\(\int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} (\frac {1}{x^2})^{2 x/3} x^4+2^{2 x/3} (\frac {1}{x^2})^{x/3} (51 x^2+70 x^3+x^3 \log (\frac {4}{x^2}))}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} (\frac {1}{x^2})^{2 x/3} x^4+2^{2 x/3} (\frac {1}{x^2})^{x/3} (48 x^2+72 x^3)} \, dx\) [713]

Optimal result

Integrand size = 135, antiderivative size = 33 \[ \int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx=x-\frac {x}{4 \left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )} \] Output:

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

Mathematica [F]

\[ \int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx=\int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx \] Input:

Integrate[(42 + 144*x + 108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 
 + 2^((2*x)/3)*(x^(-2))^(x/3)*(51*x^2 + 70*x^3 + x^3*Log[4/x^2]))/(48 + 14 
4*x + 108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 + 2^((2*x)/3)*(x^ 
(-2))^(x/3)*(48*x^2 + 72*x^3)),x]
 

Output:

Integrate[(42 + 144*x + 108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 
 + 2^((2*x)/3)*(x^(-2))^(x/3)*(51*x^2 + 70*x^3 + x^3*Log[4/x^2]))/(48 + 14 
4*x + 108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 + 2^((2*x)/3)*(x^ 
(-2))^(x/3)*(48*x^2 + 72*x^3)), 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.

\(\displaystyle \int \frac {108 x^2+3\ 2^{\frac {4 x}{3}+2} x^4 \left (\frac {1}{x^2}\right )^{2 x/3}+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (70 x^3+51 x^2+x^3 \log \left (\frac {4}{x^2}\right )\right )+144 x+42}{108 x^2+3\ 2^{\frac {4 x}{3}+2} x^4 \left (\frac {1}{x^2}\right )^{2 x/3}+2^{2 x/3} \left (72 x^3+48 x^2\right ) \left (\frac {1}{x^2}\right )^{x/3}+144 x+48} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {108 x^2+3\ 2^{\frac {4 x}{3}+2} x^4 \left (\frac {1}{x^2}\right )^{2 x/3}+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (70 x^3+51 x^2+x^3 \log \left (\frac {4}{x^2}\right )\right )+144 x+42}{12 \left (2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{12} \int \frac {2^{2 x/3} \left (\log \left (\frac {4}{x^2}\right ) x^3+70 x^3+51 x^2\right ) \left (\frac {1}{x^2}\right )^{x/3}+3\ 2^{\frac {4 x}{3}+2} x^4 \left (\frac {1}{x^2}\right )^{2 x/3}+108 x^2+144 x+42}{\left (2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{12} \int \left (\frac {\log \left (\frac {4}{x^2}\right ) x-2 x+3}{2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2}-\frac {3 \log \left (\frac {4}{x^2}\right ) x^2-6 x^2+2 \log \left (\frac {4}{x^2}\right ) x+5 x+12}{\left (2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2\right )^2}+12\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{12} \left (-12 \int \frac {1}{\left (2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2\right )^2}dx-5 \int \frac {x}{\left (2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2\right )^2}dx+6 \int \frac {x^2}{\left (2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2\right )^2}dx+3 \int \frac {1}{2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2}dx-2 \int \frac {x}{2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2}dx-4 \int \frac {\int \frac {x}{\left (2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2\right )^2}dx}{x}dx-6 \int \frac {\int \frac {x^2}{\left (2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2\right )^2}dx}{x}dx+2 \int \frac {\int \frac {x}{2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2}dx}{x}dx-2 \log \left (\frac {4}{x^2}\right ) \int \frac {x}{\left (2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2\right )^2}dx-3 \log \left (\frac {4}{x^2}\right ) \int \frac {x^2}{\left (2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2\right )^2}dx+\log \left (\frac {4}{x^2}\right ) \int \frac {x}{2^{2 x/3} \left (\frac {1}{x^2}\right )^{\frac {x}{3}-1}+3 x+2}dx+12 x\right )\)

Input:

Int[(42 + 144*x + 108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 + 2^( 
(2*x)/3)*(x^(-2))^(x/3)*(51*x^2 + 70*x^3 + x^3*Log[4/x^2]))/(48 + 144*x + 
108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 + 2^((2*x)/3)*(x^(-2))^ 
(x/3)*(48*x^2 + 72*x^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79

method	result	size
risch	\(x -\frac {x}{4 \left (3 x +\left (\frac {4}{x^{2}}\right )^{\frac {x}{3}} x^{2}+2\right )}\)	\(26\)
parallelrisch	\(\frac {12 \,{\mathrm e}^{\frac {x \ln \left (\frac {4}{x^{2}}\right )}{3}} x^{3}+36 x^{2}+21 x}{36 x +12 \,{\mathrm e}^{\frac {x \ln \left (\frac {4}{x^{2}}\right )}{3}} x^{2}+24}\)	\(48\)

