Integrand size = 103, antiderivative size = 20 \[ \int \frac {324 x^4+195 x^5+26 x^6+x^7+\left (141 x+24 x^2+x^3+1296 x^4+780 x^5+104 x^6+4 x^7\right ) \log (x)+\left (324+195 x+26 x^2+x^3\right ) \log \left (\frac {27+14 x+x^2}{12+x}\right )}{324 x+195 x^2+26 x^3+x^4} \, dx=\log (x) \left (x^4+\log \left (2+\frac {1}{4+\frac {x}{3}}+x\right )\right ) \]
Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {324 x^4+195 x^5+26 x^6+x^7+\left (141 x+24 x^2+x^3+1296 x^4+780 x^5+104 x^6+4 x^7\right ) \log (x)+\left (324+195 x+26 x^2+x^3\right ) \log \left (\frac {27+14 x+x^2}{12+x}\right )}{324 x+195 x^2+26 x^3+x^4} \, dx=x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right ) \]
Integrate[(324*x^4 + 195*x^5 + 26*x^6 + x^7 + (141*x + 24*x^2 + x^3 + 1296 *x^4 + 780*x^5 + 104*x^6 + 4*x^7)*Log[x] + (324 + 195*x + 26*x^2 + x^3)*Lo g[(27 + 14*x + x^2)/(12 + x)])/(324*x + 195*x^2 + 26*x^3 + x^4),x]
Leaf count is larger than twice the leaf count of optimal. \(209\) vs. \(2(20)=40\).
Time = 2.41 (sec) , antiderivative size = 209, normalized size of antiderivative = 10.45, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, 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.
\(\displaystyle \int \frac {x^7+26 x^6+195 x^5+324 x^4+\left (x^3+26 x^2+195 x+324\right ) \log \left (\frac {x^2+14 x+27}{x+12}\right )+\left (4 x^7+104 x^6+780 x^5+1296 x^4+x^3+24 x^2+141 x\right ) \log (x)}{x^4+26 x^3+195 x^2+324 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {x^7+26 x^6+195 x^5+324 x^4+\left (x^3+26 x^2+195 x+324\right ) \log \left (\frac {x^2+14 x+27}{x+12}\right )+\left (4 x^7+104 x^6+780 x^5+1296 x^4+x^3+24 x^2+141 x\right ) \log (x)}{x \left (x^3+26 x^2+195 x+324\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {x^7+26 x^6+195 x^5+324 x^4+\left (x^3+26 x^2+195 x+324\right ) \log \left (\frac {x^2+14 x+27}{x+12}\right )+\left (4 x^7+104 x^6+780 x^5+1296 x^4+x^3+24 x^2+141 x\right ) \log (x)}{3 x (x+12)}+\frac {(-x-2) \left (x^7+26 x^6+195 x^5+324 x^4+\left (x^3+26 x^2+195 x+324\right ) \log \left (\frac {x^2+14 x+27}{x+12}\right )+\left (4 x^7+104 x^6+780 x^5+1296 x^4+x^3+24 x^2+141 x\right ) \log (x)\right )}{3 x \left (x^2+14 x+27\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^4 \log (x)+\log (x) \log \left (\frac {x^2+14 x+27}{x+12}\right )+\frac {81}{11} \left (12100-2581 \sqrt {22}\right ) \log \left (x-\sqrt {22}+7\right )-\frac {195}{44} \left (141482-30167 \sqrt {22}\right ) \log \left (x-\sqrt {22}+7\right )+\frac {13}{22} \left (1654048-352651 \sqrt {22}\right ) \log \left (x-\sqrt {22}+7\right )-\frac {1}{44} \left (19336658-4122605 \sqrt {22}\right ) \log \left (x-\sqrt {22}+7\right )-\frac {1}{44} \left (19336658+4122605 \sqrt {22}\right ) \log \left (x+\sqrt {22}+7\right )+\frac {13}{22} \left (1654048+352651 \sqrt {22}\right ) \log \left (x+\sqrt {22}+7\right )-\frac {195}{44} \left (141482+30167 \sqrt {22}\right ) \log \left (x+\sqrt {22}+7\right )+\frac {81}{11} \left (12100+2581 \sqrt {22}\right ) \log \left (x+\sqrt {22}+7\right )\) |
Int[(324*x^4 + 195*x^5 + 26*x^6 + x^7 + (141*x + 24*x^2 + x^3 + 1296*x^4 + 780*x^5 + 104*x^6 + 4*x^7)*Log[x] + (324 + 195*x + 26*x^2 + x^3)*Log[(27 + 14*x + x^2)/(12 + x)])/(324*x + 195*x^2 + 26*x^3 + x^4),x]
x^4*Log[x] - ((19336658 - 4122605*Sqrt[22])*Log[7 - Sqrt[22] + x])/44 + (1 3*(1654048 - 352651*Sqrt[22])*Log[7 - Sqrt[22] + x])/22 - (195*(141482 - 3 0167*Sqrt[22])*Log[7 - Sqrt[22] + x])/44 + (81*(12100 - 2581*Sqrt[22])*Log [7 - Sqrt[22] + x])/11 + (81*(12100 + 2581*Sqrt[22])*Log[7 + Sqrt[22] + x] )/11 - (195*(141482 + 30167*Sqrt[22])*Log[7 + Sqrt[22] + x])/44 + (13*(165 4048 + 352651*Sqrt[22])*Log[7 + Sqrt[22] + x])/22 - ((19336658 + 4122605*S qrt[22])*Log[7 + Sqrt[22] + x])/44 + Log[x]*Log[(27 + 14*x + x^2)/(12 + x) ]
3.21.1.3.1 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])
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 = 1.86 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30
method | result | size |
