Integrand size = 95, antiderivative size = 27 \[ \int \frac {-2600-1747 x+146 x^2-3 x^3+\left (-204-145 x+6 x^2\right ) \log (4+3 x)+(-4-3 x) \log ^2(4+3 x)}{7500+5025 x-438 x^2+9 x^3+\left (600+426 x-18 x^2\right ) \log (4+3 x)+(12+9 x) \log ^2(4+3 x)} \, dx=\log (2)+\frac {1}{3} \left (1-x+\frac {x}{-25+x-\log (4+3 x)}\right ) \]
Time = 0.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-2600-1747 x+146 x^2-3 x^3+\left (-204-145 x+6 x^2\right ) \log (4+3 x)+(-4-3 x) \log ^2(4+3 x)}{7500+5025 x-438 x^2+9 x^3+\left (600+426 x-18 x^2\right ) \log (4+3 x)+(12+9 x) \log ^2(4+3 x)} \, dx=-\frac {1}{3} x \left (1+\frac {1}{25-x+\log (4+3 x)}\right ) \]
Integrate[(-2600 - 1747*x + 146*x^2 - 3*x^3 + (-204 - 145*x + 6*x^2)*Log[4 + 3*x] + (-4 - 3*x)*Log[4 + 3*x]^2)/(7500 + 5025*x - 438*x^2 + 9*x^3 + (6 00 + 426*x - 18*x^2)*Log[4 + 3*x] + (12 + 9*x)*Log[4 + 3*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.
\(\displaystyle \int \frac {-3 x^3+146 x^2+\left (6 x^2-145 x-204\right ) \log (3 x+4)-1747 x+(-3 x-4) \log ^2(3 x+4)-2600}{9 x^3-438 x^2+\left (-18 x^2+426 x+600\right ) \log (3 x+4)+5025 x+(9 x+12) \log ^2(3 x+4)+7500} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-3 x^3+146 x^2+\left (6 x^2-145 x-204\right ) \log (3 x+4)-1747 x+(-3 x-4) \log ^2(3 x+4)-2600}{3 (3 x+4) (-x+\log (3 x+4)+25)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int -\frac {3 x^3-146 x^2+1747 x+(3 x+4) \log ^2(3 x+4)+\left (-6 x^2+145 x+204\right ) \log (3 x+4)+2600}{(3 x+4) (-x+\log (3 x+4)+25)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {3 x^3-146 x^2+1747 x+(3 x+4) \log ^2(3 x+4)+\left (-6 x^2+145 x+204\right ) \log (3 x+4)+2600}{(3 x+4) (-x+\log (3 x+4)+25)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{3} \int \left (\frac {x (3 x+1)}{(3 x+4) (x-\log (3 x+4)-25)^2}+\frac {1}{-x+\log (3 x+4)+25}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\int \frac {1}{(x-\log (3 x+4)-25)^2}dx-\int \frac {x}{(x-\log (3 x+4)-25)^2}dx-4 \int \frac {1}{(3 x+4) (x-\log (3 x+4)-25)^2}dx-\int \frac {1}{-x+\log (3 x+4)+25}dx-x\right )\) |
Int[(-2600 - 1747*x + 146*x^2 - 3*x^3 + (-204 - 145*x + 6*x^2)*Log[4 + 3*x ] + (-4 - 3*x)*Log[4 + 3*x]^2)/(7500 + 5025*x - 438*x^2 + 9*x^3 + (600 + 4 26*x - 18*x^2)*Log[4 + 3*x] + (12 + 9*x)*Log[4 + 3*x]^2),x]
3.20.42.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.49 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {x}{3}+\frac {x}{3 x -3 \ln \left (4+3 x \right )-75}\) | \(21\) |
norman | \(\frac {\frac {650}{3}+\frac {26 \ln \left (4+3 x \right )}{3}+\frac {\ln \left (4+3 x \right ) x}{3}-\frac {x^{2}}{3}}{x -\ln \left (4+3 x \right )-25}\) | \(39\) |
parallelrisch | \(\frac {-600-9 x^{2}+9 \ln \left (4+3 x \right ) x +258 x -24 \ln \left (4+3 x \right )}{27 x -27 \ln \left (4+3 x \right )-675}\) | \(43\) |
derivativedivides | \(-\frac {316+246 x -\left (4+3 x \right )^{2}+3 \ln \left (4+3 x \right ) \left (4+3 x \right )}{9 \left (3 \ln \left (4+3 x \right )+75-3 x \right )}\) | \(45\) |