Input:

int((12*x^4*exp(1/3*x*ln(4/x^2))^2+(x^3*ln(4/x^2)+70*x^3+51*x^2)*exp(1/3*x 
*ln(4/x^2))+108*x^2+144*x+42)/(12*x^4*exp(1/3*x*ln(4/x^2))^2+(72*x^3+48*x^ 
2)*exp(1/3*x*ln(4/x^2))+108*x^2+144*x+48),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx=\frac {4 \, x^{3} \left (\frac {4}{x^{2}}\right )^{\frac {1}{3} \, x} + 12 \, x^{2} + 7 \, x}{4 \, {\left (x^{2} \left (\frac {4}{x^{2}}\right )^{\frac {1}{3} \, x} + 3 \, x + 2\right )}} \] Input:

integrate((12*x^4*exp(1/3*x*log(4/x^2))^2+(x^3*log(4/x^2)+70*x^3+51*x^2)*e 
xp(1/3*x*log(4/x^2))+108*x^2+144*x+42)/(12*x^4*exp(1/3*x*log(4/x^2))^2+(72 
*x^3+48*x^2)*exp(1/3*x*log(4/x^2))+108*x^2+144*x+48),x, algorithm="fricas" 
)
 

Output:

1/4*(4*x^3*(4/x^2)^(1/3*x) + 12*x^2 + 7*x)/(x^2*(4/x^2)^(1/3*x) + 3*x + 2)
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx=x - \frac {x}{4 x^{2} e^{\frac {x \log {\left (\frac {4}{x^{2}} \right )}}{3}} + 12 x + 8} \] Input:

integrate((12*x**4*exp(1/3*x*ln(4/x**2))**2+(x**3*ln(4/x**2)+70*x**3+51*x* 
*2)*exp(1/3*x*ln(4/x**2))+108*x**2+144*x+42)/(12*x**4*exp(1/3*x*ln(4/x**2) 
)**2+(72*x**3+48*x**2)*exp(1/3*x*ln(4/x**2))+108*x**2+144*x+48),x)
 

Output:

x - x/(4*x**2*exp(x*log(4/x**2)/3) + 12*x + 8)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (25) = 50\).

Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx=\frac {{\left (12 \, x^{3} + x^{2}\right )} 2^{\frac {2}{3} \, x} + 2 \, {\left (18 \, x^{2} + 12 \, x + 1\right )} x^{\frac {2}{3} \, x}}{12 \, {\left (2^{\frac {2}{3} \, x} x^{2} + {\left (3 \, x + 2\right )} x^{\frac {2}{3} \, x}\right )}} \] Input:

integrate((12*x^4*exp(1/3*x*log(4/x^2))^2+(x^3*log(4/x^2)+70*x^3+51*x^2)*e 
xp(1/3*x*log(4/x^2))+108*x^2+144*x+42)/(12*x^4*exp(1/3*x*log(4/x^2))^2+(72 
*x^3+48*x^2)*exp(1/3*x*log(4/x^2))+108*x^2+144*x+48),x, algorithm="maxima" 
)
 

Output:

1/12*((12*x^3 + x^2)*2^(2/3*x) + 2*(18*x^2 + 12*x + 1)*x^(2/3*x))/(2^(2/3* 
x)*x^2 + (3*x + 2)*x^(2/3*x))
 

Giac [F]

\[ \int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx=\int { \frac {12 \, x^{4} \left (\frac {4}{x^{2}}\right )^{\frac {2}{3} \, x} + 108 \, x^{2} + {\left (x^{3} \log \left (\frac {4}{x^{2}}\right ) + 70 \, x^{3} + 51 \, x^{2}\right )} \left (\frac {4}{x^{2}}\right )^{\frac {1}{3} \, x} + 144 \, x + 42}{12 \, {\left (x^{4} \left (\frac {4}{x^{2}}\right )^{\frac {2}{3} \, x} + 9 \, x^{2} + 2 \, {\left (3 \, x^{3} + 2 \, x^{2}\right )} \left (\frac {4}{x^{2}}\right )^{\frac {1}{3} \, x} + 12 \, x + 4\right )}} \,d x } \] Input:

integrate((12*x^4*exp(1/3*x*log(4/x^2))^2+(x^3*log(4/x^2)+70*x^3+51*x^2)*e 
xp(1/3*x*log(4/x^2))+108*x^2+144*x+42)/(12*x^4*exp(1/3*x*log(4/x^2))^2+(72 
*x^3+48*x^2)*exp(1/3*x*log(4/x^2))+108*x^2+144*x+48),x, algorithm="giac")
 

Output:

integrate(1/12*(12*x^4*(4/x^2)^(2/3*x) + 108*x^2 + (x^3*log(4/x^2) + 70*x^ 
3 + 51*x^2)*(4/x^2)^(1/3*x) + 144*x + 42)/(x^4*(4/x^2)^(2/3*x) + 9*x^2 + 2 
*(3*x^3 + 2*x^2)*(4/x^2)^(1/3*x) + 12*x + 4), x)
 

