\(\int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+(750-30 x+780 x^2-530 x^3+30 x^4) \log (x)+(75-150 x+150 x^2-150 x^3+75 x^4) \log ^2(x)}{x^2+(-10 x+10 x^2) \log (x)+(25-50 x+25 x^2) \log ^2(x)} \, dx\) [9860]

Optimal result

Integrand size = 102, antiderivative size = 28 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\left (3+x^2\right ) \left (x-\frac {2 x}{\frac {x}{25}+\frac {1}{5} (-1+x) \log (x)}\right ) \]

[Out]

(x-2/(1/5*(-1+x)*ln(x)+1/25*x)*x)*(x^2+3)

Rubi [F]

\[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx \]

[In]

Int[(-750 + 750*x - 247*x^2 + 150*x^3 + 3*x^4 + (750 - 30*x + 780*x^2 - 530*x^3 + 30*x^4)*Log[x] + (75 - 150*x
+ 150*x^2 - 150*x^3 + 75*x^4)*Log[x]^2)/(x^2 + (-10*x + 10*x^2)*Log[x] + (25 - 50*x + 25*x^2)*Log[x]^2),x]

[Out]

3*x + x^3 - 950*Defer[Int][(x - 5*Log[x] + 5*x*Log[x])^(-2), x] - 200*Defer[Int][1/((-1 + x)*(x - 5*Log[x] + 5
*x*Log[x])^2), x] + 700*Defer[Int][x/(x - 5*Log[x] + 5*x*Log[x])^2, x] - 300*Defer[Int][x^2/(x - 5*Log[x] + 5*
x*Log[x])^2, x] + 250*Defer[Int][x^3/(x - 5*Log[x] + 5*x*Log[x])^2, x] + 50*Defer[Int][(x - 5*Log[x] + 5*x*Log
[x])^(-1), x] + 200*Defer[Int][1/((-1 + x)*(x - 5*Log[x] + 5*x*Log[x])), x] + 50*Defer[Int][x/(x - 5*Log[x] +
5*x*Log[x]), x] - 100*Defer[Int][x^2/(x - 5*Log[x] + 5*x*Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+10 \left (75-3 x+78 x^2-53 x^3+3 x^4\right ) \log (x)+75 (-1+x)^2 \left (1+x^2\right ) \log ^2(x)}{(x+5 (-1+x) \log (x))^2} \, dx \\ & = \int \left (3 \left (1+x^2\right )+\frac {50 \left (15-33 x+20 x^2-11 x^3+5 x^4\right )}{(-1+x) (x-5 \log (x)+5 x \log (x))^2}-\frac {50 \left (-3-3 x^2+2 x^3\right )}{(-1+x) (x-5 \log (x)+5 x \log (x))}\right ) \, dx \\ & = 3 \int \left (1+x^2\right ) \, dx+50 \int \frac {15-33 x+20 x^2-11 x^3+5 x^4}{(-1+x) (x-5 \log (x)+5 x \log (x))^2} \, dx-50 \int \frac {-3-3 x^2+2 x^3}{(-1+x) (x-5 \log (x)+5 x \log (x))} \, dx \\ & = 3 x+x^3+50 \int \left (-\frac {19}{(x-5 \log (x)+5 x \log (x))^2}-\frac {4}{(-1+x) (x-5 \log (x)+5 x \log (x))^2}+\frac {14 x}{(x-5 \log (x)+5 x \log (x))^2}-\frac {6 x^2}{(x-5 \log (x)+5 x \log (x))^2}+\frac {5 x^3}{(x-5 \log (x)+5 x \log (x))^2}\right ) \, dx-50 \int \left (-\frac {1}{x-5 \log (x)+5 x \log (x)}-\frac {4}{(-1+x) (x-5 \log (x)+5 x \log (x))}-\frac {x}{x-5 \log (x)+5 x \log (x)}+\frac {2 x^2}{x-5 \log (x)+5 x \log (x)}\right ) \, dx \\ & = 3 x+x^3+50 \int \frac {1}{x-5 \log (x)+5 x \log (x)} \, dx+50 \int \frac {x}{x-5 \log (x)+5 x \log (x)} \, dx-100 \int \frac {x^2}{x-5 \log (x)+5 x \log (x)} \, dx-200 \int \frac {1}{(-1+x) (x-5 \log (x)+5 x \log (x))^2} \, dx+200 \int \frac {1}{(-1+x) (x-5 \log (x)+5 x \log (x))} \, dx+250 \int \frac {x^3}{(x-5 \log (x)+5 x \log (x))^2} \, dx-300 \int \frac {x^2}{(x-5 \log (x)+5 x \log (x))^2} \, dx+700 \int \frac {x}{(x-5 \log (x)+5 x \log (x))^2} \, dx-950 \int \frac {1}{(x-5 \log (x)+5 x \log (x))^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x \left (3+x^2-\frac {50 \left (3+x^2\right )}{x+5 (-1+x) \log (x)}\right ) \]

[In]

Integrate[(-750 + 750*x - 247*x^2 + 150*x^3 + 3*x^4 + (750 - 30*x + 780*x^2 - 530*x^3 + 30*x^4)*Log[x] + (75 -
150*x + 150*x^2 - 150*x^3 + 75*x^4)*Log[x]^2)/(x^2 + (-10*x + 10*x^2)*Log[x] + (25 - 50*x + 25*x^2)*Log[x]^2)
,x]

[Out]

x*(3 + x^2 - (50*(3 + x^2))/(x + 5*(-1 + x)*Log[x]))

Maple [A] (verified)

