3.23.54 \(\int \frac {(4+x^2+x^5) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 (-2-x^2+2 x^5)} \, dx\) [2254]

3.23.54.1 Optimal result

Integrand size = 47, antiderivative size = 170 \[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\frac {2 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x}-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{\sqrt {2} x^2-\sqrt {-2-x^2-2 x^4+2 x^5}}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{2 x^2+\sqrt {2} \sqrt {-2-x^2-2 x^4+2 x^5}}\right )}{\sqrt [4]{2}} \]

output

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

3.23.54.2 Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.94 \[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\frac {2 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x}-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{\sqrt {2} x^2-\sqrt {-2-x^2-2 x^4+2 x^5}}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {2 x \sqrt [4]{-4-2 x^2-4 x^4+4 x^5}}{2 x^2+\sqrt {-4-2 x^2-4 x^4+4 x^5}}\right )}{\sqrt [4]{2}} \]

input

Integrate[((4 + x^2 + x^5)*(-2 - x^2 - 2*x^4 + 2*x^5)^(1/4))/(x^2*(-2 - x^ 
2 + 2*x^5)),x]
 

output

(2*(-2 - x^2 - 2*x^4 + 2*x^5)^(1/4))/x - ArcTan[(2^(3/4)*x*(-2 - x^2 - 2*x 
^4 + 2*x^5)^(1/4))/(Sqrt[2]*x^2 - Sqrt[-2 - x^2 - 2*x^4 + 2*x^5])]/2^(1/4) 
 - ArcTanh[(2*x*(-4 - 2*x^2 - 4*x^4 + 4*x^5)^(1/4))/(2*x^2 + Sqrt[-4 - 2*x 
^2 - 4*x^4 + 4*x^5])]/2^(1/4)
 

3.23.54.3 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.

\(\displaystyle \int \frac {\left (x^5+x^2+4\right ) \sqrt [4]{2 x^5-2 x^4-x^2-2}}{x^2 \left (2 x^5-x^2-2\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (5 x^3-1\right ) \sqrt [4]{2 x^5-2 x^4-x^2-2}}{2 x^5-x^2-2}-\frac {2 \sqrt [4]{2 x^5-2 x^4-x^2-2}}{x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \int \frac {\sqrt [4]{2 x^5-2 x^4-x^2-2}}{x^2}dx+\int \frac {\sqrt [4]{2 x^5-2 x^4-x^2-2}}{-2 x^5+x^2+2}dx+5 \int \frac {x^3 \sqrt [4]{2 x^5-2 x^4-x^2-2}}{2 x^5-x^2-2}dx\)

input

Int[((4 + x^2 + x^5)*(-2 - x^2 - 2*x^4 + 2*x^5)^(1/4))/(x^2*(-2 - x^2 + 2* 
x^5)),x]
 

output

$Aborted
 

3.23.54.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.23.54.4 Maple [F(-1)]

Timed out.

\[\int \frac {\left (x^{5}+x^{2}+4\right ) \left (2 x^{5}-2 x^{4}-x^{2}-2\right )^{\frac {1}{4}}}{x^{2} \left (2 x^{5}-x^{2}-2\right )}d x\]

input

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

output

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

3.23.54.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 49.95 (sec) , antiderivative size = 543, normalized size of antiderivative = 3.19 \[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\frac {\left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x \log \left (\frac {\left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} - 8 i \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (-\left (2 i - 2\right ) \, x^{5} + \left (4 i - 4\right ) \, x^{4} + \left (i - 1\right ) \, x^{2} + 2 i - 2\right )} - 8 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{5} - x^{2} - 2}\right ) - \left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x \log \left (\frac {-\left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} + 8 i \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (\left (2 i + 2\right ) \, x^{5} - \left (4 i + 4\right ) \, x^{4} - \left (i + 1\right ) \, x^{2} - 2 i - 2\right )} - 8 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{5} - x^{2} - 2}\right ) + \left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x \log \left (\frac {\left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} + 8 i \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (-\left (2 i + 2\right ) \, x^{5} + \left (4 i + 4\right ) \, x^{4} + \left (i + 1\right ) \, x^{2} + 2 i + 2\right )} - 8 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{5} - x^{2} - 2}\right ) - \left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x \log \left (\frac {-\left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} - 8 i \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (\left (2 i - 2\right ) \, x^{5} - \left (4 i - 4\right ) \, x^{4} - \left (i - 1\right ) \, x^{2} - 2 i + 2\right )} - 8 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{5} - x^{2} - 2}\right ) + 64 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}}}{32 \, x} \]

