Integrand size = 66, antiderivative size = 22 \[ \int \frac {1171875+777500 x+3750 x^3+2503 x^4+3 x^6+2 x^7}{11328125+1171875 x+388750 x^2+36250 x^3+3750 x^4+1247 x^5+29 x^6+3 x^7+x^8} \, dx=\log \left (-25+x-(2+x)^2+\frac {3}{\frac {625}{x^2}+x}\right ) \] Output:
ln(x-(2+x)^2-25+3/(625/x^2+x))
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {1171875+777500 x+3750 x^3+2503 x^4+3 x^6+2 x^7}{11328125+1171875 x+388750 x^2+36250 x^3+3750 x^4+1247 x^5+29 x^6+3 x^7+x^8} \, dx=-\log \left (625+x^3\right )+\log \left (18125+1875 x+622 x^2+29 x^3+3 x^4+x^5\right ) \] Input:
Integrate[(1171875 + 777500*x + 3750*x^3 + 2503*x^4 + 3*x^6 + 2*x^7)/(1132 8125 + 1171875*x + 388750*x^2 + 36250*x^3 + 3750*x^4 + 1247*x^5 + 29*x^6 + 3*x^7 + x^8),x]
Output:
-Log[625 + x^3] + Log[18125 + 1875*x + 622*x^2 + 29*x^3 + 3*x^4 + x^5]
Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2462, 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.
\(\displaystyle \int \frac {2 x^7+3 x^6+2503 x^4+3750 x^3+777500 x+1171875}{x^8+3 x^7+29 x^6+1247 x^5+3750 x^4+36250 x^3+388750 x^2+1171875 x+11328125} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {5 x^4+12 x^3+87 x^2+1244 x+1875}{x^5+3 x^4+29 x^3+622 x^2+1875 x+18125}-\frac {3 x^2}{x^3+625}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (x^5+3 x^4+29 x^3+622 x^2+1875 x+18125\right )-\log \left (x^3+625\right )\) |
Input:
Int[(1171875 + 777500*x + 3750*x^3 + 2503*x^4 + 3*x^6 + 2*x^7)/(11328125 + 1171875*x + 388750*x^2 + 36250*x^3 + 3750*x^4 + 1247*x^5 + 29*x^6 + 3*x^7 + x^8),x]
Output:
-Log[625 + x^3] + Log[18125 + 1875*x + 622*x^2 + 29*x^3 + 3*x^4 + x^5]
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] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55
method | result | size |
default | \(\ln \left (x^{5}+3 x^{4}+29 x^{3}+622 x^{2}+1875 x +18125\right )-\ln \left (x^{3}+625\right )\) | \(34\) |
norman | \(\ln \left (x^{5}+3 x^{4}+29 x^{3}+622 x^{2}+1875 x +18125\right )-\ln \left (x^{3}+625\right )\) | \(34\) |
risch | \(\ln \left (x^{5}+3 x^{4}+29 x^{3}+622 x^{2}+1875 x +18125\right )-\ln \left (x^{3}+625\right )\) | \(34\) |
parallelrisch | \(\ln \left (x^{5}+3 x^{4}+29 x^{3}+622 x^{2}+1875 x +18125\right )-\ln \left (x^{3}+625\right )\) | \(34\) |
Input:
int((2*x^7+3*x^6+2503*x^4+3750*x^3+777500*x+1171875)/(x^8+3*x^7+29*x^6+124 7*x^5+3750*x^4+36250*x^3+388750*x^2+1171875*x+11328125),x,method=_RETURNVE RBOSE)
Output:
ln(x^5+3*x^4+29*x^3+622*x^2+1875*x+18125)-ln(x^3+625)
