\(\int \frac {250000-45000 x^5-187500 x^6+e^{5 x} (-1562500 x^3+1171875 x^9)+e^{5 x} (-187500 x^2+140625 x^8) \log (-4+3 x^6)+e^{5 x} (-7500 x+5625 x^7) \log ^2(-4+3 x^6)+e^{5 x} (-100+75 x^6) \log ^3(-4+3 x^6)}{-62500 x^3+46875 x^9+(-7500 x^2+5625 x^8) \log (-4+3 x^6)+(-300 x+225 x^7) \log ^2(-4+3 x^6)+(-4+3 x^6) \log ^3(-4+3 x^6)} \, dx\) [1438]

Optimal result

Integrand size = 174, antiderivative size = 26 \[ \int \frac {250000-45000 x^5-187500 x^6+e^{5 x} \left (-1562500 x^3+1171875 x^9\right )+e^{5 x} \left (-187500 x^2+140625 x^8\right ) \log \left (-4+3 x^6\right )+e^{5 x} \left (-7500 x+5625 x^7\right ) \log ^2\left (-4+3 x^6\right )+e^{5 x} \left (-100+75 x^6\right ) \log ^3\left (-4+3 x^6\right )}{-62500 x^3+46875 x^9+\left (-7500 x^2+5625 x^8\right ) \log \left (-4+3 x^6\right )+\left (-300 x+225 x^7\right ) \log ^2\left (-4+3 x^6\right )+\left (-4+3 x^6\right ) \log ^3\left (-4+3 x^6\right )} \, dx=5 e^{5 x}+\frac {2}{\left (x+\frac {1}{25} \log \left (-4+3 x^6\right )\right )^2} \] Output:

5*exp(5*x)+2/(1/25*ln(3*x^6-4)+x)^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {250000-45000 x^5-187500 x^6+e^{5 x} \left (-1562500 x^3+1171875 x^9\right )+e^{5 x} \left (-187500 x^2+140625 x^8\right ) \log \left (-4+3 x^6\right )+e^{5 x} \left (-7500 x+5625 x^7\right ) \log ^2\left (-4+3 x^6\right )+e^{5 x} \left (-100+75 x^6\right ) \log ^3\left (-4+3 x^6\right )}{-62500 x^3+46875 x^9+\left (-7500 x^2+5625 x^8\right ) \log \left (-4+3 x^6\right )+\left (-300 x+225 x^7\right ) \log ^2\left (-4+3 x^6\right )+\left (-4+3 x^6\right ) \log ^3\left (-4+3 x^6\right )} \, dx=25 \left (\frac {e^{5 x}}{5}+\frac {50}{\left (25 x+\log \left (-4+3 x^6\right )\right )^2}\right ) \] Input:

Integrate[(250000 - 45000*x^5 - 187500*x^6 + E^(5*x)*(-1562500*x^3 + 11718 
75*x^9) + E^(5*x)*(-187500*x^2 + 140625*x^8)*Log[-4 + 3*x^6] + E^(5*x)*(-7 
500*x + 5625*x^7)*Log[-4 + 3*x^6]^2 + E^(5*x)*(-100 + 75*x^6)*Log[-4 + 3*x 
^6]^3)/(-62500*x^3 + 46875*x^9 + (-7500*x^2 + 5625*x^8)*Log[-4 + 3*x^6] + 
(-300*x + 225*x^7)*Log[-4 + 3*x^6]^2 + (-4 + 3*x^6)*Log[-4 + 3*x^6]^3),x]
 

Output:

25*(E^(5*x)/5 + 50/(25*x + Log[-4 + 3*x^6])^2)
 

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.

