Integrand size = 89, antiderivative size = 33 \[ \int \frac {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{108 x-252 x^2+39 x^3+126 x^4+27 x^5} \, dx=\frac {\frac {1}{9} \left (1+e^x\right )^2-\log (2 x)}{\left (\frac {2}{3}-x\right ) (3+x)} \] Output:
1/(3+x)*(1/3*(1+exp(x))*(1/3*exp(x)+1/3)-ln(2*x))/(2/3-x)
Time = 0.63 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{108 x-252 x^2+39 x^3+126 x^4+27 x^5} \, dx=\frac {-\left (1+e^x\right )^2+9 \log (2 x)}{3 \left (-6+7 x+3 x^2\right )} \] Input:
Integrate[(-54 + 70*x + 33*x^2 + E^(2*x)*(19*x - 8*x^2 - 6*x^3) + E^x*(26* x - 2*x^2 - 6*x^3) + (-63*x - 54*x^2)*Log[2*x])/(108*x - 252*x^2 + 39*x^3 + 126*x^4 + 27*x^5),x]
Output:
(-(1 + E^x)^2 + 9*Log[2*x])/(3*(-6 + 7*x + 3*x^2))
Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(33)=66\).
Time = 3.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.42, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, 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.
\(\displaystyle \int \frac {33 x^2+\left (-54 x^2-63 x\right ) \log (2 x)+e^{2 x} \left (-6 x^3-8 x^2+19 x\right )+e^x \left (-6 x^3-2 x^2+26 x\right )+70 x-54}{27 x^5+126 x^4+39 x^3-252 x^2+108 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {33 x^2+\left (-54 x^2-63 x\right ) \log (2 x)+e^{2 x} \left (-6 x^3-8 x^2+19 x\right )+e^x \left (-6 x^3-2 x^2+26 x\right )+70 x-54}{x \left (27 x^4+126 x^3+39 x^2-252 x+108\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {2 \left (33 x^2+\left (-54 x^2-63 x\right ) \log (2 x)+e^{2 x} \left (-6 x^3-8 x^2+19 x\right )+e^x \left (-6 x^3-2 x^2+26 x\right )+70 x-54\right )}{1331 x (x+3)}-\frac {6 \left (33 x^2+\left (-54 x^2-63 x\right ) \log (2 x)+e^{2 x} \left (-6 x^3-8 x^2+19 x\right )+e^x \left (-6 x^3-2 x^2+26 x\right )+70 x-54\right )}{1331 x (3 x-2)}+\frac {33 x^2+\left (-54 x^2-63 x\right ) \log (2 x)+e^{2 x} \left (-6 x^3-8 x^2+19 x\right )+e^x \left (-6 x^3-2 x^2+26 x\right )+70 x-54}{363 x (x+3)^2}+\frac {3 \left (33 x^2+\left (-54 x^2-63 x\right ) \log (2 x)+e^{2 x} \left (-6 x^3-8 x^2+19 x\right )+e^x \left (-6 x^3-2 x^2+26 x\right )+70 x-54\right )}{121 x (3 x-2)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 e^x}{11 (2-3 x)}+\frac {e^{2 x}}{11 (2-3 x)}+\frac {1}{11 (2-3 x)}+\frac {2 e^x}{33 (x+3)}+\frac {e^{2 x}}{33 (x+3)}+\frac {1}{33 (x+3)}-\frac {\log (x)}{2}-\frac {27 x \log (2 x)}{22 (2-3 x)}+\frac {x \log (2 x)}{11 (x+3)}\) |
Input:
Int[(-54 + 70*x + 33*x^2 + E^(2*x)*(19*x - 8*x^2 - 6*x^3) + E^x*(26*x - 2* x^2 - 6*x^3) + (-63*x - 54*x^2)*Log[2*x])/(108*x - 252*x^2 + 39*x^3 + 126* x^4 + 27*x^5),x]
Output:
1/(11*(2 - 3*x)) + (2*E^x)/(11*(2 - 3*x)) + E^(2*x)/(11*(2 - 3*x)) + 1/(33 *(3 + x)) + (2*E^x)/(33*(3 + x)) + E^(2*x)/(33*(3 + x)) - Log[x]/2 - (27*x *Log[2*x])/(22*(2 - 3*x)) + (x*Log[2*x])/(11*(3 + x))
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 = 0.84 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21
method | result | size |
parallelrisch | \(\frac {-3 x^{2}-6 \,{\mathrm e}^{2 x}-7 x -12 \,{\mathrm e}^{x}+54 \ln \left (2 x \right )}{54 x^{2}+126 x -108}\) | \(40\) |
