((x**3+7*x**2+12*x+10)*ln(5+x)-5*x**5-65*x**4-301*x**3-585*x**2- 426*x-102)/(5*x**5+65*x**4+300*x**3+580*x**2+420*x+100),x)
Output:
-x + (-x - 1)*log(x + 5)/(5*x**2 + 20*x + 10)
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (23) = 46\).
Time = 0.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 5.52 \[ \int \frac {-102-426 x-585 x^2-301 x^3-65 x^4-5 x^5+\left (10+12 x+7 x^2+x^3\right ) \log (5+x)}{100+420 x+580 x^2+300 x^3+65 x^4+5 x^5} \, dx=-x - \frac {{\left (4 \, x^{2} + 23 \, x + 15\right )} \log \left (x + 5\right )}{35 \, {\left (x^{2} + 4 \, x + 2\right )}} - \frac {2 \, {\left (128 \, x + 75\right )}}{7 \, {\left (x^{2} + 4 \, x + 2\right )}} + \frac {13 \, {\left (75 \, x + 44\right )}}{7 \, {\left (x^{2} + 4 \, x + 2\right )}} - \frac {43 \, {\left (22 \, x + 13\right )}}{5 \, {\left (x^{2} + 4 \, x + 2\right )}} + \frac {117 \, {\left (13 \, x + 8\right )}}{14 \, {\left (x^{2} + 4 \, x + 2\right )}} - \frac {213 \, {\left (4 \, x + 3\right )}}{35 \, {\left (x^{2} + 4 \, x + 2\right )}} + \frac {51 \, {\left (3 \, x + 4\right )}}{70 \, {\left (x^{2} + 4 \, x + 2\right )}} + \frac {4}{35} \, \log \left (x + 5\right ) \] Input:
integrate(((x^3+7*x^2+12*x+10)*log(5+x)-5*x^5-65*x^4-301*x^3-585*x^2-426*x -102)/(5*x^5+65*x^4+300*x^3+580*x^2+420*x+100),x, algorithm="maxima")
Output:
-x - 1/35*(4*x^2 + 23*x + 15)*log(x + 5)/(x^2 + 4*x + 2) - 2/7*(128*x + 75 )/(x^2 + 4*x + 2) + 13/7*(75*x + 44)/(x^2 + 4*x + 2) - 43/5*(22*x + 13)/(x ^2 + 4*x + 2) + 117/14*(13*x + 8)/(x^2 + 4*x + 2) - 213/35*(4*x + 3)/(x^2 + 4*x + 2) + 51/70*(3*x + 4)/(x^2 + 4*x + 2) + 4/35*log(x + 5)
Time = 0.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {-102-426 x-585 x^2-301 x^3-65 x^4-5 x^5+\left (10+12 x+7 x^2+x^3\right ) \log (5+x)}{100+420 x+580 x^2+300 x^3+65 x^4+5 x^5} \, dx=-x - \frac {{\left (x + 1\right )} \log \left (x + 5\right )}{5 \, {\left (x^{2} + 4 \, x + 2\right )}} \] Input:
integrate(((x^3+7*x^2+12*x+10)*log(5+x)-5*x^5-65*x^4-301*x^3-585*x^2-426*x -102)/(5*x^5+65*x^4+300*x^3+580*x^2+420*x+100),x, algorithm="giac")
Output:
-x - 1/5*(x + 1)*log(x + 5)/(x^2 + 4*x + 2)
Time = 3.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-102-426 x-585 x^2-301 x^3-65 x^4-5 x^5+\left (10+12 x+7 x^2+x^3\right ) \log (5+x)}{100+420 x+580 x^2+300 x^3+65 x^4+5 x^5} \, dx=-x-\frac {\ln \left (x+5\right )\,\left (\frac {x}{5}+\frac {1}{5}\right )}{x^2+4\,x+2} \] Input:
int(-(426*x + 585*x^2 + 301*x^3 + 65*x^4 + 5*x^5 - log(x + 5)*(12*x + 7*x^ 2 + x^3 + 10) + 102)/(420*x + 580*x^2 + 300*x^3 + 65*x^4 + 5*x^5 + 100),x)
Output:
- x - (log(x + 5)*(x/5 + 1/5))/(4*x + x^2 + 2)
Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {-102-426 x-585 x^2-301 x^3-65 x^4-5 x^5+\left (10+12 x+7 x^2+x^3\right ) \log (5+x)}{100+420 x+580 x^2+300 x^3+65 x^4+5 x^5} \, dx=\frac {-\mathrm {log}\left (x +5\right ) x -\mathrm {log}\left (x +5\right )-5 x^{3}-20 x^{2}-10 x}{5 x^{2}+20 x +10} \] Input:
int(((x^3+7*x^2+12*x+10)*log(5+x)-5*x^5-65*x^4-301*x^3-585*x^2-426*x-102)/ (5*x^5+65*x^4+300*x^3+580*x^2+420*x+100),x)
Output:
( - log(x + 5)*x - log(x + 5) - 5*x**3 - 20*x**2 - 10*x)/(5*(x**2 + 4*x + 2))