\(\int \frac {45 x-40 x^2-9 x^3+(25-45 x) \log (\frac {1}{5} (5-9 x))}{-5 x^4+9 x^5+(50 x^2-140 x^3+90 x^4) \log (\frac {1}{5} (5-9 x))+(-125+475 x-575 x^2+225 x^3) \log ^2(\frac {1}{5} (5-9 x))} \, dx\) [6158]

Optimal result

Integrand size = 99, antiderivative size = 29 \[ \int \frac {45 x-40 x^2-9 x^3+(25-45 x) \log \left (\frac {1}{5} (5-9 x)\right )}{-5 x^4+9 x^5+\left (50 x^2-140 x^3+90 x^4\right ) \log \left (\frac {1}{5} (5-9 x)\right )+\left (-125+475 x-575 x^2+225 x^3\right ) \log ^2\left (\frac {1}{5} (5-9 x)\right )} \, dx=4+\frac {x}{x^2-5 (1-x) \log \left (-2 x+\frac {5+x}{5}\right )} \]

[Out]

x/(-5*(1-x)*ln(-9/5*x+1)+x^2)+4

Rubi [F]

\[ \int \frac {45 x-40 x^2-9 x^3+(25-45 x) \log \left (\frac {1}{5} (5-9 x)\right )}{-5 x^4+9 x^5+\left (50 x^2-140 x^3+90 x^4\right ) \log \left (\frac {1}{5} (5-9 x)\right )+\left (-125+475 x-575 x^2+225 x^3\right ) \log ^2\left (\frac {1}{5} (5-9 x)\right )} \, dx=\int \frac {45 x-40 x^2-9 x^3+(25-45 x) \log \left (\frac {1}{5} (5-9 x)\right )}{-5 x^4+9 x^5+\left (50 x^2-140 x^3+90 x^4\right ) \log \left (\frac {1}{5} (5-9 x)\right )+\left (-125+475 x-575 x^2+225 x^3\right ) \log ^2\left (\frac {1}{5} (5-9 x)\right )} \, dx \]

[In]

Int[(45*x - 40*x^2 - 9*x^3 + (25 - 45*x)*Log[(5 - 9*x)/5])/(-5*x^4 + 9*x^5 + (50*x^2 - 140*x^3 + 90*x^4)*Log[(
5 - 9*x)/5] + (-125 + 475*x - 575*x^2 + 225*x^3)*Log[(5 - 9*x)/5]^2),x]

[Out]

(29*Defer[Int][(x^2 + 5*(-1 + x)*Log[1 - (9*x)/5])^(-2), x])/9 + Defer[Int][1/((-1 + x)*(x^2 + 5*(-1 + x)*Log[
1 - (9*x)/5])^2), x] - 4*Defer[Int][x/(x^2 + 5*(-1 + x)*Log[1 - (9*x)/5])^2, x] - Defer[Int][x^2/(x^2 + 5*(-1
+ x)*Log[1 - (9*x)/5])^2, x] + (100*Defer[Int][1/((-5 + 9*x)*(x^2 + 5*(-1 + x)*Log[1 - (9*x)/5])^2), x])/9 - D
efer[Int][1/((-1 + x)*(x^2 + 5*(-1 + x)*Log[1 - (9*x)/5])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-x \left (45-40 x-9 x^2\right )+5 (-5+9 x) \log \left (1-\frac {9 x}{5}\right )}{(5-9 x) \left (x^2+5 (-1+x) \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx \\ & = \int \left (-\frac {x \left (45-80 x+22 x^2+9 x^3\right )}{(-1+x) (-5+9 x) \left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2}-\frac {1}{(-1+x) \left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )}\right ) \, dx \\ & = -\int \frac {x \left (45-80 x+22 x^2+9 x^3\right )}{(-1+x) (-5+9 x) \left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx-\int \frac {1}{(-1+x) \left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )} \, dx \\ & = -\int \frac {1}{(-1+x) \left (x^2+5 (-1+x) \log \left (1-\frac {9 x}{5}\right )\right )} \, dx-\int \left (-\frac {29}{9 \left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2}-\frac {1}{(-1+x) \left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2}+\frac {4 x}{\left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2}+\frac {x^2}{\left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2}-\frac {100}{9 (-5+9 x) \left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2}\right ) \, dx \\ & = \frac {29}{9} \int \frac {1}{\left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx-4 \int \frac {x}{\left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx+\frac {100}{9} \int \frac {1}{(-5+9 x) \left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx-\int \frac {1}{(-1+x) \left (x^2+5 (-1+x) \log \left (1-\frac {9 x}{5}\right )\right )} \, dx+\int \frac {1}{(-1+x) \left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx-\int \frac {x^2}{\left (x^2-5 \log \left (1-\frac {9 x}{5}\right )+5 x \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx \\ & = \frac {29}{9} \int \frac {1}{\left (x^2+5 (-1+x) \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx-4 \int \frac {x}{\left (x^2+5 (-1+x) \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx+\frac {100}{9} \int \frac {1}{(-5+9 x) \left (x^2+5 (-1+x) \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx+\int \frac {1}{(-1+x) \left (x^2+5 (-1+x) \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx-\int \frac {x^2}{\left (x^2+5 (-1+x) \log \left (1-\frac {9 x}{5}\right )\right )^2} \, dx-\int \frac {1}{(-1+x) \left (x^2+5 (-1+x) \log \left (1-\frac {9 x}{5}\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {45 x-40 x^2-9 x^3+(25-45 x) \log \left (\frac {1}{5} (5-9 x)\right )}{-5 x^4+9 x^5+\left (50 x^2-140 x^3+90 x^4\right ) \log \left (\frac {1}{5} (5-9 x)\right )+\left (-125+475 x-575 x^2+225 x^3\right ) \log ^2\left (\frac {1}{5} (5-9 x)\right )} \, dx=\frac {x}{x^2+5 (-1+x) \log \left (1-\frac {9 x}{5}\right )} \]

