Integrand size = 101, antiderivative size = 28 \[ \int \frac {-9-63 x+x^2+7 x^3+\left (84 x^3-28 x^4\right ) \log (1+7 x)+\left (6 x^2+38 x^3-28 x^4\right ) \log ^2(1+7 x)+28 x^5 \log ^3(1+7 x)+\left (3 x^4+21 x^5\right ) \log ^4(1+7 x)}{x^2+7 x^3} \, dx=\frac {\left (-3+x-(x-x (1-\log (1+7 x)))^2\right )^2}{x} \]
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {-9-63 x+x^2+7 x^3+\left (84 x^3-28 x^4\right ) \log (1+7 x)+\left (6 x^2+38 x^3-28 x^4\right ) \log ^2(1+7 x)+28 x^5 \log ^3(1+7 x)+\left (3 x^4+21 x^5\right ) \log ^4(1+7 x)}{x^2+7 x^3} \, dx=\frac {9}{x}+x-2 (-3+x) x \log ^2(1+7 x)+x^3 \log ^4(1+7 x) \]
Integrate[(-9 - 63*x + x^2 + 7*x^3 + (84*x^3 - 28*x^4)*Log[1 + 7*x] + (6*x ^2 + 38*x^3 - 28*x^4)*Log[1 + 7*x]^2 + 28*x^5*Log[1 + 7*x]^3 + (3*x^4 + 21 *x^5)*Log[1 + 7*x]^4)/(x^2 + 7*x^3),x]
Leaf count is larger than twice the leaf count of optimal. \(193\) vs. \(2(28)=56\).
Time = 1.11 (sec) , antiderivative size = 193, normalized size of antiderivative = 6.89, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2026, 7293, 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.
\(\displaystyle \int \frac {28 x^5 \log ^3(7 x+1)+7 x^3+x^2+\left (21 x^5+3 x^4\right ) \log ^4(7 x+1)+\left (84 x^3-28 x^4\right ) \log (7 x+1)+\left (-28 x^4+38 x^3+6 x^2\right ) \log ^2(7 x+1)-63 x-9}{7 x^3+x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {28 x^5 \log ^3(7 x+1)+7 x^3+x^2+\left (21 x^5+3 x^4\right ) \log ^4(7 x+1)+\left (84 x^3-28 x^4\right ) \log (7 x+1)+\left (-28 x^4+38 x^3+6 x^2\right ) \log ^2(7 x+1)-63 x-9}{x^2 (7 x+1)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {28 x^3 \log ^3(7 x+1)}{7 x+1}+\frac {x^2-9}{x^2}+3 x^2 \log ^4(7 x+1)-2 (2 x-3) \log ^2(7 x+1)-\frac {28 (x-3) x \log (7 x+1)}{7 x+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^2-2 x^2 \log (7 x+1)-\frac {1}{49} (7 x+1)^2+\frac {9 x}{7}+\frac {9}{x}+\frac {1}{343} (7 x+1)^3 \log ^4(7 x+1)-\frac {3}{343} (7 x+1)^2 \log ^4(7 x+1)+\frac {3}{343} (7 x+1) \log ^4(7 x+1)-\frac {1}{343} \log ^4(7 x+1)-\frac {2}{49} (7 x+1)^2 \log ^2(7 x+1)+\frac {46}{49} (7 x+1) \log ^2(7 x+1)-\frac {44}{49} \log ^2(7 x+1)+\frac {2}{49} (7 x+1)^2 \log (7 x+1)-\frac {4}{49} (7 x+1) \log (7 x+1)+\frac {2}{49} \log (7 x+1)\) |
Int[(-9 - 63*x + x^2 + 7*x^3 + (84*x^3 - 28*x^4)*Log[1 + 7*x] + (6*x^2 + 3 8*x^3 - 28*x^4)*Log[1 + 7*x]^2 + 28*x^5*Log[1 + 7*x]^3 + (3*x^4 + 21*x^5)* Log[1 + 7*x]^4)/(x^2 + 7*x^3),x]
