Integrand size = 177, antiderivative size = 32 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=\frac {x^2}{5-x^3-\log \left (\frac {4 e^{3+\frac {2}{4+x}}}{x}\right )} \]
Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=-\frac {x^2}{-5+x^3+\log \left (\frac {4 e^{3+\frac {2}{4+x}}}{x}\right )} \]
Integrate[(144*x + 70*x^2 + 9*x^3 + 16*x^4 + 8*x^5 + x^6 + (-32*x - 16*x^2 - 2*x^3)*Log[(4*E^((14 + 3*x)/(4 + x)))/x])/(400 + 200*x + 25*x^2 - 160*x ^3 - 80*x^4 - 10*x^5 + 16*x^6 + 8*x^7 + x^8 + (-160 - 80*x - 10*x^2 + 32*x ^3 + 16*x^4 + 2*x^5)*Log[(4*E^((14 + 3*x)/(4 + x)))/x] + (16 + 8*x + x^2)* Log[(4*E^((14 + 3*x)/(4 + x)))/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.
\(\displaystyle \int \frac {x^6+8 x^5+16 x^4+9 x^3+70 x^2+\left (-2 x^3-16 x^2-32 x\right ) \log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )+144 x}{x^8+8 x^7+16 x^6-10 x^5-80 x^4-160 x^3+25 x^2+\left (x^2+8 x+16\right ) \log ^2\left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )+\left (2 x^5+16 x^4+32 x^3-10 x^2-80 x-160\right ) \log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )+200 x+400} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {x \left (x^5+8 x^4+16 x^3+9 x^2+70 x-2 (x+4)^2 \log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )+144\right )}{(x+4)^2 \left (-x^3-\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )+5\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x \left (3 x^5+24 x^4+48 x^3-x^2-10 x-16\right )}{(x+4)^2 \left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}-\frac {2 x}{x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \int \frac {1}{\left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}dx-\int \frac {x}{\left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}dx-32 \int \frac {1}{(x+4)^2 \left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}dx+16 \int \frac {1}{(x+4) \left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}dx-2 \int \frac {x}{x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5}dx+3 \int \frac {x^4}{\left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}dx\) |
Int[(144*x + 70*x^2 + 9*x^3 + 16*x^4 + 8*x^5 + x^6 + (-32*x - 16*x^2 - 2*x ^3)*Log[(4*E^((14 + 3*x)/(4 + x)))/x])/(400 + 200*x + 25*x^2 - 160*x^3 - 8 0*x^4 - 10*x^5 + 16*x^6 + 8*x^7 + x^8 + (-160 - 80*x - 10*x^2 + 32*x^3 + 1 6*x^4 + 2*x^5)*Log[(4*E^((14 + 3*x)/(4 + x)))/x] + (16 + 8*x + x^2)*Log[(4 *E^((14 + 3*x)/(4 + x)))/x]^2),x]
3.21.32.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 2.55 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34
method | result | size |
parallelrisch | \(\frac {-x^{3}-4 x^{2}}{\left (x^{3}+\ln \left (\frac {4 \,{\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )-5\right ) \left (4+x \right )}\) | \(43\) |
default | \(\frac {x^{3}+4 x^{2}}{-x^{4}-4 x^{3}+x \ln \left (x \right )-\left (\ln \left (\frac {4 \,{\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )-\frac {3 x +14}{4+x}+\ln \left (x \right )\right ) x +2 x -4 \ln \left (\frac {4 \,{\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )+\frac {12 x +56}{4+x}+6}\) | \(100\) |