[In]

Integrate[(45*x - 40*x^2 - 9*x^3 + (25 - 45*x)*Log[(5 - 9*x)/5])/(-5*x^4 + 9*x^5 + (50*x^2 - 140*x^3 + 90*x^4)
*Log[(5 - 9*x)/5] + (-125 + 475*x - 575*x^2 + 225*x^3)*Log[(5 - 9*x)/5]^2),x]

[Out]

x/(x^2 + 5*(-1 + x)*Log[1 - (9*x)/5])

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90

method	result	size
norman	\(\frac {x}{5 \ln \left (-\frac {9 x}{5}+1\right ) x +x^{2}-5 \ln \left (-\frac {9 x}{5}+1\right )}\)	\(26\)
risch	\(\frac {x}{5 \ln \left (-\frac {9 x}{5}+1\right ) x +x^{2}-5 \ln \left (-\frac {9 x}{5}+1\right )}\)	\(26\)
parallelrisch	\(\frac {x}{5 \ln \left (-\frac {9 x}{5}+1\right ) x +x^{2}-5 \ln \left (-\frac {9 x}{5}+1\right )}\)	\(26\)
derivativedivides	\(-\frac {81 x}{5 \left (45 \ln \left (-\frac {9 x}{5}+1\right ) \left (-\frac {9 x}{5}+1\right )-5 \left (-\frac {9 x}{5}+1\right )^{2}+36 \ln \left (-\frac {9 x}{5}+1\right )-18 x +5\right )}\)	\(41\)
default	\(-\frac {81 x}{5 \left (45 \ln \left (-\frac {9 x}{5}+1\right ) \left (-\frac {9 x}{5}+1\right )-5 \left (-\frac {9 x}{5}+1\right )^{2}+36 \ln \left (-\frac {9 x}{5}+1\right )-18 x +5\right )}\)	\(41\)

[In]

int(((-45*x+25)*ln(-9/5*x+1)-9*x^3-40*x^2+45*x)/((225*x^3-575*x^2+475*x-125)*ln(-9/5*x+1)^2+(90*x^4-140*x^3+50
*x^2)*ln(-9/5*x+1)+9*x^5-5*x^4),x,method=_RETURNVERBOSE)

[Out]

x/(5*ln(-9/5*x+1)*x+x^2-5*ln(-9/5*x+1))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {45 x-40 x^2-9 x^3+(25-45 x) \log \left (\frac {1}{5} (5-9 x)\right )}{-5 x^4+9 x^5+\left (50 x^2-140 x^3+90 x^4\right ) \log \left (\frac {1}{5} (5-9 x)\right )+\left (-125+475 x-575 x^2+225 x^3\right ) \log ^2\left (\frac {1}{5} (5-9 x)\right )} \, dx=\frac {x}{x^{2} + 5 \, {\left (x - 1\right )} \log \left (-\frac {9}{5} \, x + 1\right )} \]

[In]

integrate(((-45*x+25)*log(-9/5*x+1)-9*x^3-40*x^2+45*x)/((225*x^3-575*x^2+475*x-125)*log(-9/5*x+1)^2+(90*x^4-14
0*x^3+50*x^2)*log(-9/5*x+1)+9*x^5-5*x^4),x, algorithm="fricas")

[Out]

x/(x^2 + 5*(x - 1)*log(-9/5*x + 1))