Time = 1.57 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04

method	result	size
risch	\(x^{3}+3 x -\frac {50 \left (x^{2}+3\right ) x}{5 x \ln \left (x \right )-5 \ln \left (x \right )+x}\)	\(29\)
default	\(\frac {x^{4}-147 x -15 \ln \left (x \right )+3 x^{2}-50 x^{3}+15 x^{2} \ln \left (x \right )-5 x^{3} \ln \left (x \right )+5 x^{4} \ln \left (x \right )}{5 x \ln \left (x \right )-5 \ln \left (x \right )+x}\)	\(57\)
norman	\(\frac {x^{4}-147 x -15 \ln \left (x \right )+3 x^{2}-50 x^{3}+15 x^{2} \ln \left (x \right )-5 x^{3} \ln \left (x \right )+5 x^{4} \ln \left (x \right )}{5 x \ln \left (x \right )-5 \ln \left (x \right )+x}\)	\(57\)
parallelrisch	\(\frac {25 x^{4} \ln \left (x \right )+5 x^{4}-25 x^{3} \ln \left (x \right )-250 x^{3}+75 x^{2} \ln \left (x \right )+15 x^{2}-735 x -75 \ln \left (x \right )}{25 x \ln \left (x \right )-25 \ln \left (x \right )+5 x}\)	\(60\)

[In]

int(((75*x^4-150*x^3+150*x^2-150*x+75)*ln(x)^2+(30*x^4-530*x^3+780*x^2-30*x+750)*ln(x)+3*x^4+150*x^3-247*x^2+7
50*x-750)/((25*x^2-50*x+25)*ln(x)^2+(10*x^2-10*x)*ln(x)+x^2),x,method=_RETURNVERBOSE)

[Out]

x^3+3*x-50*(x^2+3)*x/(5*x*ln(x)-5*ln(x)+x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).

Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\frac {x^{4} - 50 \, x^{3} + 3 \, x^{2} + 5 \, {\left (x^{4} - x^{3} + 3 \, x^{2} - 3 \, x\right )} \log \left (x\right ) - 150 \, x}{5 \, {\left (x - 1\right )} \log \left (x\right ) + x} \]

[In]

integrate(((75*x^4-150*x^3+150*x^2-150*x+75)*log(x)^2+(30*x^4-530*x^3+780*x^2-30*x+750)*log(x)+3*x^4+150*x^3-2
47*x^2+750*x-750)/((25*x^2-50*x+25)*log(x)^2+(10*x^2-10*x)*log(x)+x^2),x, algorithm="fricas")

[Out]

(x^4 - 50*x^3 + 3*x^2 + 5*(x^4 - x^3 + 3*x^2 - 3*x)*log(x) - 150*x)/(5*(x - 1)*log(x) + x)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x^{3} + 3 x + \frac {- 50 x^{3} - 150 x}{x + \left (5 x - 5\right ) \log {\left (x \right )}} \]

[In]

integrate(((75*x**4-150*x**3+150*x**2-150*x+75)*ln(x)**2+(30*x**4-530*x**3+780*x**2-30*x+750)*ln(x)+3*x**4+150
*x**3-247*x**2+750*x-750)/((25*x**2-50*x+25)*ln(x)**2+(10*x**2-10*x)*ln(x)+x**2),x)

[Out]

x**3 + 3*x + (-50*x**3 - 150*x)/(x + (5*x - 5)*log(x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).

Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\frac {x^{4} - 50 \, x^{3} + 3 \, x^{2} + 5 \, {\left (x^{4} - x^{3} + 3 \, x^{2} - 3 \, x\right )} \log \left (x\right ) - 150 \, x}{5 \, {\left (x - 1\right )} \log \left (x\right ) + x} \]

[In]

integrate(((75*x^4-150*x^3+150*x^2-150*x+75)*log(x)^2+(30*x^4-530*x^3+780*x^2-30*x+750)*log(x)+3*x^4+150*x^3-2
47*x^2+750*x-750)/((25*x^2-50*x+25)*log(x)^2+(10*x^2-10*x)*log(x)+x^2),x, algorithm="maxima")

[Out]

(x^4 - 50*x^3 + 3*x^2 + 5*(x^4 - x^3 + 3*x^2 - 3*x)*log(x) - 150*x)/(5*(x - 1)*log(x) + x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x^{3} + 3 \, x - \frac {50 \, {\left (x^{3} + 3 \, x\right )}}{5 \, x \log \left (x\right ) + x - 5 \, \log \left (x\right )} \]

[In]

integrate(((75*x^4-150*x^3+150*x^2-150*x+75)*log(x)^2+(30*x^4-530*x^3+780*x^2-30*x+750)*log(x)+3*x^4+150*x^3-2
47*x^2+750*x-750)/((25*x^2-50*x+25)*log(x)^2+(10*x^2-10*x)*log(x)+x^2),x, algorithm="giac")

[Out]

x^3 + 3*x - 50*(x^3 + 3*x)/(5*x*log(x) + x - 5*log(x))

Mupad [B] (verification not implemented)

Time = 15.74 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x\,\left (x^2+3\right )-\frac {x^2\,\left (x^2+3\right )-x\,\left (x^2+3\right )\,\left (x-50\right )}{x-5\,\ln \left (x\right )+5\,x\,\ln \left (x\right )} \]

[In]

int((750*x + log(x)*(780*x^2 - 30*x - 530*x^3 + 30*x^4 + 750) + log(x)^2*(150*x^2 - 150*x - 150*x^3 + 75*x^4 +
75) - 247*x^2 + 150*x^3 + 3*x^4 - 750)/(log(x)^2*(25*x^2 - 50*x + 25) - log(x)*(10*x - 10*x^2) + x^2),x)

[Out]

x*(x^2 + 3) - (x^2*(x^2 + 3) - x*(x^2 + 3)*(x - 50))/(x - 5*log(x) + 5*x*log(x))