Integrand size = 57, antiderivative size = 28 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=x^2-x \left (-\frac {x^2 \log (x)}{3 (3+x)}-\log \left (x^2\right )\right ) \] Output:
x^2-(-1/3*x^2/(3+x)*ln(x)-ln(x^2))*x
Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=\frac {1}{3} \left (\frac {x^3 \log (x)}{3+x}+3 x \left (x+\log \left (x^2\right )\right )\right ) \] Input:
Integrate[(54 + 90*x + 45*x^2 + 7*x^3 + (9*x^2 + 2*x^3)*Log[x] + (27 + 18* x + 3*x^2)*Log[x^2])/(27 + 18*x + 3*x^2),x]
Output:
((x^3*Log[x])/(3 + x) + 3*x*(x + Log[x^2]))/3
Time = 0.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2007, 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.
\(\displaystyle \int \frac {7 x^3+45 x^2+\left (3 x^2+18 x+27\right ) \log \left (x^2\right )+\left (2 x^3+9 x^2\right ) \log (x)+90 x+54}{3 x^2+18 x+27} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {7 x^3+45 x^2+\left (3 x^2+18 x+27\right ) \log \left (x^2\right )+\left (2 x^3+9 x^2\right ) \log (x)+90 x+54}{\left (\sqrt {3} x+3 \sqrt {3}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {7 x^3}{3 (x+3)^2}+\frac {15 x^2}{(x+3)^2}+\frac {(2 x+9) x^2 \log (x)}{3 (x+3)^2}+\log \left (x^2\right )+\frac {30 x}{(x+3)^2}+\frac {18}{(x+3)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^2+\frac {1}{3} x^2 \log (x)+x \log \left (x^2\right )+\frac {3 x \log (x)}{x+3}-x \log (x)\) |
Input:
Int[(54 + 90*x + 45*x^2 + 7*x^3 + (9*x^2 + 2*x^3)*Log[x] + (27 + 18*x + 3* x^2)*Log[x^2])/(27 + 18*x + 3*x^2),x]
Output:
x^2 - x*Log[x] + (x^2*Log[x])/3 + (3*x*Log[x])/(3 + x) + x*Log[x^2]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 0.64 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18
method | result | size |
default | \(x^{2}+\frac {x^{2} \ln \left (x \right )}{3}-x \ln \left (x \right )+\frac {3 \ln \left (x \right ) x}{3+x}+x \ln \left (x^{2}\right )\) | \(33\) |
parts | \(x^{2}+\frac {x^{2} \ln \left (x \right )}{3}-x \ln \left (x \right )+\frac {3 \ln \left (x \right ) x}{3+x}+x \ln \left (x^{2}\right )\) | \(33\) |
parallelrisch | \(\frac {3 x^{2} \ln \left (x^{2}\right )+x^{3} \ln \left (x \right )+18 x \ln \left (x \right )+54 \ln \left (x \right )+9 x^{2}+3 x^{3}-27 \ln \left (x^{2}\right )}{3 x +9}\) | \(49\) |
risch | \(\frac {\left (x^{3}+6 x^{2}+9 x -27\right ) \ln \left (x \right )}{3 x +9}-\frac {i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}+i x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} \pi -\frac {i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}+x^{2}+3 \ln \left (x \right )\) | \(83\) |
orering | \(\frac {3 x \left (x^{5}+19 x^{4}+84 x^{3}-972 x +324\right ) \left (\left (3 x^{2}+18 x +27\right ) \ln \left (x^{2}\right )+\left (2 x^{3}+9 x^{2}\right ) \ln \left (x \right )+7 x^{3}+45 x^{2}+90 x +54\right )}{\left (4 x^{5}+63 x^{4}+324 x^{3}+1404 x^{2}+1944 x +972\right ) \left (3 x^{2}+18 x +27\right )}-\frac {\left (x^{5}+21 x^{4}-972 x^{2}-2916 x -4860\right ) x \left (3+x \right ) \left (\frac {\left (6 x +18\right ) \ln \left (x^{2}\right )+\frac {6 x^{2}+36 x +54}{x}+\left (6 x^{2}+18 x \right ) \ln \left (x \right )+\frac {2 x^{3}+9 x^{2}}{x}+21 x^{2}+90 x +90}{3 x^{2}+18 x +27}-\frac {\left (\left (3 x^{2}+18 x +27\right ) \ln \left (x^{2}\right )+\left (2 x^{3}+9 x^{2}\right ) \ln \left (x \right )+7 x^{3}+45 x^{2}+90 x +54\right ) \left (6 x +18\right )}{\left (3 x^{2}+18 x +27\right )^{2}}\right )}{4 x^{5}+63 x^{4}+324 x^{3}+1404 x^{2}+1944 x +972}\) | \(296\) |