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {1171875+777500 x+3750 x^3+2503 x^4+3 x^6+2 x^7}{11328125+1171875 x+388750 x^2+36250 x^3+3750 x^4+1247 x^5+29 x^6+3 x^7+x^8} \, dx=\log \left (x^{5} + 3 \, x^{4} + 29 \, x^{3} + 622 \, x^{2} + 1875 \, x + 18125\right ) - \log \left (x^{3} + 625\right ) \] Input:
integrate((2*x^7+3*x^6+2503*x^4+3750*x^3+777500*x+1171875)/(x^8+3*x^7+29*x ^6+1247*x^5+3750*x^4+36250*x^3+388750*x^2+1171875*x+11328125),x, algorithm ="fricas")
Output:
log(x^5 + 3*x^4 + 29*x^3 + 622*x^2 + 1875*x + 18125) - log(x^3 + 625)
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {1171875+777500 x+3750 x^3+2503 x^4+3 x^6+2 x^7}{11328125+1171875 x+388750 x^2+36250 x^3+3750 x^4+1247 x^5+29 x^6+3 x^7+x^8} \, dx=- \log {\left (x^{3} + 625 \right )} + \log {\left (x^{5} + 3 x^{4} + 29 x^{3} + 622 x^{2} + 1875 x + 18125 \right )} \] Input:
integrate((2*x**7+3*x**6+2503*x**4+3750*x**3+777500*x+1171875)/(x**8+3*x** 7+29*x**6+1247*x**5+3750*x**4+36250*x**3+388750*x**2+1171875*x+11328125),x )
Output:
-log(x**3 + 625) + log(x**5 + 3*x**4 + 29*x**3 + 622*x**2 + 1875*x + 18125 )
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {1171875+777500 x+3750 x^3+2503 x^4+3 x^6+2 x^7}{11328125+1171875 x+388750 x^2+36250 x^3+3750 x^4+1247 x^5+29 x^6+3 x^7+x^8} \, dx=\log \left (x^{5} + 3 \, x^{4} + 29 \, x^{3} + 622 \, x^{2} + 1875 \, x + 18125\right ) - \log \left (x^{3} + 625\right ) \] Input:
integrate((2*x^7+3*x^6+2503*x^4+3750*x^3+777500*x+1171875)/(x^8+3*x^7+29*x ^6+1247*x^5+3750*x^4+36250*x^3+388750*x^2+1171875*x+11328125),x, algorithm ="maxima")
Output:
log(x^5 + 3*x^4 + 29*x^3 + 622*x^2 + 1875*x + 18125) - log(x^3 + 625)
Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {1171875+777500 x+3750 x^3+2503 x^4+3 x^6+2 x^7}{11328125+1171875 x+388750 x^2+36250 x^3+3750 x^4+1247 x^5+29 x^6+3 x^7+x^8} \, dx=\log \left ({\left | x^{5} + 3 \, x^{4} + 29 \, x^{3} + 622 \, x^{2} + 1875 \, x + 18125 \right |}\right ) - \log \left ({\left | x^{3} + 625 \right |}\right ) \] Input:
integrate((2*x^7+3*x^6+2503*x^4+3750*x^3+777500*x+1171875)/(x^8+3*x^7+29*x ^6+1247*x^5+3750*x^4+36250*x^3+388750*x^2+1171875*x+11328125),x, algorithm ="giac")
Output:
log(abs(x^5 + 3*x^4 + 29*x^3 + 622*x^2 + 1875*x + 18125)) - log(abs(x^3 + 625))
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {1171875+777500 x+3750 x^3+2503 x^4+3 x^6+2 x^7}{11328125+1171875 x+388750 x^2+36250 x^3+3750 x^4+1247 x^5+29 x^6+3 x^7+x^8} \, dx=\ln \left (x^5+3\,x^4+29\,x^3+622\,x^2+1875\,x+18125\right )-\ln \left (x^3+625\right ) \] Input:
int((777500*x + 3750*x^3 + 2503*x^4 + 3*x^6 + 2*x^7 + 1171875)/(1171875*x + 388750*x^2 + 36250*x^3 + 3750*x^4 + 1247*x^5 + 29*x^6 + 3*x^7 + x^8 + 11 328125),x)
Output:
log(1875*x + 622*x^2 + 29*x^3 + 3*x^4 + x^5 + 18125) - log(x^3 + 625)
\[ \int \frac {1171875+777500 x+3750 x^3+2503 x^4+3 x^6+2 x^7}{11328125+1171875 x+388750 x^2+36250 x^3+3750 x^4+1247 x^5+29 x^6+3 x^7+x^8} \, dx=-\frac {9 \left (\int \frac {x^{6}}{x^{8}+3 x^{7}+29 x^{6}+1247 x^{5}+3750 x^{4}+36250 x^{3}+388750 x^{2}+1171875 x +11328125}d x \right )}{4}-\frac {87 \left (\int \frac {x^{5}}{x^{8}+3 x^{7}+29 x^{6}+1247 x^{5}+3750 x^{4}+36250 x^{3}+388750 x^{2}+1171875 x +11328125}d x \right )}{2}+\frac {3777 \left (\int \frac {x^{4}}{x^{8}+3 x^{7}+29 x^{6}+1247 x^{5}+3750 x^{4}+36250 x^{3}+388750 x^{2}+1171875 x +11328125}d x \right )}{4}-\frac {54375 \left (\int \frac {x^{2}}{x^{8}+3 x^{7}+29 x^{6}+1247 x^{5}+3750 x^{4}+36250 x^{3}+388750 x^{2}+1171875 x +11328125}d x \right )}{2}+583125 \left (\int \frac {x}{x^{8}+3 x^{7}+29 x^{6}+1247 x^{5}+3750 x^{4}+36250 x^{3}+388750 x^{2}+1171875 x +11328125}d x \right )+\frac {3515625 \left (\int \frac {1}{x^{8}+3 x^{7}+29 x^{6}+1247 x^{5}+3750 x^{4}+36250 x^{3}+388750 x^{2}+1171875 x +11328125}d x \right )}{4}+\frac {\mathrm {log}\left (x^{8}+3 x^{7}+29 x^{6}+1247 x^{5}+3750 x^{4}+36250 x^{3}+388750 x^{2}+1171875 x +11328125\right )}{4} \] Input:
int((2*x^7+3*x^6+2503*x^4+3750*x^3+777500*x+1171875)/(x^8+3*x^7+29*x^6+124 7*x^5+3750*x^4+36250*x^3+388750*x^2+1171875*x+11328125),x)
Output:
( - 9*int(x**6/(x**8 + 3*x**7 + 29*x**6 + 1247*x**5 + 3750*x**4 + 36250*x* *3 + 388750*x**2 + 1171875*x + 11328125),x) - 174*int(x**5/(x**8 + 3*x**7 + 29*x**6 + 1247*x**5 + 3750*x**4 + 36250*x**3 + 388750*x**2 + 1171875*x + 11328125),x) + 3777*int(x**4/(x**8 + 3*x**7 + 29*x**6 + 1247*x**5 + 3750* x**4 + 36250*x**3 + 388750*x**2 + 1171875*x + 11328125),x) - 108750*int(x* *2/(x**8 + 3*x**7 + 29*x**6 + 1247*x**5 + 3750*x**4 + 36250*x**3 + 388750* x**2 + 1171875*x + 11328125),x) + 2332500*int(x/(x**8 + 3*x**7 + 29*x**6 + 1247*x**5 + 3750*x**4 + 36250*x**3 + 388750*x**2 + 1171875*x + 11328125), x) + 3515625*int(1/(x**8 + 3*x**7 + 29*x**6 + 1247*x**5 + 3750*x**4 + 3625 0*x**3 + 388750*x**2 + 1171875*x + 11328125),x) + log(x**8 + 3*x**7 + 29*x **6 + 1247*x**5 + 3750*x**4 + 36250*x**3 + 388750*x**2 + 1171875*x + 11328 125))/4