\(\int \frac {300-120 x^2+12 x^4+2^{-\frac {x}{-5+x^2}} (-75+150 x-45 x^2-60 x^3+27 x^4+6 x^5-3 x^6+(-15 x+30 x^2-18 x^3+6 x^4-3 x^5) \log (2))}{100-200 x+60 x^2+80 x^3-36 x^4-8 x^5+4 x^6} \, dx\) [1186]

Optimal result

Integrand size = 115, antiderivative size = 34 \[ \int \frac {300-120 x^2+12 x^4+2^{-\frac {x}{-5+x^2}} \left (-75+150 x-45 x^2-60 x^3+27 x^4+6 x^5-3 x^6+\left (-15 x+30 x^2-18 x^3+6 x^4-3 x^5\right ) \log (2)\right )}{100-200 x+60 x^2+80 x^3-36 x^4-8 x^5+4 x^6} \, dx=3 \left (6-2^{-2+\frac {1}{\frac {5}{x}-x}} x-\frac {x}{-x+x^2}\right ) \] Output:

18-3*x/(x^2-x)-3/4*x*exp(ln(2)/(5/x-x))
 

Mathematica [F]

\[ \int \frac {300-120 x^2+12 x^4+2^{-\frac {x}{-5+x^2}} \left (-75+150 x-45 x^2-60 x^3+27 x^4+6 x^5-3 x^6+\left (-15 x+30 x^2-18 x^3+6 x^4-3 x^5\right ) \log (2)\right )}{100-200 x+60 x^2+80 x^3-36 x^4-8 x^5+4 x^6} \, dx=\int \frac {300-120 x^2+12 x^4+2^{-\frac {x}{-5+x^2}} \left (-75+150 x-45 x^2-60 x^3+27 x^4+6 x^5-3 x^6+\left (-15 x+30 x^2-18 x^3+6 x^4-3 x^5\right ) \log (2)\right )}{100-200 x+60 x^2+80 x^3-36 x^4-8 x^5+4 x^6} \, dx \] Input:

Integrate[(300 - 120*x^2 + 12*x^4 + (-75 + 150*x - 45*x^2 - 60*x^3 + 27*x^ 
4 + 6*x^5 - 3*x^6 + (-15*x + 30*x^2 - 18*x^3 + 6*x^4 - 3*x^5)*Log[2])/2^(x 
/(-5 + x^2)))/(100 - 200*x + 60*x^2 + 80*x^3 - 36*x^4 - 8*x^5 + 4*x^6),x]
 

Output:

Integrate[(300 - 120*x^2 + 12*x^4 + (-75 + 150*x - 45*x^2 - 60*x^3 + 27*x^ 
4 + 6*x^5 - 3*x^6 + (-15*x + 30*x^2 - 18*x^3 + 6*x^4 - 3*x^5)*Log[2])/2^(x 
/(-5 + x^2)))/(100 - 200*x + 60*x^2 + 80*x^3 - 36*x^4 - 8*x^5 + 4*x^6), x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {12 x^4-120 x^2+2^{-\frac {x}{x^2-5}} \left (-3 x^6+6 x^5+27 x^4-60 x^3-45 x^2+\left (-3 x^5+6 x^4-18 x^3+30 x^2-15 x\right ) \log (2)+150 x-75\right )+300}{4 x^6-8 x^5-36 x^4+80 x^3+60 x^2-200 x+100} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {12 x^4-120 x^2+2^{-\frac {x}{x^2-5}} \left (-3 x^6+6 x^5+27 x^4-60 x^3-45 x^2+\left (-3 x^5+6 x^4-18 x^3+30 x^2-15 x\right ) \log (2)+150 x-75\right )+300}{64 (x-1)}+\frac {(-x-2) \left (12 x^4-120 x^2+2^{-\frac {x}{x^2-5}} \left (-3 x^6+6 x^5+27 x^4-60 x^3-45 x^2+\left (-3 x^5+6 x^4-18 x^3+30 x^2-15 x\right ) \log (2)+150 x-75\right )+300\right )}{64 \left (x^2-5\right )}+\frac {12 x^4-120 x^2+2^{-\frac {x}{x^2-5}} \left (-3 x^6+6 x^5+27 x^4-60 x^3-45 x^2+\left (-3 x^5+6 x^4-18 x^3+30 x^2-15 x\right ) \log (2)+150 x-75\right )+300}{64 (x-1)^2}+\frac {(x+3) \left (12 x^4-120 x^2+2^{-\frac {x}{x^2-5}} \left (-3 x^6+6 x^5+27 x^4-60 x^3-45 x^2+\left (-3 x^5+6 x^4-18 x^3+30 x^2-15 x\right ) \log (2)+150 x-75\right )+300\right )}{32 \left (x^2-5\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {15}{64} \int 2^{1-\frac {x}{x^2-5}} x^2dx-\frac {3}{32} (3+\log (1024)) \int 2^{-\frac {x}{x^2-5}}dx-\frac {3}{64} (5-\log (1024)) \int 2^{1-\frac {x}{x^2-5}}dx-\frac {15}{32} \left (1+\sqrt {5}\right ) \log (2) \int \frac {2^{-\frac {x}{x^2-5}}}{\sqrt {5}-x}dx+\frac {3}{32} \left (9+5 \sqrt {5}\right ) \log (4) \int \frac {2^{-\frac {x}{x^2-5}-1}}{\sqrt {5}-x}dx-\frac {3}{64} \log (32) \int 2^{-\frac {x}{x^2-5}} xdx-\frac {3}{64} (25-\log (32)) \int 2^{-\frac {x}{x^2-5}} xdx+\frac {3}{64} (15+\log (4)) \int 2^{-\frac {x}{x^2-5}} xdx+\frac {3}{32} (5-\log (2)) \int 2^{-\frac {x}{x^2-5}} xdx+\frac {3}{64} (2+\log (128)) \int 2^{-\frac {x}{x^2-5}} x^2dx-\frac {3}{64} (10+\log (32)) \int 2^{-\frac {x}{x^2-5}} x^2dx-\frac {3}{32} (1+\log (2)) \int 2^{-\frac {x}{x^2-5}} x^2dx+\frac {15}{32} \left (1-\sqrt {5}\right ) \log (2) \int \frac {2^{-\frac {x}{x^2-5}}}{x+\sqrt {5}}dx-\frac {3}{32} \left (9-5 \sqrt {5}\right ) \log (4) \int \frac {2^{-\frac {x}{x^2-5}-1}}{x+\sqrt {5}}dx-\frac {3}{32} \log (1024) \int \frac {2^{3-\frac {x}{x^2-5}} x}{\left (x^2-5\right )^2}dx-\frac {3}{64} \int 2^{-\frac {x}{x^2-5}} x^4dx+\frac {3}{64} \log (2) \int 2^{-\frac {x}{x^2-5}} x^4dx+\frac {3}{64} (1-\log (2)) \int 2^{-\frac {x}{x^2-5}} x^4dx-\frac {3}{32} \int 2^{-\frac {x}{x^2-5}} x^3dx-\frac {3}{64} \int 2^{3-\frac {x}{x^2-5}} x^3dx+\frac {3}{64} (10+\log (2)) \int 2^{-\frac {x}{x^2-5}} x^3dx-\frac {3}{64} \log (2) \int 2^{-\frac {x}{x^2-5}} x^3dx+\frac {3 x^4}{64}-\frac {21 x^2}{32}-\frac {3}{64} \left (5-x^2\right )^2-\frac {9 x}{8}+\frac {3}{16} (x+3)^2+\frac {3}{1-x}\)

Input:

Int[(300 - 120*x^2 + 12*x^4 + (-75 + 150*x - 45*x^2 - 60*x^3 + 27*x^4 + 6* 
x^5 - 3*x^6 + (-15*x + 30*x^2 - 18*x^3 + 6*x^4 - 3*x^5)*Log[2])/2^(x/(-5 + 
 x^2)))/(100 - 200*x + 60*x^2 + 80*x^3 - 36*x^4 - 8*x^5 + 4*x^6),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 5.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68

method	result	size
risch	\(-\frac {3}{-1+x}-\frac {3 \left (\frac {1}{2}\right )^{\frac {x}{x^{2}-5}} x}{4}\)	\(23\)
parallelrisch	\(-\frac {3 \,{\mathrm e}^{-\frac {x \ln \left (2\right )}{x^{2}-5}} x^{2}-3 \,{\mathrm e}^{-\frac {x \ln \left (2\right )}{x^{2}-5}} x +12}{4 \left (-1+x \right )}\)	\(44\)
parts	\(-\frac {3}{-1+x}+\frac {\frac {15 \,{\mathrm e}^{-\frac {x \ln \left (2\right )}{x^{2}-5}} x}{4}-\frac {3 \,{\mathrm e}^{-\frac {x \ln \left (2\right )}{x^{2}-5}} x^{3}}{4}}{x^{2}-5}\)	\(52\)
norman	\(\frac {-3 x^{2}+15-\frac {15 \,{\mathrm e}^{-\frac {x \ln \left (2\right )}{x^{2}-5}} x}{4}+\frac {15 \,{\mathrm e}^{-\frac {x \ln \left (2\right )}{x^{2}-5}} x^{2}}{4}+\frac {3 \,{\mathrm e}^{-\frac {x \ln \left (2\right )}{x^{2}-5}} x^{3}}{4}-\frac {3 \,{\mathrm e}^{-\frac {x \ln \left (2\right )}{x^{2}-5}} x^{4}}{4}}{x^{3}-x^{2}-5 x +5}\)	\(94\)