default | \(-\frac {316+246 x -\left (4+3 x \right )^{2}+3 \ln \left (4+3 x \right ) \left (4+3 x \right )}{9 \left (3 \ln \left (4+3 x \right )+75-3 x \right )}\) | \(45\) |
int(((-3*x-4)*ln(4+3*x)^2+(6*x^2-145*x-204)*ln(4+3*x)-3*x^3+146*x^2-1747*x -2600)/((9*x+12)*ln(4+3*x)^2+(-18*x^2+426*x+600)*ln(4+3*x)+9*x^3-438*x^2+5 025*x+7500),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-2600-1747 x+146 x^2-3 x^3+\left (-204-145 x+6 x^2\right ) \log (4+3 x)+(-4-3 x) \log ^2(4+3 x)}{7500+5025 x-438 x^2+9 x^3+\left (600+426 x-18 x^2\right ) \log (4+3 x)+(12+9 x) \log ^2(4+3 x)} \, dx=-\frac {x^{2} - x \log \left (3 \, x + 4\right ) - 26 \, x}{3 \, {\left (x - \log \left (3 \, x + 4\right ) - 25\right )}} \]
integrate(((-3*x-4)*log(4+3*x)^2+(6*x^2-145*x-204)*log(4+3*x)-3*x^3+146*x^ 2-1747*x-2600)/((9*x+12)*log(4+3*x)^2+(-18*x^2+426*x+600)*log(4+3*x)+9*x^3 -438*x^2+5025*x+7500),x, algorithm=\
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {-2600-1747 x+146 x^2-3 x^3+\left (-204-145 x+6 x^2\right ) \log (4+3 x)+(-4-3 x) \log ^2(4+3 x)}{7500+5025 x-438 x^2+9 x^3+\left (600+426 x-18 x^2\right ) \log (4+3 x)+(12+9 x) \log ^2(4+3 x)} \, dx=- \frac {x}{3} - \frac {x}{- 3 x + 3 \log {\left (3 x + 4 \right )} + 75} \]
integrate(((-3*x-4)*ln(4+3*x)**2+(6*x**2-145*x-204)*ln(4+3*x)-3*x**3+146*x **2-1747*x-2600)/((9*x+12)*ln(4+3*x)**2+(-18*x**2+426*x+600)*ln(4+3*x)+9*x **3-438*x**2+5025*x+7500),x)
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-2600-1747 x+146 x^2-3 x^3+\left (-204-145 x+6 x^2\right ) \log (4+3 x)+(-4-3 x) \log ^2(4+3 x)}{7500+5025 x-438 x^2+9 x^3+\left (600+426 x-18 x^2\right ) \log (4+3 x)+(12+9 x) \log ^2(4+3 x)} \, dx=-\frac {x^{2} - x \log \left (3 \, x + 4\right ) - 26 \, x}{3 \, {\left (x - \log \left (3 \, x + 4\right ) - 25\right )}} \]
integrate(((-3*x-4)*log(4+3*x)^2+(6*x^2-145*x-204)*log(4+3*x)-3*x^3+146*x^ 2-1747*x-2600)/((9*x+12)*log(4+3*x)^2+(-18*x^2+426*x+600)*log(4+3*x)+9*x^3 -438*x^2+5025*x+7500),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-2600-1747 x+146 x^2-3 x^3+\left (-204-145 x+6 x^2\right ) \log (4+3 x)+(-4-3 x) \log ^2(4+3 x)}{7500+5025 x-438 x^2+9 x^3+\left (600+426 x-18 x^2\right ) \log (4+3 x)+(12+9 x) \log ^2(4+3 x)} \, dx=-\frac {1}{3} \, x + \frac {x}{3 \, {\left (x - \log \left (3 \, x + 4\right ) - 25\right )}} \]
integrate(((-3*x-4)*log(4+3*x)^2+(6*x^2-145*x-204)*log(4+3*x)-3*x^3+146*x^ 2-1747*x-2600)/((9*x+12)*log(4+3*x)^2+(-18*x^2+426*x+600)*log(4+3*x)+9*x^3 -438*x^2+5025*x+7500),x, algorithm=\
Time = 0.32 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {-2600-1747 x+146 x^2-3 x^3+\left (-204-145 x+6 x^2\right ) \log (4+3 x)+(-4-3 x) \log ^2(4+3 x)}{7500+5025 x-438 x^2+9 x^3+\left (600+426 x-18 x^2\right ) \log (4+3 x)+(12+9 x) \log ^2(4+3 x)} \, dx=-\frac {25\,x+\ln \left (3\,x+4\right )+x\,\ln \left (3\,x+4\right )-x^2+25}{3\,\left (\ln \left (3\,x+4\right )-x+25\right )} \]