Input:
int(((3*x^2+18*x+27)*ln(x^2)+(2*x^3+9*x^2)*ln(x)+7*x^3+45*x^2+90*x+54)/(3* x^2+18*x+27),x,method=_RETURNVERBOSE)
Output:
x^2+1/3*x^2*ln(x)-x*ln(x)+3*ln(x)*x/(3+x)+x*ln(x^2)
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=\frac {3 \, x^{3} + 9 \, x^{2} + {\left (x^{3} + 6 \, x^{2} + 18 \, x\right )} \log \left (x\right )}{3 \, {\left (x + 3\right )}} \] Input:
integrate(((3*x^2+18*x+27)*log(x^2)+(2*x^3+9*x^2)*log(x)+7*x^3+45*x^2+90*x +54)/(3*x^2+18*x+27),x, algorithm="fricas")
Output:
1/3*(3*x^3 + 9*x^2 + (x^3 + 6*x^2 + 18*x)*log(x))/(x + 3)
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=x^{2} + 3 \log {\left (x \right )} + \frac {\left (x^{3} + 6 x^{2} + 9 x - 27\right ) \log {\left (x \right )}}{3 x + 9} \] Input:
integrate(((3*x**2+18*x+27)*ln(x**2)+(2*x**3+9*x**2)*ln(x)+7*x**3+45*x**2+ 90*x+54)/(3*x**2+18*x+27),x)
Output:
x**2 + 3*log(x) + (x**3 + 6*x**2 + 9*x - 27)*log(x)/(3*x + 9)
Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=\frac {7}{6} \, x^{2} + x - \frac {x^{3} + 9 \, x^{2} - 2 \, {\left (x^{3} + 6 \, x^{2} + 9 \, x - 27\right )} \log \left (x\right ) + 18 \, x}{6 \, {\left (x + 3\right )}} + 3 \, \log \left (x\right ) \] Input:
integrate(((3*x^2+18*x+27)*log(x^2)+(2*x^3+9*x^2)*log(x)+7*x^3+45*x^2+90*x +54)/(3*x^2+18*x+27),x, algorithm="maxima")
Output:
7/6*x^2 + x - 1/6*(x^3 + 9*x^2 - 2*(x^3 + 6*x^2 + 9*x - 27)*log(x) + 18*x) /(x + 3) + 3*log(x)
Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=x^{2} + \frac {1}{3} \, {\left (x^{2} + 3 \, x - \frac {27}{x + 3}\right )} \log \left (x\right ) + 3 \, \log \left (x\right ) \] Input:
integrate(((3*x^2+18*x+27)*log(x^2)+(2*x^3+9*x^2)*log(x)+7*x^3+45*x^2+90*x +54)/(3*x^2+18*x+27),x, algorithm="giac")
Output:
x^2 + 1/3*(x^2 + 3*x - 27/(x + 3))*log(x) + 3*log(x)
Time = 3.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=\frac {x\,\left (9\,x+9\,\ln \left (x^2\right )+3\,x\,\ln \left (x^2\right )+x^2\,\ln \left (x\right )+3\,x^2\right )}{3\,\left (x+3\right )} \] Input:
int((90*x + log(x)*(9*x^2 + 2*x^3) + log(x^2)*(18*x + 3*x^2 + 27) + 45*x^2 + 7*x^3 + 54)/(18*x + 3*x^2 + 27),x)
Output:
(x*(9*x + 9*log(x^2) + 3*x*log(x^2) + x^2*log(x) + 3*x^2))/(3*(x + 3))
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=\frac {x \left (3 \,\mathrm {log}\left (x^{2}\right ) x +9 \,\mathrm {log}\left (x^{2}\right )+\mathrm {log}\left (x \right ) x^{2}+3 x^{2}+9 x \right )}{3 x +9} \] Input:
int(((3*x^2+18*x+27)*log(x^2)+(2*x^3+9*x^2)*log(x)+7*x^3+45*x^2+90*x+54)/( 3*x^2+18*x+27),x)
Output:
(x*(3*log(x**2)*x + 9*log(x**2) + log(x)*x**2 + 3*x**2 + 9*x))/(3*(x + 3))