risch | \(-\frac {2 x^{2}}{-10-i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\frac {3 x +14}{4+x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )^{2}+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{\frac {3 x +14}{4+x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )^{3}+2 x^{3}+4 \ln \left (2\right )-2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{\frac {3 x +14}{4+x}}\right )}\) | \(182\) |
int(((-2*x^3-16*x^2-32*x)*ln(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16*x^4+9*x ^3+70*x^2+144*x)/((x^2+8*x+16)*ln(4*exp((3*x+14)/(4+x))/x)^2+(2*x^5+16*x^4 +32*x^3-10*x^2-80*x-160)*ln(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+16*x^6-10*x ^5-80*x^4-160*x^3+25*x^2+200*x+400),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=-\frac {x^{2}}{x^{3} + \log \left (\frac {4 \, e^{\left (\frac {3 \, x + 14}{x + 4}\right )}}{x}\right ) - 5} \]
integrate(((-2*x^3-16*x^2-32*x)*log(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16* x^4+9*x^3+70*x^2+144*x)/((x^2+8*x+16)*log(4*exp((3*x+14)/(4+x))/x)^2+(2*x^ 5+16*x^4+32*x^3-10*x^2-80*x-160)*log(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+16 *x^6-10*x^5-80*x^4-160*x^3+25*x^2+200*x+400),x, algorithm=\
Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=- \frac {x^{2}}{x^{3} + \log {\left (\frac {4 e^{\frac {3 x + 14}{x + 4}}}{x} \right )} - 5} \]
integrate(((-2*x**3-16*x**2-32*x)*ln(4*exp((3*x+14)/(4+x))/x)+x**6+8*x**5+ 16*x**4+9*x**3+70*x**2+144*x)/((x**2+8*x+16)*ln(4*exp((3*x+14)/(4+x))/x)** 2+(2*x**5+16*x**4+32*x**3-10*x**2-80*x-160)*ln(4*exp((3*x+14)/(4+x))/x)+x* *8+8*x**7+16*x**6-10*x**5-80*x**4-160*x**3+25*x**2+200*x+400),x)
Time = 0.33 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=-\frac {x^{3} + 4 \, x^{2}}{x^{4} + 4 \, x^{3} + 2 \, x {\left (\log \left (2\right ) - 1\right )} - {\left (x + 4\right )} \log \left (x\right ) + 8 \, \log \left (2\right ) - 6} \]
integrate(((-2*x^3-16*x^2-32*x)*log(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16* x^4+9*x^3+70*x^2+144*x)/((x^2+8*x+16)*log(4*exp((3*x+14)/(4+x))/x)^2+(2*x^ 5+16*x^4+32*x^3-10*x^2-80*x-160)*log(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+16 *x^6-10*x^5-80*x^4-160*x^3+25*x^2+200*x+400),x, algorithm=\
Time = 0.41 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=-\frac {x^{3} + 4 \, x^{2}}{x^{4} + 4 \, x^{3} + 2 \, x \log \left (2\right ) - x \log \left (x\right ) - 2 \, x + 8 \, \log \left (2\right ) - 4 \, \log \left (x\right ) - 6} \]
integrate(((-2*x^3-16*x^2-32*x)*log(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16* x^4+9*x^3+70*x^2+144*x)/((x^2+8*x+16)*log(4*exp((3*x+14)/(4+x))/x)^2+(2*x^ 5+16*x^4+32*x^3-10*x^2-80*x-160)*log(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+16 *x^6-10*x^5-80*x^4-160*x^3+25*x^2+200*x+400),x, algorithm=\
Time = 11.67 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=-\frac {x^2}{\ln \left (\frac {4}{x}\right )+\frac {3\,x}{x+4}+\frac {14}{x+4}+x^3-5} \]
int((144*x + 70*x^2 + 9*x^3 + 16*x^4 + 8*x^5 + x^6 - log((4*exp((3*x + 14) /(x + 4)))/x)*(32*x + 16*x^2 + 2*x^3))/(200*x - log((4*exp((3*x + 14)/(x + 4)))/x)*(80*x + 10*x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 160) + log((4*exp((3*x + 14)/(x + 4)))/x)^2*(8*x + x^2 + 16) + 25*x^2 - 160*x^3 - 80*x^4 - 10*x^ 5 + 16*x^6 + 8*x^7 + x^8 + 400),x)