\(\displaystyle \int \frac {-187500 x^6+e^{5 x} \left (75 x^6-100\right ) \log ^3\left (3 x^6-4\right )-45000 x^5+e^{5 x} \left (1171875 x^9-1562500 x^3\right )+e^{5 x} \left (5625 x^7-7500 x\right ) \log ^2\left (3 x^6-4\right )+e^{5 x} \left (140625 x^8-187500 x^2\right ) \log \left (3 x^6-4\right )+250000}{46875 x^9+\left (3 x^6-4\right ) \log ^3\left (3 x^6-4\right )-62500 x^3+\left (225 x^7-300 x\right ) \log ^2\left (3 x^6-4\right )+\left (5625 x^8-7500 x^2\right ) \log \left (3 x^6-4\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {187500 x^6-e^{5 x} \left (75 x^6-100\right ) \log ^3\left (3 x^6-4\right )+45000 x^5-e^{5 x} \left (1171875 x^9-1562500 x^3\right )-e^{5 x} \left (5625 x^7-7500 x\right ) \log ^2\left (3 x^6-4\right )-e^{5 x} \left (140625 x^8-187500 x^2\right ) \log \left (3 x^6-4\right )-250000}{\left (4-3 x^6\right ) \left (\log \left (3 x^6-4\right )+25 x\right )^3}dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (-\frac {187500 x^6}{\left (3 x^6-4\right ) \left (\log \left (3 x^6-4\right )+25 x\right )^3}+\frac {250000}{\left (3 x^6-4\right ) \left (\log \left (3 x^6-4\right )+25 x\right )^3}-\frac {45000 x^5}{\left (3 x^6-4\right ) \left (\log \left (3 x^6-4\right )+25 x\right )^3}+25 e^{5 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -62500 \int \frac {1}{\left (25 x+\log \left (3 x^6-4\right )\right )^3}dx+2500 \sqrt [6]{3} \int \frac {1}{\left (-\sqrt [6]{3} x-\sqrt [3]{-2}\right ) \left (25 x+\log \left (3 x^6-4\right )\right )^3}dx+2500 \sqrt [6]{3} \int \frac {1}{\left (\sqrt [3]{2}-\sqrt [6]{3} x\right ) \left (25 x+\log \left (3 x^6-4\right )\right )^3}dx+2500 \sqrt [6]{3} \int \frac {1}{\left ((-1)^{2/3} \sqrt [3]{2}-\sqrt [6]{3} x\right ) \left (25 x+\log \left (3 x^6-4\right )\right )^3}dx-2500 \sqrt [6]{3} \int \frac {1}{\left (\sqrt [6]{3} x-\sqrt [3]{-2}\right ) \left (25 x+\log \left (3 x^6-4\right )\right )^3}dx-2500 \sqrt [6]{3} \int \frac {1}{\left (\sqrt [6]{3} x+\sqrt [3]{2}\right ) \left (25 x+\log \left (3 x^6-4\right )\right )^3}dx-2500 \sqrt [6]{3} \int \frac {1}{\left (\sqrt [6]{3} x+(-1)^{2/3} \sqrt [3]{2}\right ) \left (25 x+\log \left (3 x^6-4\right )\right )^3}dx+5 e^{5 x}\)

Input:

Int[(250000 - 45000*x^5 - 187500*x^6 + E^(5*x)*(-1562500*x^3 + 1171875*x^9 
) + E^(5*x)*(-187500*x^2 + 140625*x^8)*Log[-4 + 3*x^6] + E^(5*x)*(-7500*x 
+ 5625*x^7)*Log[-4 + 3*x^6]^2 + E^(5*x)*(-100 + 75*x^6)*Log[-4 + 3*x^6]^3) 
/(-62500*x^3 + 46875*x^9 + (-7500*x^2 + 5625*x^8)*Log[-4 + 3*x^6] + (-300* 
x + 225*x^7)*Log[-4 + 3*x^6]^2 + (-4 + 3*x^6)*Log[-4 + 3*x^6]^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 52.70 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

method	result	size
risch	\(5 \,{\mathrm e}^{5 x}+\frac {1250}{{\left (\ln \left (3 x^{6}-4\right )+25 x \right )}^{2}}\)	\(24\)
parallelrisch	\(\frac {37500000+93750000 x^{2} {\mathrm e}^{5 x}+7500000 x \,{\mathrm e}^{5 x} \ln \left (3 x^{6}-4\right )+150000 \,{\mathrm e}^{5 x} \ln \left (3 x^{6}-4\right )^{2}}{18750000 x^{2}+1500000 x \ln \left (3 x^{6}-4\right )+30000 \ln \left (3 x^{6}-4\right )^{2}}\)	\(74\)

Input:

int(((75*x^6-100)*exp(5*x)*ln(3*x^6-4)^3+(5625*x^7-7500*x)*exp(5*x)*ln(3*x 
^6-4)^2+(140625*x^8-187500*x^2)*exp(5*x)*ln(3*x^6-4)+(1171875*x^9-1562500* 
x^3)*exp(5*x)-187500*x^6-45000*x^5+250000)/((3*x^6-4)*ln(3*x^6-4)^3+(225*x 
^7-300*x)*ln(3*x^6-4)^2+(5625*x^8-7500*x^2)*ln(3*x^6-4)+46875*x^9-62500*x^ 
3),x,method=_RETURNVERBOSE)
 

Output:

5*exp(5*x)+1250/(ln(3*x^6-4)+25*x)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (23) = 46\).

Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77 \[ \int \frac {250000-45000 x^5-187500 x^6+e^{5 x} \left (-1562500 x^3+1171875 x^9\right )+e^{5 x} \left (-187500 x^2+140625 x^8\right ) \log \left (-4+3 x^6\right )+e^{5 x} \left (-7500 x+5625 x^7\right ) \log ^2\left (-4+3 x^6\right )+e^{5 x} \left (-100+75 x^6\right ) \log ^3\left (-4+3 x^6\right )}{-62500 x^3+46875 x^9+\left (-7500 x^2+5625 x^8\right ) \log \left (-4+3 x^6\right )+\left (-300 x+225 x^7\right ) \log ^2\left (-4+3 x^6\right )+\left (-4+3 x^6\right ) \log ^3\left (-4+3 x^6\right )} \, dx=\frac {5 \, {\left (625 \, x^{2} e^{\left (5 \, x\right )} + 50 \, x e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right ) + e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right )^{2} + 250\right )}}{625 \, x^{2} + 50 \, x \log \left (3 \, x^{6} - 4\right ) + \log \left (3 \, x^{6} - 4\right )^{2}} \] Input:

integrate(((75*x^6-100)*exp(5*x)*log(3*x^6-4)^3+(5625*x^7-7500*x)*exp(5*x) 
*log(3*x^6-4)^2+(140625*x^8-187500*x^2)*exp(5*x)*log(3*x^6-4)+(1171875*x^9 
-1562500*x^3)*exp(5*x)-187500*x^6-45000*x^5+250000)/((3*x^6-4)*log(3*x^6-4 
)^3+(225*x^7-300*x)*log(3*x^6-4)^2+(5625*x^8-7500*x^2)*log(3*x^6-4)+46875* 
x^9-62500*x^3),x, algorithm="fricas")
 

Output:

5*(625*x^2*e^(5*x) + 50*x*e^(5*x)*log(3*x^6 - 4) + e^(5*x)*log(3*x^6 - 4)^ 
2 + 250)/(625*x^2 + 50*x*log(3*x^6 - 4) + log(3*x^6 - 4)^2)
 

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {250000-45000 x^5-187500 x^6+e^{5 x} \left (-1562500 x^3+1171875 x^9\right )+e^{5 x} \left (-187500 x^2+140625 x^8\right ) \log \left (-4+3 x^6\right )+e^{5 x} \left (-7500 x+5625 x^7\right ) \log ^2\left (-4+3 x^6\right )+e^{5 x} \left (-100+75 x^6\right ) \log ^3\left (-4+3 x^6\right )}{-62500 x^3+46875 x^9+\left (-7500 x^2+5625 x^8\right ) \log \left (-4+3 x^6\right )+\left (-300 x+225 x^7\right ) \log ^2\left (-4+3 x^6\right )+\left (-4+3 x^6\right ) \log ^3\left (-4+3 x^6\right )} \, dx=5 e^{5 x} + \frac {1250}{625 x^{2} + 50 x \log {\left (3 x^{6} - 4 \right )} + \log {\left (3 x^{6} - 4 \right )}^{2}} \] Input:

integrate(((75*x**6-100)*exp(5*x)*ln(3*x**6-4)**3+(5625*x**7-7500*x)*exp(5 
*x)*ln(3*x**6-4)**2+(140625*x**8-187500*x**2)*exp(5*x)*ln(3*x**6-4)+(11718 
75*x**9-1562500*x**3)*exp(5*x)-187500*x**6-45000*x**5+250000)/((3*x**6-4)* 
ln(3*x**6-4)**3+(225*x**7-300*x)*ln(3*x**6-4)**2+(5625*x**8-7500*x**2)*ln( 
3*x**6-4)+46875*x**9-62500*x**3),x)
 

Output:

5*exp(5*x) + 1250/(625*x**2 + 50*x*log(3*x**6 - 4) + log(3*x**6 - 4)**2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (23) = 46\).

Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77 \[ \int \frac {250000-45000 x^5-187500 x^6+e^{5 x} \left (-1562500 x^3+1171875 x^9\right )+e^{5 x} \left (-187500 x^2+140625 x^8\right ) \log \left (-4+3 x^6\right )+e^{5 x} \left (-7500 x+5625 x^7\right ) \log ^2\left (-4+3 x^6\right )+e^{5 x} \left (-100+75 x^6\right ) \log ^3\left (-4+3 x^6\right )}{-62500 x^3+46875 x^9+\left (-7500 x^2+5625 x^8\right ) \log \left (-4+3 x^6\right )+\left (-300 x+225 x^7\right ) \log ^2\left (-4+3 x^6\right )+\left (-4+3 x^6\right ) \log ^3\left (-4+3 x^6\right )} \, dx=\frac {5 \, {\left (625 \, x^{2} e^{\left (5 \, x\right )} + 50 \, x e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right ) + e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right )^{2} + 250\right )}}{625 \, x^{2} + 50 \, x \log \left (3 \, x^{6} - 4\right ) + \log \left (3 \, x^{6} - 4\right )^{2}} \] Input:

integrate(((75*x^6-100)*exp(5*x)*log(3*x^6-4)^3+(5625*x^7-7500*x)*exp(5*x) 
*log(3*x^6-4)^2+(140625*x^8-187500*x^2)*exp(5*x)*log(3*x^6-4)+(1171875*x^9 
-1562500*x^3)*exp(5*x)-187500*x^6-45000*x^5+250000)/((3*x^6-4)*log(3*x^6-4 
)^3+(225*x^7-300*x)*log(3*x^6-4)^2+(5625*x^8-7500*x^2)*log(3*x^6-4)+46875* 
x^9-62500*x^3),x, algorithm="maxima")
 

Output:

5*(625*x^2*e^(5*x) + 50*x*e^(5*x)*log(3*x^6 - 4) + e^(5*x)*log(3*x^6 - 4)^ 
2 + 250)/(625*x^2 + 50*x*log(3*x^6 - 4) + log(3*x^6 - 4)^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (23) = 46\).

Time = 4.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77 \[ \int \frac {250000-45000 x^5-187500 x^6+e^{5 x} \left (-1562500 x^3+1171875 x^9\right )+e^{5 x} \left (-187500 x^2+140625 x^8\right ) \log \left (-4+3 x^6\right )+e^{5 x} \left (-7500 x+5625 x^7\right ) \log ^2\left (-4+3 x^6\right )+e^{5 x} \left (-100+75 x^6\right ) \log ^3\left (-4+3 x^6\right )}{-62500 x^3+46875 x^9+\left (-7500 x^2+5625 x^8\right ) \log \left (-4+3 x^6\right )+\left (-300 x+225 x^7\right ) \log ^2\left (-4+3 x^6\right )+\left (-4+3 x^6\right ) \log ^3\left (-4+3 x^6\right )} \, dx=\frac {5 \, {\left (625 \, x^{2} e^{\left (5 \, x\right )} + 50 \, x e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right ) + e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right )^{2} + 250\right )}}{625 \, x^{2} + 50 \, x \log \left (3 \, x^{6} - 4\right ) + \log \left (3 \, x^{6} - 4\right )^{2}} \] Input:

integrate(((75*x^6-100)*exp(5*x)*log(3*x^6-4)^3+(5625*x^7-7500*x)*exp(5*x) 
*log(3*x^6-4)^2+(140625*x^8-187500*x^2)*exp(5*x)*log(3*x^6-4)+(1171875*x^9 
-1562500*x^3)*exp(5*x)-187500*x^6-45000*x^5+250000)/((3*x^6-4)*log(3*x^6-4 
)^3+(225*x^7-300*x)*log(3*x^6-4)^2+(5625*x^8-7500*x^2)*log(3*x^6-4)+46875* 
x^9-62500*x^3),x, algorithm="giac")
 

Output:

5*(625*x^2*e^(5*x) + 50*x*e^(5*x)*log(3*x^6 - 4) + e^(5*x)*log(3*x^6 - 4)^ 
2 + 250)/(625*x^2 + 50*x*log(3*x^6 - 4) + log(3*x^6 - 4)^2)
 

Mupad [B] (verification not implemented)