9/x + (9*x)/7 + x^2 - (1 + 7*x)^2/49 + (2*Log[1 + 7*x])/49 - 2*x^2*Log[1 + 7*x] - (4*(1 + 7*x)*Log[1 + 7*x])/49 + (2*(1 + 7*x)^2*Log[1 + 7*x])/49 - (44*Log[1 + 7*x]^2)/49 + (46*(1 + 7*x)*Log[1 + 7*x]^2)/49 - (2*(1 + 7*x)^2 *Log[1 + 7*x]^2)/49 - Log[1 + 7*x]^4/343 + (3*(1 + 7*x)*Log[1 + 7*x]^4)/34 3 - (3*(1 + 7*x)^2*Log[1 + 7*x]^4)/343 + ((1 + 7*x)^3*Log[1 + 7*x]^4)/343
3.3.31.3.1 Defintions of rubi rules used
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])
Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46
method | result | size |
risch | \(x^{3} \ln \left (7 x +1\right )^{4}+\left (-2 x^{2}+6 x \right ) \ln \left (7 x +1\right )^{2}+\frac {x^{2}+9}{x}\) | \(41\) |
parallelrisch | \(-\frac {-98 \ln \left (7 x +1\right )^{4} x^{4}+196 \ln \left (7 x +1\right )^{2} x^{3}-588 \ln \left (7 x +1\right )^{2} x^{2}-882-98 x^{2}+3094 x}{98 x}\) | \(55\) |
parts | \(x +\frac {9}{x}+\frac {3 \ln \left (7 x +1\right )^{4} \left (7 x +1\right )}{343}-\frac {2 \ln \left (7 x +1\right )^{2} \left (7 x +1\right )^{2}}{49}+\frac {46 \ln \left (7 x +1\right )^{2} \left (7 x +1\right )}{49}+\frac {\ln \left (7 x +1\right )^{4} \left (7 x +1\right )^{3}}{343}-\frac {3 \ln \left (7 x +1\right )^{4} \left (7 x +1\right )^{2}}{343}-\frac {\ln \left (7 x +1\right )^{4}}{343}-\frac {44 \ln \left (7 x +1\right )^{2}}{49}\) | \(109\) |
derivativedivides | \(\frac {3 \ln \left (7 x +1\right )^{4} \left (7 x +1\right )}{343}-\frac {2 \ln \left (7 x +1\right )^{2} \left (7 x +1\right )^{2}}{49}+\frac {46 \ln \left (7 x +1\right )^{2} \left (7 x +1\right )}{49}+\frac {\ln \left (7 x +1\right )^{4} \left (7 x +1\right )^{3}}{343}-\frac {3 \ln \left (7 x +1\right )^{4} \left (7 x +1\right )^{2}}{343}+x +\frac {1}{7}+\frac {9}{x}-\frac {44 \ln \left (7 x +1\right )^{2}}{49}-\frac {\ln \left (7 x +1\right )^{4}}{343}\) | \(110\) |
default | \(\frac {3 \ln \left (7 x +1\right )^{4} \left (7 x +1\right )}{343}-\frac {2 \ln \left (7 x +1\right )^{2} \left (7 x +1\right )^{2}}{49}+\frac {46 \ln \left (7 x +1\right )^{2} \left (7 x +1\right )}{49}+\frac {\ln \left (7 x +1\right )^{4} \left (7 x +1\right )^{3}}{343}-\frac {3 \ln \left (7 x +1\right )^{4} \left (7 x +1\right )^{2}}{343}+x +\frac {1}{7}+\frac {9}{x}-\frac {44 \ln \left (7 x +1\right )^{2}}{49}-\frac {\ln \left (7 x +1\right )^{4}}{343}\) | \(110\) |
int(((21*x^5+3*x^4)*ln(7*x+1)^4+28*x^5*ln(7*x+1)^3+(-28*x^4+38*x^3+6*x^2)* ln(7*x+1)^2+(-28*x^4+84*x^3)*ln(7*x+1)+7*x^3+x^2-63*x-9)/(7*x^3+x^2),x,met hod=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-9-63 x+x^2+7 x^3+\left (84 x^3-28 x^4\right ) \log (1+7 x)+\left (6 x^2+38 x^3-28 x^4\right ) \log ^2(1+7 x)+28 x^5 \log ^3(1+7 x)+\left (3 x^4+21 x^5\right ) \log ^4(1+7 x)}{x^2+7 x^3} \, dx=\frac {x^{4} \log \left (7 \, x + 1\right )^{4} - 2 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (7 \, x + 1\right )^{2} + x^{2} + 9}{x} \]