Input:

int((((-3*x^5+6*x^4-18*x^3+30*x^2-15*x)*ln(2)-3*x^6+6*x^5+27*x^4-60*x^3-45 
*x^2+150*x-75)*exp(-x*ln(2)/(x^2-5))+12*x^4-120*x^2+300)/(4*x^6-8*x^5-36*x 
^4+80*x^3+60*x^2-200*x+100),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.18 \[ \int \frac {300-120 x^2+12 x^4+2^{-\frac {x}{-5+x^2}} \left (-75+150 x-45 x^2-60 x^3+27 x^4+6 x^5-3 x^6+\left (-15 x+30 x^2-18 x^3+6 x^4-3 x^5\right ) \log (2)\right )}{100-200 x+60 x^2+80 x^3-36 x^4-8 x^5+4 x^6} \, dx=-\frac {3 \, {\left (x^{2} + 4 \cdot 2^{\frac {x}{x^{2} - 5}} - x\right )}}{4 \cdot 2^{\frac {x}{x^{2} - 5}} {\left (x - 1\right )}} \] Input:

integrate((((-3*x^5+6*x^4-18*x^3+30*x^2-15*x)*log(2)-3*x^6+6*x^5+27*x^4-60 
*x^3-45*x^2+150*x-75)*exp(-x*log(2)/(x^2-5))+12*x^4-120*x^2+300)/(4*x^6-8* 
x^5-36*x^4+80*x^3+60*x^2-200*x+100),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {300-120 x^2+12 x^4+2^{-\frac {x}{-5+x^2}} \left (-75+150 x-45 x^2-60 x^3+27 x^4+6 x^5-3 x^6+\left (-15 x+30 x^2-18 x^3+6 x^4-3 x^5\right ) \log (2)\right )}{100-200 x+60 x^2+80 x^3-36 x^4-8 x^5+4 x^6} \, dx=- \frac {3 x e^{- \frac {x \log {\left (2 \right )}}{x^{2} - 5}}}{4} - \frac {3}{x - 1} \] Input:

integrate((((-3*x**5+6*x**4-18*x**3+30*x**2-15*x)*ln(2)-3*x**6+6*x**5+27*x 
**4-60*x**3-45*x**2+150*x-75)*exp(-x*ln(2)/(x**2-5))+12*x**4-120*x**2+300) 
/(4*x**6-8*x**5-36*x**4+80*x**3+60*x**2-200*x+100),x)
 

Output:

-3*x*exp(-x*log(2)/(x**2 - 5))/4 - 3/(x - 1)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (30) = 60\).

Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.76 \[ \int \frac {300-120 x^2+12 x^4+2^{-\frac {x}{-5+x^2}} \left (-75+150 x-45 x^2-60 x^3+27 x^4+6 x^5-3 x^6+\left (-15 x+30 x^2-18 x^3+6 x^4-3 x^5\right ) \log (2)\right )}{100-200 x+60 x^2+80 x^3-36 x^4-8 x^5+4 x^6} \, dx=-\frac {3 \, {\left (8 \, x^{2} + 5 \, x - 15\right )}}{8 \, {\left (x^{3} - x^{2} - 5 \, x + 5\right )}} - \frac {15 \, {\left (4 \, x^{2} + x - 15\right )}}{8 \, {\left (x^{3} - x^{2} - 5 \, x + 5\right )}} + \frac {15 \, {\left (2 \, x^{2} + x - 5\right )}}{4 \, {\left (x^{3} - x^{2} - 5 \, x + 5\right )}} - \frac {3 \, x}{4 \cdot 2^{\frac {x}{x^{2} - 5}}} \] Input:

integrate((((-3*x^5+6*x^4-18*x^3+30*x^2-15*x)*log(2)-3*x^6+6*x^5+27*x^4-60 
*x^3-45*x^2+150*x-75)*exp(-x*log(2)/(x^2-5))+12*x^4-120*x^2+300)/(4*x^6-8* 
x^5-36*x^4+80*x^3+60*x^2-200*x+100),x, algorithm="maxima")
 

Output:

