Integrand size = 56, antiderivative size = 29 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=-4-x+(9-x)^2 x-x \log \left (7+\frac {3}{5-x}\right ) \] Output:
(9-x)^2*x-x-4-ln(7+3/(5-x))*x
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=80 x-18 x^2+x^3-\frac {38}{7} \log (38-7 x)+\frac {38}{7} \log (5-x)-\frac {1}{7} (-38+7 x) \log \left (\frac {-38+7 x}{-5+x}\right ) \] Input:
Integrate[(15200 - 12683*x + 3758*x^2 - 471*x^3 + 21*x^4 + (-190 + 73*x - 7*x^2)*Log[(-38 + 7*x)/(-5 + x)])/(190 - 73*x + 7*x^2),x]
Output:
80*x - 18*x^2 + x^3 - (38*Log[38 - 7*x])/7 + (38*Log[5 - x])/7 - ((-38 + 7 *x)*Log[(-38 + 7*x)/(-5 + x)])/7
Time = 0.39 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {7279, 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.
\(\displaystyle \int \frac {21 x^4-471 x^3+3758 x^2+\left (-7 x^2+73 x-190\right ) \log \left (\frac {7 x-38}{x-5}\right )-12683 x+15200}{7 x^2-73 x+190} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {3758 x^2}{7 x^2-73 x+190}-\frac {12683 x}{7 x^2-73 x+190}+\frac {15200}{7 x^2-73 x+190}+\frac {21 x^4}{7 x^2-73 x+190}-\frac {471 x^3}{7 x^2-73 x+190}-\log \left (\frac {7 x-38}{x-5}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^3-18 x^2+80 x-\frac {38}{7} \log (38-7 x)+\frac {1}{7} (38-7 x) \log \left (\frac {38-7 x}{5-x}\right )+\frac {38}{7} \log (5-x)\) |
Input:
Int[(15200 - 12683*x + 3758*x^2 - 471*x^3 + 21*x^4 + (-190 + 73*x - 7*x^2) *Log[(-38 + 7*x)/(-5 + x)])/(190 - 73*x + 7*x^2),x]
Output:
80*x - 18*x^2 + x^3 - (38*Log[38 - 7*x])/7 + ((38 - 7*x)*Log[(38 - 7*x)/(5 - x)])/7 + (38*Log[5 - x])/7
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.59 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97
method | result | size |
norman | \(x^{3}+80 x -18 x^{2}-\ln \left (\frac {7 x -38}{-5+x}\right ) x\) | \(28\) |
risch | \(x^{3}+80 x -18 x^{2}-\ln \left (\frac {7 x -38}{-5+x}\right ) x\) | \(28\) |
parallelrisch | \(x^{3}+\frac {36186}{7}-18 x^{2}-\ln \left (\frac {7 x -38}{-5+x}\right ) x +80 x\) | \(29\) |
derivativedivides | \(-\frac {\ln \left (7-\frac {3}{-5+x}\right ) \left (7-\frac {3}{-5+x}\right ) \left (-5+x \right )}{7}+\left (-5+x \right )^{3}-3 \left (-5+x \right )^{2}+125-25 x -\frac {38 \ln \left (7-\frac {3}{-5+x}\right )}{7}\) | \(54\) |
default | \(-\frac {\ln \left (7-\frac {3}{-5+x}\right ) \left (7-\frac {3}{-5+x}\right ) \left (-5+x \right )}{7}+\left (-5+x \right )^{3}-3 \left (-5+x \right )^{2}+125-25 x -\frac {38 \ln \left (7-\frac {3}{-5+x}\right )}{7}\) | \(54\) |
parts | \(x^{3}-18 x^{2}+80 x +5 \ln \left (-5+x \right )-\frac {38 \ln \left (7 x -38\right )}{7}-\frac {3 \ln \left (-\frac {3}{-5+x}\right )}{7}-\frac {\ln \left (7-\frac {3}{-5+x}\right ) \left (7-\frac {3}{-5+x}\right ) \left (-5+x \right )}{7}\) | \(61\) |
orering | \(\frac {x \left (\left (-7 x^{2}+73 x -190\right ) \ln \left (\frac {7 x -38}{-5+x}\right )+21 x^{4}-471 x^{3}+3758 x^{2}-12683 x +15200\right )}{7 x^{2}-73 x +190}-\frac {\left (14 x^{5}-272 x^{4}+1694 x^{3}-35697 x +92910\right ) \left (7 x -38\right ) \left (-5+x \right ) \left (\frac {\left (-14 x +73\right ) \ln \left (\frac {7 x -38}{-5+x}\right )+\frac {\left (-7 x^{2}+73 x -190\right ) \left (\frac {7}{-5+x}-\frac {7 x -38}{\left (-5+x \right )^{2}}\right ) \left (-5+x \right )}{7 x -38}+84 x^{3}-1413 x^{2}+7516 x -12683}{7 x^{2}-73 x +190}-\frac {\left (\left (-7 x^{2}+73 x -190\right ) \ln \left (\frac {7 x -38}{-5+x}\right )+21 x^{4}-471 x^{3}+3758 x^{2}-12683 x +15200\right ) \left (14 x -73\right )}{\left (7 x^{2}-73 x +190\right )^{2}}\right )}{3 \left (98 x^{5}-2632 x^{4}+28242 x^{3}-151348 x^{2}+405153 x -433580\right )}\) | \(266\) |
Input:
int(((-7*x^2+73*x-190)*ln((7*x-38)/(-5+x))+21*x^4-471*x^3+3758*x^2-12683*x +15200)/(7*x^2-73*x+190),x,method=_RETURNVERBOSE)