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59 \[ \int \frac {45 x-40 x^2-9 x^3+(25-45 x) \log \left (\frac {1}{5} (5-9 x)\right )}{-5 x^4+9 x^5+\left (50 x^2-140 x^3+90 x^4\right ) \log \left (\frac {1}{5} (5-9 x)\right )+\left (-125+475 x-575 x^2+225 x^3\right ) \log ^2\left (\frac {1}{5} (5-9 x)\right )} \, dx=\frac {x}{x^{2} + \left (5 x - 5\right ) \log {\left (1 - \frac {9 x}{5} \right )}} \]

[In]

integrate(((-45*x+25)*ln(-9/5*x+1)-9*x**3-40*x**2+45*x)/((225*x**3-575*x**2+475*x-125)*ln(-9/5*x+1)**2+(90*x**
4-140*x**3+50*x**2)*ln(-9/5*x+1)+9*x**5-5*x**4),x)

[Out]

x/(x**2 + (5*x - 5)*log(1 - 9*x/5))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {45 x-40 x^2-9 x^3+(25-45 x) \log \left (\frac {1}{5} (5-9 x)\right )}{-5 x^4+9 x^5+\left (50 x^2-140 x^3+90 x^4\right ) \log \left (\frac {1}{5} (5-9 x)\right )+\left (-125+475 x-575 x^2+225 x^3\right ) \log ^2\left (\frac {1}{5} (5-9 x)\right )} \, dx=\frac {x}{x^{2} - 5 \, x \log \left (5\right ) + 5 \, {\left (x - 1\right )} \log \left (-9 \, x + 5\right ) + 5 \, \log \left (5\right )} \]

[In]

integrate(((-45*x+25)*log(-9/5*x+1)-9*x^3-40*x^2+45*x)/((225*x^3-575*x^2+475*x-125)*log(-9/5*x+1)^2+(90*x^4-14
0*x^3+50*x^2)*log(-9/5*x+1)+9*x^5-5*x^4),x, algorithm="maxima")

[Out]

x/(x^2 - 5*x*log(5) + 5*(x - 1)*log(-9*x + 5) + 5*log(5))

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {45 x-40 x^2-9 x^3+(25-45 x) \log \left (\frac {1}{5} (5-9 x)\right )}{-5 x^4+9 x^5+\left (50 x^2-140 x^3+90 x^4\right ) \log \left (\frac {1}{5} (5-9 x)\right )+\left (-125+475 x-575 x^2+225 x^3\right ) \log ^2\left (\frac {1}{5} (5-9 x)\right )} \, dx=\frac {x}{x^{2} + 5 \, x \log \left (-\frac {9}{5} \, x + 1\right ) - 5 \, \log \left (-\frac {9}{5} \, x + 1\right )} \]

[In]

integrate(((-45*x+25)*log(-9/5*x+1)-9*x^3-40*x^2+45*x)/((225*x^3-575*x^2+475*x-125)*log(-9/5*x+1)^2+(90*x^4-14
0*x^3+50*x^2)*log(-9/5*x+1)+9*x^5-5*x^4),x, algorithm="giac")

[Out]

x/(x^2 + 5*x*log(-9/5*x + 1) - 5*log(-9/5*x + 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {45 x-40 x^2-9 x^3+(25-45 x) \log \left (\frac {1}{5} (5-9 x)\right )}{-5 x^4+9 x^5+\left (50 x^2-140 x^3+90 x^4\right ) \log \left (\frac {1}{5} (5-9 x)\right )+\left (-125+475 x-575 x^2+225 x^3\right ) \log ^2\left (\frac {1}{5} (5-9 x)\right )} \, dx=\int -\frac {\ln \left (1-\frac {9\,x}{5}\right )\,\left (45\,x-25\right )-45\,x+40\,x^2+9\,x^3}{\ln \left (1-\frac {9\,x}{5}\right )\,\left (90\,x^4-140\,x^3+50\,x^2\right )+{\ln \left (1-\frac {9\,x}{5}\right )}^2\,\left (225\,x^3-575\,x^2+475\,x-125\right )-5\,x^4+9\,x^5} \,d x \]

[In]

int(-(log(1 - (9*x)/5)*(45*x - 25) - 45*x + 40*x^2 + 9*x^3)/(log(1 - (9*x)/5)*(50*x^2 - 140*x^3 + 90*x^4) + lo
g(1 - (9*x)/5)^2*(475*x - 575*x^2 + 225*x^3 - 125) - 5*x^4 + 9*x^5),x)

[Out]

int(-(log(1 - (9*x)/5)*(45*x - 25) - 45*x + 40*x^2 + 9*x^3)/(log(1 - (9*x)/5)*(50*x^2 - 140*x^3 + 90*x^4) + lo
g(1 - (9*x)/5)^2*(475*x - 575*x^2 + 225*x^3 - 125) - 5*x^4 + 9*x^5), x)