Mupad [B] (verification not implemented)

Time = 4.46 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx=\frac {x\,\left (12\,x+2^{\frac {2\,x}{3}+2}\,x^2\,{\left (\frac {1}{x^2}\right )}^{x/3}+7\right )}{4\,\left (3\,x+2^{\frac {2\,x}{3}}\,x^2\,{\left (\frac {1}{x^2}\right )}^{x/3}+2\right )} \] Input:

int((144*x + 12*x^4*exp((2*x*log(4/x^2))/3) + 108*x^2 + exp((x*log(4/x^2)) 
/3)*(51*x^2 + 70*x^3 + x^3*log(4/x^2)) + 42)/(144*x + exp((x*log(4/x^2))/3 
)*(48*x^2 + 72*x^3) + 12*x^4*exp((2*x*log(4/x^2))/3) + 108*x^2 + 48),x)
 

Output:

(x*(12*x + 2^((2*x)/3 + 2)*x^2*(1/x^2)^(x/3) + 7))/(4*(3*x + 2^((2*x)/3)*x 
^2*(1/x^2)^(x/3) + 2))
 

Reduce [F]

\[ \int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx=\text {too large to display} \] Input:

int((12*x^4*exp(1/3*x*log(4/x^2))^2+(x^3*log(4/x^2)+70*x^3+51*x^2)*exp(1/3 
*x*log(4/x^2))+108*x^2+144*x+42)/(12*x^4*exp(1/3*x*log(4/x^2))^2+(72*x^3+4 
8*x^2)*exp(1/3*x*log(4/x^2))+108*x^2+144*x+48),x)
 

Output:

( - 162*x**((2*x)/3)*int((x**((2*x)/3)*4**(x/3)*x**3)/(9*x**((4*x)/3)*x**2 
 + 12*x**((4*x)/3)*x + 4*x**((4*x)/3) + 6*x**((2*x)/3)*4**(x/3)*x**3 + 4*x 
**((2*x)/3)*4**(x/3)*x**2 + 4**((2*x)/3)*x**4),x)*x - 108*x**((2*x)/3)*int 
((x**((2*x)/3)*4**(x/3)*x**3)/(9*x**((4*x)/3)*x**2 + 12*x**((4*x)/3)*x + 4 
*x**((4*x)/3) + 6*x**((2*x)/3)*4**(x/3)*x**3 + 4*x**((2*x)/3)*4**(x/3)*x** 
2 + 4**((2*x)/3)*x**4),x) + 243*x**((2*x)/3)*int((x**((2*x)/3)*4**(x/3)*x* 
*2)/(9*x**((4*x)/3)*x**2 + 12*x**((4*x)/3)*x + 4*x**((4*x)/3) + 6*x**((2*x 
)/3)*4**(x/3)*x**3 + 4*x**((2*x)/3)*4**(x/3)*x**2 + 4**((2*x)/3)*x**4),x)* 
x + 162*x**((2*x)/3)*int((x**((2*x)/3)*4**(x/3)*x**2)/(9*x**((4*x)/3)*x**2 
 + 12*x**((4*x)/3)*x + 4*x**((4*x)/3) + 6*x**((2*x)/3)*4**(x/3)*x**3 + 4*x 
**((2*x)/3)*4**(x/3)*x**2 + 4**((2*x)/3)*x**4),x) + 81*x**((2*x)/3)*int((x 
**((2*x)/3)*4**(x/3)*log(4/x**2)*x**3)/(9*x**((4*x)/3)*x**2 + 12*x**((4*x) 
/3)*x + 4*x**((4*x)/3) + 6*x**((2*x)/3)*4**(x/3)*x**3 + 4*x**((2*x)/3)*4** 
(x/3)*x**2 + 4**((2*x)/3)*x**4),x)*x + 54*x**((2*x)/3)*int((x**((2*x)/3)*4 
**(x/3)*log(4/x**2)*x**3)/(9*x**((4*x)/3)*x**2 + 12*x**((4*x)/3)*x + 4*x** 
((4*x)/3) + 6*x**((2*x)/3)*4**(x/3)*x**3 + 4*x**((2*x)/3)*4**(x/3)*x**2 + 
4**((2*x)/3)*x**4),x) - 48*x**((2*x)/3)*int((x**((2*x)/3)*4**(x/3)*log(x)) 
/(9*x**((4*x)/3)*x**2 + 12*x**((4*x)/3)*x + 4*x**((4*x)/3) + 6*x**((2*x)/3 
)*4**(x/3)*x**3 + 4*x**((2*x)/3)*4**(x/3)*x**2 + 4**((2*x)/3)*x**4),x)*x - 
 32*x**((2*x)/3)*int((x**((2*x)/3)*4**(x/3)*log(x))/(9*x**((4*x)/3)*x**...