default | \(\ln \left (x \right ) \ln \left (\frac {x^{2}+14 x +27}{x +12}\right )+x^{4} \ln \left (x \right )\) | \(26\) |
parallelrisch | \(\ln \left (x \right ) \ln \left (\frac {x^{2}+14 x +27}{x +12}\right )+x^{4} \ln \left (x \right )\) | \(26\) |
parts | \(\ln \left (x \right ) \ln \left (\frac {x^{2}+14 x +27}{x +12}\right )+x^{4} \ln \left (x \right )\) | \(26\) |
risch | \(\ln \left (x \right ) \ln \left (x^{2}+14 x +27\right )-\ln \left (x \right ) \ln \left (x +12\right )+x^{4} \ln \left (x \right )-\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{x +12}\right ) \operatorname {csgn}\left (i \left (x^{2}+14 x +27\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )}{2}+\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{x +12}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )}^{2}}{2}+\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (i \left (x^{2}+14 x +27\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )}^{2}}{2}-\frac {i \pi \ln \left (x \right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )}^{3}}{2}\) | \(168\) |
int(((x^3+26*x^2+195*x+324)*ln((x^2+14*x+27)/(x+12))+(4*x^7+104*x^6+780*x^ 5+1296*x^4+x^3+24*x^2+141*x)*ln(x)+x^7+26*x^6+195*x^5+324*x^4)/(x^4+26*x^3 +195*x^2+324*x),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {324 x^4+195 x^5+26 x^6+x^7+\left (141 x+24 x^2+x^3+1296 x^4+780 x^5+104 x^6+4 x^7\right ) \log (x)+\left (324+195 x+26 x^2+x^3\right ) \log \left (\frac {27+14 x+x^2}{12+x}\right )}{324 x+195 x^2+26 x^3+x^4} \, dx=x^{4} \log \left (x\right ) + \log \left (x\right ) \log \left (\frac {x^{2} + 14 \, x + 27}{x + 12}\right ) \]
integrate(((x^3+26*x^2+195*x+324)*log((x^2+14*x+27)/(x+12))+(4*x^7+104*x^6 +780*x^5+1296*x^4+x^3+24*x^2+141*x)*log(x)+x^7+26*x^6+195*x^5+324*x^4)/(x^ 4+26*x^3+195*x^2+324*x),x, algorithm=\
Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {324 x^4+195 x^5+26 x^6+x^7+\left (141 x+24 x^2+x^3+1296 x^4+780 x^5+104 x^6+4 x^7\right ) \log (x)+\left (324+195 x+26 x^2+x^3\right ) \log \left (\frac {27+14 x+x^2}{12+x}\right )}{324 x+195 x^2+26 x^3+x^4} \, dx=x^{4} \log {\left (x \right )} + \log {\left (x \right )} \log {\left (\frac {x^{2} + 14 x + 27}{x + 12} \right )} \]
integrate(((x**3+26*x**2+195*x+324)*ln((x**2+14*x+27)/(x+12))+(4*x**7+104* x**6+780*x**5+1296*x**4+x**3+24*x**2+141*x)*ln(x)+x**7+26*x**6+195*x**5+32 4*x**4)/(x**4+26*x**3+195*x**2+324*x),x)
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {324 x^4+195 x^5+26 x^6+x^7+\left (141 x+24 x^2+x^3+1296 x^4+780 x^5+104 x^6+4 x^7\right ) \log (x)+\left (324+195 x+26 x^2+x^3\right ) \log \left (\frac {27+14 x+x^2}{12+x}\right )}{324 x+195 x^2+26 x^3+x^4} \, dx=x^{4} \log \left (x\right ) + \log \left (x^{2} + 14 \, x + 27\right ) \log \left (x\right ) - \log \left (x + 12\right ) \log \left (x\right ) \]
integrate(((x^3+26*x^2+195*x+324)*log((x^2+14*x+27)/(x+12))+(4*x^7+104*x^6 +780*x^5+1296*x^4+x^3+24*x^2+141*x)*log(x)+x^7+26*x^6+195*x^5+324*x^4)/(x^ 4+26*x^3+195*x^2+324*x),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {324 x^4+195 x^5+26 x^6+x^7+\left (141 x+24 x^2+x^3+1296 x^4+780 x^5+104 x^6+4 x^7\right ) \log (x)+\left (324+195 x+26 x^2+x^3\right ) \log \left (\frac {27+14 x+x^2}{12+x}\right )}{324 x+195 x^2+26 x^3+x^4} \, dx=x^{4} \log \left (x\right ) + \log \left (x^{2} + 14 \, x + 27\right ) \log \left (x\right ) - \log \left (x + 12\right ) \log \left (x\right ) \]
integrate(((x^3+26*x^2+195*x+324)*log((x^2+14*x+27)/(x+12))+(4*x^7+104*x^6 +780*x^5+1296*x^4+x^3+24*x^2+141*x)*log(x)+x^7+26*x^6+195*x^5+324*x^4)/(x^ 4+26*x^3+195*x^2+324*x),x, algorithm=\
Time = 12.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {324 x^4+195 x^5+26 x^6+x^7+\left (141 x+24 x^2+x^3+1296 x^4+780 x^5+104 x^6+4 x^7\right ) \log (x)+\left (324+195 x+26 x^2+x^3\right ) \log \left (\frac {27+14 x+x^2}{12+x}\right )}{324 x+195 x^2+26 x^3+x^4} \, dx=\ln \left (x\right )\,\left (\ln \left (\frac {x^2+14\,x+27}{x+12}\right )+x^4\right ) \]