risch | \(\frac {3 \ln \left (2 x \right )}{3 x^{2}+7 x -6}-\frac {{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}+1}{3 \left (3 x^{2}+7 x -6\right )}\) | \(44\) |
default | \(\frac {1}{99+33 x}+\frac {\ln \left (3+x \right )}{11}-\frac {\ln \left (x \right )}{2}-\frac {1}{11 \left (-2+3 x \right )}+\frac {9 \ln \left (-2+3 x \right )}{22}-\frac {9 \ln \left (6 x -4\right )}{22}+\frac {27 \ln \left (2 x \right ) x}{11 \left (6 x -4\right )}-\frac {\ln \left (2 x +6\right )}{11}+\frac {2 \ln \left (2 x \right ) x}{11 \left (2 x +6\right )}-\frac {2 \,{\mathrm e}^{x}}{33 \left (-\frac {2}{3}+x \right )}+\frac {2 \,{\mathrm e}^{x}}{33 \left (3+x \right )}-\frac {{\mathrm e}^{2 x}}{33 \left (-\frac {2}{3}+x \right )}+\frac {{\mathrm e}^{2 x}}{99+33 x}\) | \(120\) |
parts | \(\frac {1}{99+33 x}+\frac {\ln \left (3+x \right )}{11}-\frac {\ln \left (x \right )}{2}-\frac {1}{11 \left (-2+3 x \right )}+\frac {9 \ln \left (-2+3 x \right )}{22}-\frac {9 \ln \left (6 x -4\right )}{22}+\frac {27 \ln \left (2 x \right ) x}{11 \left (6 x -4\right )}-\frac {\ln \left (2 x +6\right )}{11}+\frac {2 \ln \left (2 x \right ) x}{11 \left (2 x +6\right )}-\frac {2 \,{\mathrm e}^{x}}{33 \left (-\frac {2}{3}+x \right )}+\frac {2 \,{\mathrm e}^{x}}{33 \left (3+x \right )}-\frac {{\mathrm e}^{2 x}}{33 \left (-\frac {2}{3}+x \right )}+\frac {{\mathrm e}^{2 x}}{99+33 x}\) | \(120\) |
orering | \(\text {Expression too large to display}\) | \(1911\) |
Input:
int(((-54*x^2-63*x)*ln(2*x)+(-6*x^3-8*x^2+19*x)*exp(x)^2+(-6*x^3-2*x^2+26* x)*exp(x)+33*x^2+70*x-54)/(27*x^5+126*x^4+39*x^3-252*x^2+108*x),x,method=_ RETURNVERBOSE)
Output:
1/18*(-3*x^2-6*exp(x)^2-7*x-12*exp(x)+54*ln(2*x))/(3*x^2+7*x-6)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{108 x-252 x^2+39 x^3+126 x^4+27 x^5} \, dx=-\frac {e^{\left (2 \, x\right )} + 2 \, e^{x} - 9 \, \log \left (2 \, x\right ) + 1}{3 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} \] Input:
integrate(((-54*x^2-63*x)*log(2*x)+(-6*x^3-8*x^2+19*x)*exp(x)^2+(-6*x^3-2* x^2+26*x)*exp(x)+33*x^2+70*x-54)/(27*x^5+126*x^4+39*x^3-252*x^2+108*x),x, algorithm="fricas")
Output:
-1/3*(e^(2*x) + 2*e^x - 9*log(2*x) + 1)/(3*x^2 + 7*x - 6)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (22) = 44\).
Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.30 \[ \int \frac {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{108 x-252 x^2+39 x^3+126 x^4+27 x^5} \, dx=\frac {\left (- 18 x^{2} - 42 x + 36\right ) e^{x} + \left (- 9 x^{2} - 21 x + 18\right ) e^{2 x}}{81 x^{4} + 378 x^{3} + 117 x^{2} - 756 x + 324} - \frac {1}{9 x^{2} + 21 x - 18} + \frac {3 \log {\left (2 x \right )}}{3 x^{2} + 7 x - 6} \] Input:
integrate(((-54*x**2-63*x)*ln(2*x)+(-6*x**3-8*x**2+19*x)*exp(x)**2+(-6*x** 3-2*x**2+26*x)*exp(x)+33*x**2+70*x-54)/(27*x**5+126*x**4+39*x**3-252*x**2+ 108*x),x)
Output:
((-18*x**2 - 42*x + 36)*exp(x) + (-9*x**2 - 21*x + 18)*exp(2*x))/(81*x**4 + 378*x**3 + 117*x**2 - 756*x + 324) - 1/(9*x**2 + 21*x - 18) + 3*log(2*x) /(3*x**2 + 7*x - 6)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (27) = 54\).