Time = 3.88 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {250000-45000 x^5-187500 x^6+e^{5 x} \left (-1562500 x^3+1171875 x^9\right )+e^{5 x} \left (-187500 x^2+140625 x^8\right ) \log \left (-4+3 x^6\right )+e^{5 x} \left (-7500 x+5625 x^7\right ) \log ^2\left (-4+3 x^6\right )+e^{5 x} \left (-100+75 x^6\right ) \log ^3\left (-4+3 x^6\right )}{-62500 x^3+46875 x^9+\left (-7500 x^2+5625 x^8\right ) \log \left (-4+3 x^6\right )+\left (-300 x+225 x^7\right ) \log ^2\left (-4+3 x^6\right )+\left (-4+3 x^6\right ) \log ^3\left (-4+3 x^6\right )} \, dx=5\,{\mathrm {e}}^{5\,x}+\frac {1250}{{\left (25\,x+\ln \left (3\,x^6-4\right )\right )}^2} \] Input:

int((exp(5*x)*(1562500*x^3 - 1171875*x^9) + 45000*x^5 + 187500*x^6 + log(3 
*x^6 - 4)^2*exp(5*x)*(7500*x - 5625*x^7) + log(3*x^6 - 4)*exp(5*x)*(187500 
*x^2 - 140625*x^8) - log(3*x^6 - 4)^3*exp(5*x)*(75*x^6 - 100) - 250000)/(l 
og(3*x^6 - 4)^2*(300*x - 225*x^7) + log(3*x^6 - 4)*(7500*x^2 - 5625*x^8) - 
 log(3*x^6 - 4)^3*(3*x^6 - 4) + 62500*x^3 - 46875*x^9),x)
 

Output:

5*exp(5*x) + 1250/(25*x + log(3*x^6 - 4))^2
 

Reduce [F]

\[ \int \frac {250000-45000 x^5-187500 x^6+e^{5 x} \left (-1562500 x^3+1171875 x^9\right )+e^{5 x} \left (-187500 x^2+140625 x^8\right ) \log \left (-4+3 x^6\right )+e^{5 x} \left (-7500 x+5625 x^7\right ) \log ^2\left (-4+3 x^6\right )+e^{5 x} \left (-100+75 x^6\right ) \log ^3\left (-4+3 x^6\right )}{-62500 x^3+46875 x^9+\left (-7500 x^2+5625 x^8\right ) \log \left (-4+3 x^6\right )+\left (-300 x+225 x^7\right ) \log ^2\left (-4+3 x^6\right )+\left (-4+3 x^6\right ) \log ^3\left (-4+3 x^6\right )} \, dx=\int \frac {\left (75 x^{6}-100\right ) {\mathrm e}^{5 x} \mathrm {log}\left (3 x^{6}-4\right )^{3}+\left (5625 x^{7}-7500 x \right ) {\mathrm e}^{5 x} \mathrm {log}\left (3 x^{6}-4\right )^{2}+\left (140625 x^{8}-187500 x^{2}\right ) {\mathrm e}^{5 x} \mathrm {log}\left (3 x^{6}-4\right )+\left (1171875 x^{9}-1562500 x^{3}\right ) {\mathrm e}^{5 x}-187500 x^{6}-45000 x^{5}+250000}{\left (3 x^{6}-4\right ) \mathrm {log}\left (3 x^{6}-4\right )^{3}+\left (225 x^{7}-300 x \right ) \mathrm {log}\left (3 x^{6}-4\right )^{2}+\left (5625 x^{8}-7500 x^{2}\right ) \mathrm {log}\left (3 x^{6}-4\right )+46875 x^{9}-62500 x^{3}}d x \] Input:

int(((75*x^6-100)*exp(5*x)*log(3*x^6-4)^3+(5625*x^7-7500*x)*exp(5*x)*log(3 
*x^6-4)^2+(140625*x^8-187500*x^2)*exp(5*x)*log(3*x^6-4)+(1171875*x^9-15625 
00*x^3)*exp(5*x)-187500*x^6-45000*x^5+250000)/((3*x^6-4)*log(3*x^6-4)^3+(2 
25*x^7-300*x)*log(3*x^6-4)^2+(5625*x^8-7500*x^2)*log(3*x^6-4)+46875*x^9-62 
500*x^3),x)
 

Output:

int(((75*x^6-100)*exp(5*x)*log(3*x^6-4)^3+(5625*x^7-7500*x)*exp(5*x)*log(3 
*x^6-4)^2+(140625*x^8-187500*x^2)*exp(5*x)*log(3*x^6-4)+(1171875*x^9-15625 
00*x^3)*exp(5*x)-187500*x^6-45000*x^5+250000)/((3*x^6-4)*log(3*x^6-4)^3+(2 
25*x^7-300*x)*log(3*x^6-4)^2+(5625*x^8-7500*x^2)*log(3*x^6-4)+46875*x^9-62 
500*x^3),x)