integrate(((21*x^5+3*x^4)*log(7*x+1)^4+28*x^5*log(7*x+1)^3+(-28*x^4+38*x^3 +6*x^2)*log(7*x+1)^2+(-28*x^4+84*x^3)*log(7*x+1)+7*x^3+x^2-63*x-9)/(7*x^3+ x^2),x, algorithm=\
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-9-63 x+x^2+7 x^3+\left (84 x^3-28 x^4\right ) \log (1+7 x)+\left (6 x^2+38 x^3-28 x^4\right ) \log ^2(1+7 x)+28 x^5 \log ^3(1+7 x)+\left (3 x^4+21 x^5\right ) \log ^4(1+7 x)}{x^2+7 x^3} \, dx=x^{3} \log {\left (7 x + 1 \right )}^{4} + x + \left (- 2 x^{2} + 6 x\right ) \log {\left (7 x + 1 \right )}^{2} + \frac {9}{x} \]
integrate(((21*x**5+3*x**4)*ln(7*x+1)**4+28*x**5*ln(7*x+1)**3+(-28*x**4+38 *x**3+6*x**2)*ln(7*x+1)**2+(-28*x**4+84*x**3)*ln(7*x+1)+7*x**3+x**2-63*x-9 )/(7*x**3+x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (25) = 50\).
Time = 0.20 (sec) , antiderivative size = 395, normalized size of antiderivative = 14.11 \[ \int \frac {-9-63 x+x^2+7 x^3+\left (84 x^3-28 x^4\right ) \log (1+7 x)+\left (6 x^2+38 x^3-28 x^4\right ) \log ^2(1+7 x)+28 x^5 \log ^3(1+7 x)+\left (3 x^4+21 x^5\right ) \log ^4(1+7 x)}{x^2+7 x^3} \, dx=\frac {1}{9261} \, {\left (27 \, \log \left (7 \, x + 1\right )^{4} - 36 \, \log \left (7 \, x + 1\right )^{3} + 36 \, \log \left (7 \, x + 1\right )^{2} - 24 \, \log \left (7 \, x + 1\right ) + 8\right )} {\left (7 \, x + 1\right )}^{3} + \frac {4}{9261} \, {\left (9 \, \log \left (7 \, x + 1\right )^{3} - 9 \, \log \left (7 \, x + 1\right )^{2} + 6 \, \log \left (7 \, x + 1\right ) - 2\right )} {\left (7 \, x + 1\right )}^{3} - \frac {1}{343} \, \log \left (7 \, x + 1\right )^{4} - \frac {3}{686} \, {\left (2 \, \log \left (7 \, x + 1\right )^{4} - 4 \, \log \left (7 \, x + 1\right )^{3} + 6 \, \log \left (7 \, x + 1\right )^{2} - 6 \, \log \left (7 \, x + 1\right ) + 3\right )} {\left (7 \, x + 1\right )}^{2} - \frac {3}{686} \, {\left (4 \, \log \left (7 \, x + 1\right )^{3} - 6 \, \log \left (7 \, x + 1\right )^{2} + 6 \, \log \left (7 \, x + 1\right ) - 3\right )} {\left (7 \, x + 1\right )}^{2} - \frac {1}{49} \, {\left (2 \, \log \left (7 \, x + 1\right )^{2} - 2 \, \log \left (7 \, x + 1\right ) + 1\right )} {\left (7 \, x + 1\right )}^{2} + \frac {3}{343} \, {\left (\log \left (7 \, x + 1\right )^{4} - 4 \, \log \left (7 \, x + 1\right )^{3} + 12 \, \log \left (7 \, x + 1\right )^{2} - 24 \, \log \left (7 \, x + 1\right ) + 24\right )} {\left (7 \, x + 1\right )} + \frac {12}{343} \, {\left (\log \left (7 \, x + 1\right )^{3} - 3 \, \log \left (7 \, x + 1\right )^{2} + 6 \, \log \left (7 \, x + 1\right ) - 6\right )} {\left (7 \, x + 1\right )} + \frac {46}{49} \, {\left (\log \left (7 \, x + 1\right )^{2} - 2 \, \log \left (7 \, x + 1\right ) + 2\right )} {\left (7 \, x + 1\right )} + x^{2} - \frac {2}{49} \, {\left (49 \, x^{2} - 14 \, x + 2 \, \log \left (7 \, x + 1\right )\right )} \log \left (7 \, x + 1\right ) + \frac {12}{7} \, {\left (7 \, x - \log \left (7 \, x + 1\right )\right )} \log \left (7 \, x + 1\right ) + \frac {44}{49} \, \log \left (7 \, x + 1\right )^{2} - \frac {83}{7} \, x + \frac {9}{x} + \frac {90}{49} \, \log \left (7 \, x + 1\right ) \]
integrate(((21*x^5+3*x^4)*log(7*x+1)^4+28*x^5*log(7*x+1)^3+(-28*x^4+38*x^3 +6*x^2)*log(7*x+1)^2+(-28*x^4+84*x^3)*log(7*x+1)+7*x^3+x^2-63*x-9)/(7*x^3+ x^2),x, algorithm=\
1/9261*(27*log(7*x + 1)^4 - 36*log(7*x + 1)^3 + 36*log(7*x + 1)^2 - 24*log (7*x + 1) + 8)*(7*x + 1)^3 + 4/9261*(9*log(7*x + 1)^3 - 9*log(7*x + 1)^2 + 6*log(7*x + 1) - 2)*(7*x + 1)^3 - 1/343*log(7*x + 1)^4 - 3/686*(2*log(7*x + 1)^4 - 4*log(7*x + 1)^3 + 6*log(7*x + 1)^2 - 6*log(7*x + 1) + 3)*(7*x + 1)^2 - 3/686*(4*log(7*x + 1)^3 - 6*log(7*x + 1)^2 + 6*log(7*x + 1) - 3)*( 7*x + 1)^2 - 1/49*(2*log(7*x + 1)^2 - 2*log(7*x + 1) + 1)*(7*x + 1)^2 + 3/ 343*(log(7*x + 1)^4 - 4*log(7*x + 1)^3 + 12*log(7*x + 1)^2 - 24*log(7*x + 1) + 24)*(7*x + 1) + 12/343*(log(7*x + 1)^3 - 3*log(7*x + 1)^2 + 6*log(7*x + 1) - 6)*(7*x + 1) + 46/49*(log(7*x + 1)^2 - 2*log(7*x + 1) + 2)*(7*x + 1) + x^2 - 2/49*(49*x^2 - 14*x + 2*log(7*x + 1))*log(7*x + 1) + 12/7*(7*x - log(7*x + 1))*log(7*x + 1) + 44/49*log(7*x + 1)^2 - 83/7*x + 9/x + 90/49 *log(7*x + 1)
Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-9-63 x+x^2+7 x^3+\left (84 x^3-28 x^4\right ) \log (1+7 x)+\left (6 x^2+38 x^3-28 x^4\right ) \log ^2(1+7 x)+28 x^5 \log ^3(1+7 x)+\left (3 x^4+21 x^5\right ) \log ^4(1+7 x)}{x^2+7 x^3} \, dx=x^{3} \log \left (7 \, x + 1\right )^{4} - 2 \, {\left (x^{2} - 3 \, x\right )} \log \left (7 \, x + 1\right )^{2} + x + \frac {9}{x} \]
integrate(((21*x^5+3*x^4)*log(7*x+1)^4+28*x^5*log(7*x+1)^3+(-28*x^4+38*x^3 +6*x^2)*log(7*x+1)^2+(-28*x^4+84*x^3)*log(7*x+1)+7*x^3+x^2-63*x-9)/(7*x^3+ x^2),x, algorithm=\
Time = 7.87 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {-9-63 x+x^2+7 x^3+\left (84 x^3-28 x^4\right ) \log (1+7 x)+\left (6 x^2+38 x^3-28 x^4\right ) \log ^2(1+7 x)+28 x^5 \log ^3(1+7 x)+\left (3 x^4+21 x^5\right ) \log ^4(1+7 x)}{x^2+7 x^3} \, dx=x-2\,x^2\,{\ln \left (7\,x+1\right )}^2+x^3\,{\ln \left (7\,x+1\right )}^4+\frac {9}{x}+6\,x\,{\ln \left (7\,x+1\right )}^2 \]