-3/8*(8*x^2 + 5*x - 15)/(x^3 - x^2 - 5*x + 5) - 15/8*(4*x^2 + x - 15)/(x^3 
 - x^2 - 5*x + 5) + 15/4*(2*x^2 + x - 5)/(x^3 - x^2 - 5*x + 5) - 3/4*x/2^( 
x/(x^2 - 5))
 

Giac [F]

\[ \int \frac {300-120 x^2+12 x^4+2^{-\frac {x}{-5+x^2}} \left (-75+150 x-45 x^2-60 x^3+27 x^4+6 x^5-3 x^6+\left (-15 x+30 x^2-18 x^3+6 x^4-3 x^5\right ) \log (2)\right )}{100-200 x+60 x^2+80 x^3-36 x^4-8 x^5+4 x^6} \, dx=\int { \frac {3 \, {\left (4 \, x^{4} - 40 \, x^{2} - \frac {x^{6} - 2 \, x^{5} - 9 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} + {\left (x^{5} - 2 \, x^{4} + 6 \, x^{3} - 10 \, x^{2} + 5 \, x\right )} \log \left (2\right ) - 50 \, x + 25}{2^{\frac {x}{x^{2} - 5}}} + 100\right )}}{4 \, {\left (x^{6} - 2 \, x^{5} - 9 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} - 50 \, x + 25\right )}} \,d x } \] Input:

integrate((((-3*x^5+6*x^4-18*x^3+30*x^2-15*x)*log(2)-3*x^6+6*x^5+27*x^4-60 
*x^3-45*x^2+150*x-75)*exp(-x*log(2)/(x^2-5))+12*x^4-120*x^2+300)/(4*x^6-8* 
x^5-36*x^4+80*x^3+60*x^2-200*x+100),x, algorithm="giac")
 

Output:

integrate(3/4*(4*x^4 - 40*x^2 - (x^6 - 2*x^5 - 9*x^4 + 20*x^3 + 15*x^2 + ( 
x^5 - 2*x^4 + 6*x^3 - 10*x^2 + 5*x)*log(2) - 50*x + 25)/2^(x/(x^2 - 5)) + 
100)/(x^6 - 2*x^5 - 9*x^4 + 20*x^3 + 15*x^2 - 50*x + 25), x)
 

Mupad [B] (verification not implemented)

Time = 2.96 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \frac {300-120 x^2+12 x^4+2^{-\frac {x}{-5+x^2}} \left (-75+150 x-45 x^2-60 x^3+27 x^4+6 x^5-3 x^6+\left (-15 x+30 x^2-18 x^3+6 x^4-3 x^5\right ) \log (2)\right )}{100-200 x+60 x^2+80 x^3-36 x^4-8 x^5+4 x^6} \, dx=-\frac {3}{x-1}-\frac {3\,x}{4\,2^{\frac {x}{x^2-5}}} \] Input:

int(-(exp(-(x*log(2))/(x^2 - 5))*(45*x^2 - 150*x + 60*x^3 - 27*x^4 - 6*x^5 
 + 3*x^6 + log(2)*(15*x - 30*x^2 + 18*x^3 - 6*x^4 + 3*x^5) + 75) + 120*x^2 
 - 12*x^4 - 300)/(60*x^2 - 200*x + 80*x^3 - 36*x^4 - 8*x^5 + 4*x^6 + 100), 
x)
 

Output:

- 3/(x - 1) - (3*x)/(4*2^(x/(x^2 - 5)))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {300-120 x^2+12 x^4+2^{-\frac {x}{-5+x^2}} \left (-75+150 x-45 x^2-60 x^3+27 x^4+6 x^5-3 x^6+\left (-15 x+30 x^2-18 x^3+6 x^4-3 x^5\right ) \log (2)\right )}{100-200 x+60 x^2+80 x^3-36 x^4-8 x^5+4 x^6} \, dx=\frac {3 x \left (-4 e^{\frac {\mathrm {log}\left (2\right ) x}{x^{2}-5}}-x +1\right )}{4 e^{\frac {\mathrm {log}\left (2\right ) x}{x^{2}-5}} \left (x -1\right )} \] Input:

int((((-3*x^5+6*x^4-18*x^3+30*x^2-15*x)*log(2)-3*x^6+6*x^5+27*x^4-60*x^3-4 
5*x^2+150*x-75)*exp(-x*log(2)/(x^2-5))+12*x^4-120*x^2+300)/(4*x^6-8*x^5-36 
*x^4+80*x^3+60*x^2-200*x+100),x)
 

Output:

(3*x*( - 4*e**((log(2)*x)/(x**2 - 5)) - x + 1))/(4*e**((log(2)*x)/(x**2 - 
5))*(x - 1))