Output:
x^3+80*x-18*x^2-ln((7*x-38)/(-5+x))*x
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=x^{3} - 18 \, x^{2} - x \log \left (\frac {7 \, x - 38}{x - 5}\right ) + 80 \, x \] Input:
integrate(((-7*x^2+73*x-190)*log((7*x-38)/(-5+x))+21*x^4-471*x^3+3758*x^2- 12683*x+15200)/(7*x^2-73*x+190),x, algorithm="fricas")
Output:
x^3 - 18*x^2 - x*log((7*x - 38)/(x - 5)) + 80*x
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=x^{3} - 18 x^{2} - x \log {\left (\frac {7 x - 38}{x - 5} \right )} + 80 x \] Input:
integrate(((-7*x**2+73*x-190)*ln((7*x-38)/(-5+x))+21*x**4-471*x**3+3758*x* *2-12683*x+15200)/(7*x**2-73*x+190),x)
Output:
x**3 - 18*x**2 - x*log((7*x - 38)/(x - 5)) + 80*x
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.97 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=x^{3} - 18 \, x^{2} - \frac {1}{21} \, {\left (21 \, x + 1330 \, \log \left (x - 5\right ) - 114\right )} \log \left (7 \, x - 38\right ) + \frac {190}{3} \, \log \left (7 \, x - 38\right )^{2} + {\left (x - 5\right )} \log \left (x - 5\right ) - \frac {190}{3} \, \log \left (7 \, x - 38\right ) \log \left (x - 5\right ) + \frac {190}{3} \, \log \left (x - 5\right )^{2} - \frac {190}{3} \, {\left (\log \left (7 \, x - 38\right ) - \log \left (x - 5\right )\right )} \log \left (\frac {7 \, x}{x - 5} - \frac {38}{x - 5}\right ) + 80 \, x - \frac {38}{7} \, \log \left (7 \, x - 38\right ) + 5 \, \log \left (x - 5\right ) \] Input:
integrate(((-7*x^2+73*x-190)*log((7*x-38)/(-5+x))+21*x^4-471*x^3+3758*x^2- 12683*x+15200)/(7*x^2-73*x+190),x, algorithm="maxima")
Output:
x^3 - 18*x^2 - 1/21*(21*x + 1330*log(x - 5) - 114)*log(7*x - 38) + 190/3*l og(7*x - 38)^2 + (x - 5)*log(x - 5) - 190/3*log(7*x - 38)*log(x - 5) + 190 /3*log(x - 5)^2 - 190/3*(log(7*x - 38) - log(x - 5))*log(7*x/(x - 5) - 38/ (x - 5)) + 80*x - 38/7*log(7*x - 38) + 5*log(x - 5)
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (25) = 50\).
Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 4.03 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=\frac {3 \, {\left (\frac {25 \, {\left (7 \, x - 38\right )}^{2}}{{\left (x - 5\right )}^{2}} - \frac {359 \, {\left (7 \, x - 38\right )}}{x - 5} + 1279\right )}}{\frac {{\left (7 \, x - 38\right )}^{3}}{{\left (x - 5\right )}^{3}} - \frac {21 \, {\left (7 \, x - 38\right )}^{2}}{{\left (x - 5\right )}^{2}} + \frac {147 \, {\left (7 \, x - 38\right )}}{x - 5} - 343} + \frac {3 \, \log \left (\frac {7 \, x - 38}{x - 5}\right )}{\frac {7 \, x - 38}{x - 5} - 7} - 5 \, \log \left (\frac {7 \, x - 38}{x - 5}\right ) \] Input:
integrate(((-7*x^2+73*x-190)*log((7*x-38)/(-5+x))+21*x^4-471*x^3+3758*x^2- 12683*x+15200)/(7*x^2-73*x+190),x, algorithm="giac")
Output:
3*(25*(7*x - 38)^2/(x - 5)^2 - 359*(7*x - 38)/(x - 5) + 1279)/((7*x - 38)^ 3/(x - 5)^3 - 21*(7*x - 38)^2/(x - 5)^2 + 147*(7*x - 38)/(x - 5) - 343) + 3*log((7*x - 38)/(x - 5))/((7*x - 38)/(x - 5) - 7) - 5*log((7*x - 38)/(x - 5))
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=80\,x-x\,\ln \left (\frac {7\,x-38}{x-5}\right )-18\,x^2+x^3 \] Input:
int(-(12683*x + log((7*x - 38)/(x - 5))*(7*x^2 - 73*x + 190) - 3758*x^2 + 471*x^3 - 21*x^4 - 15200)/(7*x^2 - 73*x + 190),x)
Output:
80*x - x*log((7*x - 38)/(x - 5)) - 18*x^2 + x^3
Time = 0.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=-\frac {38 \,\mathrm {log}\left (7 x -38\right )}{7}+\frac {38 \,\mathrm {log}\left (-5+x \right )}{7}-\mathrm {log}\left (\frac {7 x -38}{-5+x}\right ) x +\frac {38 \,\mathrm {log}\left (\frac {7 x -38}{-5+x}\right )}{7}+x^{3}-18 x^{2}+80 x \] Input:
int(((-7*x^2+73*x-190)*log((7*x-38)/(-5+x))+21*x^4-471*x^3+3758*x^2-12683* x+15200)/(7*x^2-73*x+190),x)
Output:
( - 38*log(7*x - 38) + 38*log(x - 5) - 7*log((7*x - 38)/(x - 5))*x + 38*lo g((7*x - 38)/(x - 5)) + 7*x**3 - 126*x**2 + 560*x)/7