Time = 0.15 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.15 \[ \int \frac {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{108 x-252 x^2+39 x^3+126 x^4+27 x^5} \, dx=\frac {3 \, {\left (3 \, x^{2} + 7 \, x\right )} \log \left (x\right ) - 2 \, e^{\left (2 \, x\right )} - 4 \, e^{x} + 18 \, \log \left (2\right )}{6 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} + \frac {3 \, {\left (21 \, x + 85\right )}}{121 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} + \frac {7 \, x - 12}{11 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} - \frac {70 \, {\left (6 \, x + 7\right )}}{363 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} - \frac {1}{2} \, \log \left (x\right ) \] Input:
integrate(((-54*x^2-63*x)*log(2*x)+(-6*x^3-8*x^2+19*x)*exp(x)^2+(-6*x^3-2* x^2+26*x)*exp(x)+33*x^2+70*x-54)/(27*x^5+126*x^4+39*x^3-252*x^2+108*x),x, algorithm="maxima")
Output:
1/6*(3*(3*x^2 + 7*x)*log(x) - 2*e^(2*x) - 4*e^x + 18*log(2))/(3*x^2 + 7*x - 6) + 3/121*(21*x + 85)/(3*x^2 + 7*x - 6) + 1/11*(7*x - 12)/(3*x^2 + 7*x - 6) - 70/363*(6*x + 7)/(3*x^2 + 7*x - 6) - 1/2*log(x)
Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{108 x-252 x^2+39 x^3+126 x^4+27 x^5} \, dx=-\frac {e^{\left (2 \, x\right )} + 2 \, e^{x} - 9 \, \log \left (2 \, x\right ) + 1}{3 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} \] Input:
integrate(((-54*x^2-63*x)*log(2*x)+(-6*x^3-8*x^2+19*x)*exp(x)^2+(-6*x^3-2* x^2+26*x)*exp(x)+33*x^2+70*x-54)/(27*x^5+126*x^4+39*x^3-252*x^2+108*x),x, algorithm="giac")
Output:
-1/3*(e^(2*x) + 2*e^x - 9*log(2*x) + 1)/(3*x^2 + 7*x - 6)
Time = 3.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.94 \[ \int \frac {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{108 x-252 x^2+39 x^3+126 x^4+27 x^5} \, dx=\frac {\ln \left (2\,x\right )}{x^2+\frac {7\,x}{3}-2}-\frac {{\mathrm {e}}^{2\,x}}{9\,\left (x^2+\frac {7\,x}{3}-2\right )}-\frac {1}{9\,\left (x^2+\frac {7\,x}{3}-2\right )}-\frac {2\,{\mathrm {e}}^x}{9\,\left (x^2+\frac {7\,x}{3}-2\right )} \] Input:
int(-(log(2*x)*(63*x + 54*x^2) - 70*x + exp(2*x)*(8*x^2 - 19*x + 6*x^3) - 33*x^2 + exp(x)*(2*x^2 - 26*x + 6*x^3) + 54)/(108*x - 252*x^2 + 39*x^3 + 1 26*x^4 + 27*x^5),x)
Output:
log(2*x)/((7*x)/3 + x^2 - 2) - exp(2*x)/(9*((7*x)/3 + x^2 - 2)) - 1/(9*((7 *x)/3 + x^2 - 2)) - (2*exp(x))/(9*((7*x)/3 + x^2 - 2))
Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.79 \[ \int \frac {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{108 x-252 x^2+39 x^3+126 x^4+27 x^5} \, dx=\frac {-2 e^{2 x}-4 e^{x}+9 \,\mathrm {log}\left (2 x \right ) x^{2}+21 \,\mathrm {log}\left (2 x \right ) x -9 \,\mathrm {log}\left (x \right ) x^{2}-21 \,\mathrm {log}\left (x \right ) x +18 \,\mathrm {log}\left (x \right )-2}{18 x^{2}+42 x -36} \] Input:
int(((-54*x^2-63*x)*log(2*x)+(-6*x^3-8*x^2+19*x)*exp(x)^2+(-6*x^3-2*x^2+26 *x)*exp(x)+33*x^2+70*x-54)/(27*x^5+126*x^4+39*x^3-252*x^2+108*x),x)
Output:
( - 2*e**(2*x) - 4*e**x + 9*log(2*x)*x**2 + 21*log(2*x)*x - 9*log(x)*x**2 - 21*log(x)*x + 18*log(x) - 2)/(6*(3*x**2 + 7*x - 6))