Integrand size = 106, antiderivative size = 23 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3 \left (e^{14 x/5}+\log (8+x)\right )}{(3+x) \log (x)} \]
Time = 0.47 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3 \left (e^{14 x/5}+\log (8+x)\right )}{(3+x) \log (x)} \]
Integrate[(E^((14*x)/5)*(-360 - 165*x - 15*x^2) + (45*x + 15*x^2 + E^((14* x)/5)*(888*x + 447*x^2 + 42*x^3))*Log[x] + (-360 - 165*x - 15*x^2 + (-120* x - 15*x^2)*Log[x])*Log[8 + x])/((360*x + 285*x^2 + 70*x^3 + 5*x^4)*Log[x] ^2),x]
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.
\(\displaystyle \int \frac {e^{14 x/5} \left (-15 x^2-165 x-360\right )+\left (-15 x^2+\left (-15 x^2-120 x\right ) \log (x)-165 x-360\right ) \log (x+8)+\left (15 x^2+e^{14 x/5} \left (42 x^3+447 x^2+888 x\right )+45 x\right ) \log (x)}{\left (5 x^4+70 x^3+285 x^2+360 x\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{14 x/5} \left (-15 x^2-165 x-360\right )+\left (-15 x^2+\left (-15 x^2-120 x\right ) \log (x)-165 x-360\right ) \log (x+8)+\left (15 x^2+e^{14 x/5} \left (42 x^3+447 x^2+888 x\right )+45 x\right ) \log (x)}{x \left (5 x^3+70 x^2+285 x+360\right ) \log ^2(x)}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {e^{14 x/5} \left (-15 x^2-165 x-360\right )+\left (-15 x^2+\left (-15 x^2-120 x\right ) \log (x)-165 x-360\right ) \log (x+8)+\left (15 x^2+e^{14 x/5} \left (42 x^3+447 x^2+888 x\right )+45 x\right ) \log (x)}{125 x (x+3) \log ^2(x)}+\frac {e^{14 x/5} \left (-15 x^2-165 x-360\right )+\left (-15 x^2+\left (-15 x^2-120 x\right ) \log (x)-165 x-360\right ) \log (x+8)+\left (15 x^2+e^{14 x/5} \left (42 x^3+447 x^2+888 x\right )+45 x\right ) \log (x)}{125 x (x+8) \log ^2(x)}+\frac {e^{14 x/5} \left (-15 x^2-165 x-360\right )+\left (-15 x^2+\left (-15 x^2-120 x\right ) \log (x)-165 x-360\right ) \log (x+8)+\left (15 x^2+e^{14 x/5} \left (42 x^3+447 x^2+888 x\right )+45 x\right ) \log (x)}{25 x (x+3)^2 \log ^2(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {e^{14 x/5}}{x \log ^2(x)}dx+\int \frac {e^{14 x/5}}{(x+3) \log ^2(x)}dx-\int \frac {\log (x+8)}{x \log ^2(x)}dx+\int \frac {\log (x+8)}{(x+3) \log ^2(x)}dx-3 \int \frac {e^{14 x/5}}{(x+3)^2 \log (x)}dx+\frac {3}{5} \int \frac {1}{(x+3) \log (x)}dx+\frac {42}{5} \int \frac {e^{14 x/5}}{(x+3) \log (x)}dx-\frac {3}{25} \int \frac {-x-3}{(x+8) \log (x)}dx-3 \int \frac {\log (x+8)}{(x+3)^2 \log (x)}dx-\frac {3 \operatorname {LogIntegral}(x)}{25}\) |
Int[(E^((14*x)/5)*(-360 - 165*x - 15*x^2) + (45*x + 15*x^2 + E^((14*x)/5)* (888*x + 447*x^2 + 42*x^3))*Log[x] + (-360 - 165*x - 15*x^2 + (-120*x - 15 *x^2)*Log[x])*Log[8 + x])/((360*x + 285*x^2 + 70*x^3 + 5*x^4)*Log[x]^2),x]
3.1.89.3.1 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 = 57.57 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35
method | result | size |
parallelrisch | \(-\frac {-285 \,{\mathrm e}^{2 x} {\mathrm e}^{\frac {4 x}{5}}-285 \ln \left (x +8\right )}{95 \left (3+x \right ) \ln \left (x \right )}\) | \(31\) |
risch | \(\frac {3 \ln \left (x +8\right )}{\left (3+x \right ) \ln \left (x \right )}+\frac {3 \,{\mathrm e}^{\frac {14 x}{5}}}{\left (3+x \right ) \ln \left (x \right )}\) | \(32\) |
int((((-15*x^2-120*x)*ln(x)-15*x^2-165*x-360)*ln(x+8)+((42*x^3+447*x^2+888 *x)*exp(2/5*x)^2*exp(x)^2+15*x^2+45*x)*ln(x)+(-15*x^2-165*x-360)*exp(2/5*x )^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/ln(x)^2,x,method=_RETURNVERBOSE )
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{{\left (x + 3\right )} \log \left (x\right )} \]
integrate((((-15*x^2-120*x)*log(x)-15*x^2-165*x-360)*log(x+8)+((42*x^3+447 *x^2+888*x)*exp(2/5*x)^2*exp(x)^2+15*x^2+45*x)*log(x)+(-15*x^2-165*x-360)* exp(2/5*x)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/log(x)^2,x, algorithm= \
Exception generated. \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \]
integrate((((-15*x**2-120*x)*ln(x)-15*x**2-165*x-360)*ln(x+8)+((42*x**3+44 7*x**2+888*x)*exp(2/5*x)**2*exp(x)**2+15*x**2+45*x)*ln(x)+(-15*x**2-165*x- 360)*exp(2/5*x)**2*exp(x)**2)/(5*x**4+70*x**3+285*x**2+360*x)/ln(x)**2,x)
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{{\left (x + 3\right )} \log \left (x\right )} \]
integrate((((-15*x^2-120*x)*log(x)-15*x^2-165*x-360)*log(x+8)+((42*x^3+447 *x^2+888*x)*exp(2/5*x)^2*exp(x)^2+15*x^2+45*x)*log(x)+(-15*x^2-165*x-360)* exp(2/5*x)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/log(x)^2,x, algorithm= \
Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{x \log \left (x\right ) + 3 \, \log \left (x\right )} \]
integrate((((-15*x^2-120*x)*log(x)-15*x^2-165*x-360)*log(x+8)+((42*x^3+447 *x^2+888*x)*exp(2/5*x)^2*exp(x)^2+15*x^2+45*x)*log(x)+(-15*x^2-165*x-360)* exp(2/5*x)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/log(x)^2,x, algorithm= \
Time = 8.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3\,\left ({\mathrm {e}}^{\frac {14\,x}{5}}+\ln \left (x+8\right )\right )}{\ln \left (x\right )\,\left (x+3\right )} \]