Integrand size = 36, antiderivative size = 25 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=x \left (e^{3 \left (-25+(4+x)^2\right )}-x+(5-\log (x))^2\right ) \] Output:
x*((5-ln(x))^2+exp(3*(4+x)^2-75)-x)
Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=x \left (25+e^{-27+3 x (8+x)}-x-10 \log (x)+\log ^2(x)\right ) \] Input:
Integrate[15 - 2*x + E^(-27 + 24*x + 3*x^2)*(1 + 24*x + 6*x^2) - 8*Log[x] + Log[x]^2,x]
Output:
x*(25 + E^(-27 + 3*x*(8 + x)) - x - 10*Log[x] + Log[x]^2)
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {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.
\(\displaystyle \int \left (e^{3 x^2+24 x-27} \left (6 x^2+24 x+1\right )-2 x+\log ^2(x)-8 \log (x)+15\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -x^2+\frac {e^{3 x^2+24 x-27} \left (x^2+4 x\right )}{x+4}+25 x+x \log ^2(x)-10 x \log (x)\) |
Input:
Int[15 - 2*x + E^(-27 + 24*x + 3*x^2)*(1 + 24*x + 6*x^2) - 8*Log[x] + Log[ x]^2,x]
Output:
25*x - x^2 + (E^(-27 + 24*x + 3*x^2)*(4*x + x^2))/(4 + x) - 10*x*Log[x] + x*Log[x]^2
Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28
method | result | size |
risch | \(x \ln \left (x \right )^{2}+{\mathrm e}^{3 \left (x +9\right ) \left (-1+x \right )} x +25 x -x^{2}-10 x \ln \left (x \right )\) | \(32\) |
norman | \(x \ln \left (x \right )^{2}+{\mathrm e}^{3 x^{2}+24 x -27} x +25 x -x^{2}-10 x \ln \left (x \right )\) | \(34\) |
parallelrisch | \(x \ln \left (x \right )^{2}+{\mathrm e}^{3 x^{2}+24 x -27} x +25 x -x^{2}-10 x \ln \left (x \right )\) | \(34\) |
default | \(25 x +x \ln \left (x \right )^{2}-10 x \ln \left (x \right )-\frac {i {\mathrm e}^{-27} \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{6}+24 \,{\mathrm e}^{-27} \left (\frac {{\mathrm e}^{3 x^{2}+24 x}}{6}+\frac {2 i \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{3}\right )+6 \,{\mathrm e}^{-27} \left (\frac {x \,{\mathrm e}^{3 x^{2}+24 x}}{6}-\frac {2 \,{\mathrm e}^{3 x^{2}+24 x}}{3}-\frac {95 i \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{36}\right )-x^{2}\) | \(148\) |
parts | \(25 x +x \ln \left (x \right )^{2}-10 x \ln \left (x \right )-\frac {i {\mathrm e}^{-27} \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{6}+24 \,{\mathrm e}^{-27} \left (\frac {{\mathrm e}^{3 x^{2}+24 x}}{6}+\frac {2 i \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{3}\right )+6 \,{\mathrm e}^{-27} \left (\frac {x \,{\mathrm e}^{3 x^{2}+24 x}}{6}-\frac {2 \,{\mathrm e}^{3 x^{2}+24 x}}{3}-\frac {95 i \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{36}\right )-x^{2}\) | \(148\) |
orering | \(\left (\ln \left (x \right )^{2}-8 \ln \left (x \right )+\left (6 x^{2}+24 x +1\right ) {\mathrm e}^{3 x^{2}+24 x -27}-2 x +15\right ) x -\frac {x^{2} \left (216 x^{8}+3456 x^{7}+26532 x^{6}+145368 x^{5}+594090 x^{4}+1491480 x^{3}+1716045 x^{2}+360592 x +14445\right ) \left (\frac {2 \ln \left (x \right )}{x}-\frac {8}{x}+\left (12 x +24\right ) {\mathrm e}^{3 x^{2}+24 x -27}+\left (6 x^{2}+24 x +1\right ) \left (24+6 x \right ) {\mathrm e}^{3 x^{2}+24 x -27}-2\right )}{4 \left (108 x^{8}+1620 x^{7}+8856 x^{6}+19206 x^{5}+4617 x^{4}-28254 x^{3}-8265 x^{2}-486 x -4\right )}+\frac {x^{2} \left (648 x^{9}+9072 x^{8}+26460 x^{7}-207792 x^{6}-1726866 x^{5}-4742784 x^{4}-4954509 x^{3}-701472 x^{2}+14775 x +1192\right ) \left (\frac {10}{x^{2}}-\frac {2 \ln \left (x \right )}{x^{2}}+12 \,{\mathrm e}^{3 x^{2}+24 x -27}+2 \left (12 x +24\right ) \left (24+6 x \right ) {\mathrm e}^{3 x^{2}+24 x -27}+6 \left (6 x^{2}+24 x +1\right ) {\mathrm e}^{3 x^{2}+24 x -27}+\left (6 x^{2}+24 x +1\right ) \left (24+6 x \right )^{2} {\mathrm e}^{3 x^{2}+24 x -27}\right )}{1296 x^{8}+19440 x^{7}+106272 x^{6}+230472 x^{5}+55404 x^{4}-339048 x^{3}-99180 x^{2}-5832 x -48}-\frac {x^{3} \left (108 x^{7}+1080 x^{6}-35352 x^{4}-146403 x^{3}-193068 x^{2}-29079 x -596\right ) \left (-\frac {22}{x^{3}}+\frac {4 \ln \left (x \right )}{x^{3}}+36 \left (24+6 x \right ) {\mathrm e}^{3 x^{2}+24 x -27}+18 \left (12 x +24\right ) {\mathrm e}^{3 x^{2}+24 x -27}+3 \left (12 x +24\right ) \left (24+6 x \right )^{2} {\mathrm e}^{3 x^{2}+24 x -27}+18 \left (6 x^{2}+24 x +1\right ) \left (24+6 x \right ) {\mathrm e}^{3 x^{2}+24 x -27}+\left (6 x^{2}+24 x +1\right ) \left (24+6 x \right )^{3} {\mathrm e}^{3 x^{2}+24 x -27}\right )}{12 \left (108 x^{8}+1620 x^{7}+8856 x^{6}+19206 x^{5}+4617 x^{4}-28254 x^{3}-8265 x^{2}-486 x -4\right )}\) | \(585\) |
Input:
int(ln(x)^2-8*ln(x)+(6*x^2+24*x+1)*exp(3*x^2+24*x-27)-2*x+15,x,method=_RET URNVERBOSE)
Output:
x*ln(x)^2+exp(3*(x+9)*(-1+x))*x+25*x-x^2-10*x*ln(x)
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=x \log \left (x\right )^{2} - x^{2} + x e^{\left (3 \, x^{2} + 24 \, x - 27\right )} - 10 \, x \log \left (x\right ) + 25 \, x \] Input:
integrate(log(x)^2-8*log(x)+(6*x^2+24*x+1)*exp(3*x^2+24*x-27)-2*x+15,x, al gorithm="fricas")
Output:
x*log(x)^2 - x^2 + x*e^(3*x^2 + 24*x - 27) - 10*x*log(x) + 25*x
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=- x^{2} + x e^{3 x^{2} + 24 x - 27} + x \log {\left (x \right )}^{2} - 10 x \log {\left (x \right )} + 25 x \] Input:
integrate(ln(x)**2-8*ln(x)+(6*x**2+24*x+1)*exp(3*x**2+24*x-27)-2*x+15,x)
Output:
-x**2 + x*exp(3*x**2 + 24*x - 27) + x*log(x)**2 - 10*x*log(x) + 25*x
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx={\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - x^{2} + x e^{\left (3 \, x^{2} + 24 \, x - 27\right )} - 8 \, x \log \left (x\right ) + 23 \, x \] Input:
integrate(log(x)^2-8*log(x)+(6*x^2+24*x+1)*exp(3*x^2+24*x-27)-2*x+15,x, al gorithm="maxima")
Output:
(log(x)^2 - 2*log(x) + 2)*x - x^2 + x*e^(3*x^2 + 24*x - 27) - 8*x*log(x) + 23*x
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=x \log \left (x\right )^{2} - x^{2} + x e^{\left (3 \, x^{2} + 24 \, x - 27\right )} - 10 \, x \log \left (x\right ) + 25 \, x \] Input:
integrate(log(x)^2-8*log(x)+(6*x^2+24*x+1)*exp(3*x^2+24*x-27)-2*x+15,x, al gorithm="giac")
Output:
x*log(x)^2 - x^2 + x*e^(3*x^2 + 24*x - 27) - 10*x*log(x) + 25*x
Time = 2.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=x\,{\mathrm {e}}^{-27}\,\left ({\mathrm {e}}^{27}\,{\ln \left (x\right )}^2-10\,{\mathrm {e}}^{27}\,\ln \left (x\right )+25\,{\mathrm {e}}^{27}+{\mathrm {e}}^{3\,x^2+24\,x}-x\,{\mathrm {e}}^{27}\right ) \] Input:
int(log(x)^2 - 8*log(x) - 2*x + exp(24*x + 3*x^2 - 27)*(24*x + 6*x^2 + 1) + 15,x)
Output:
x*exp(-27)*(25*exp(27) + exp(24*x + 3*x^2) + exp(27)*log(x)^2 - x*exp(27) - 10*exp(27)*log(x))
Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=\frac {x \left (e^{3 x^{2}+24 x}+\mathrm {log}\left (x \right )^{2} e^{27}-10 \,\mathrm {log}\left (x \right ) e^{27}-e^{27} x +25 e^{27}\right )}{e^{27}} \] Input:
int(log(x)^2-8*log(x)+(6*x^2+24*x+1)*exp(3*x^2+24*x-27)-2*x+15,x)
Output:
(x*(e**(3*x**2 + 24*x) + log(x)**2*e**27 - 10*log(x)*e**27 - e**27*x + 25* e**27))/e**27