input

integrate((x^5+x^2+4)*(2*x^5-2*x^4-x^2-2)^(1/4)/x^2/(2*x^5-x^2-2),x, algor 
ithm="fricas")
 

output

1/32*((I + 1)*8^(3/4)*sqrt(2)*x*log(((I + 1)*8^(3/4)*sqrt(2)*sqrt(2*x^5 - 
2*x^4 - x^2 - 2)*x^2 - 8*I*sqrt(2)*(2*x^5 - 2*x^4 - x^2 - 2)^(1/4)*x^3 + 8 
^(1/4)*sqrt(2)*(-(2*I - 2)*x^5 + (4*I - 4)*x^4 + (I - 1)*x^2 + 2*I - 2) - 
8*(2*x^5 - 2*x^4 - x^2 - 2)^(3/4)*x)/(2*x^5 - x^2 - 2)) - (I - 1)*8^(3/4)* 
sqrt(2)*x*log((-(I - 1)*8^(3/4)*sqrt(2)*sqrt(2*x^5 - 2*x^4 - x^2 - 2)*x^2 
+ 8*I*sqrt(2)*(2*x^5 - 2*x^4 - x^2 - 2)^(1/4)*x^3 + 8^(1/4)*sqrt(2)*((2*I 
+ 2)*x^5 - (4*I + 4)*x^4 - (I + 1)*x^2 - 2*I - 2) - 8*(2*x^5 - 2*x^4 - x^2 
 - 2)^(3/4)*x)/(2*x^5 - x^2 - 2)) + (I - 1)*8^(3/4)*sqrt(2)*x*log(((I - 1) 
*8^(3/4)*sqrt(2)*sqrt(2*x^5 - 2*x^4 - x^2 - 2)*x^2 + 8*I*sqrt(2)*(2*x^5 - 
2*x^4 - x^2 - 2)^(1/4)*x^3 + 8^(1/4)*sqrt(2)*(-(2*I + 2)*x^5 + (4*I + 4)*x 
^4 + (I + 1)*x^2 + 2*I + 2) - 8*(2*x^5 - 2*x^4 - x^2 - 2)^(3/4)*x)/(2*x^5 
- x^2 - 2)) - (I + 1)*8^(3/4)*sqrt(2)*x*log((-(I + 1)*8^(3/4)*sqrt(2)*sqrt 
(2*x^5 - 2*x^4 - x^2 - 2)*x^2 - 8*I*sqrt(2)*(2*x^5 - 2*x^4 - x^2 - 2)^(1/4 
)*x^3 + 8^(1/4)*sqrt(2)*((2*I - 2)*x^5 - (4*I - 4)*x^4 - (I - 1)*x^2 - 2*I 
 + 2) - 8*(2*x^5 - 2*x^4 - x^2 - 2)^(3/4)*x)/(2*x^5 - x^2 - 2)) + 64*(2*x^ 
5 - 2*x^4 - x^2 - 2)^(1/4))/x
 

3.23.54.6 Sympy [F]

\[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\int \frac {\left (x^{5} + x^{2} + 4\right ) \sqrt [4]{2 x^{5} - 2 x^{4} - x^{2} - 2}}{x^{2} \cdot \left (2 x^{5} - x^{2} - 2\right )}\, dx \]

input

integrate((x**5+x**2+4)*(2*x**5-2*x**4-x**2-2)**(1/4)/x**2/(2*x**5-x**2-2) 
,x)
 

output

Integral((x**5 + x**2 + 4)*(2*x**5 - 2*x**4 - x**2 - 2)**(1/4)/(x**2*(2*x* 
*5 - x**2 - 2)), x)
 

3.23.54.7 Maxima [F]

\[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} {\left (x^{5} + x^{2} + 4\right )}}{{\left (2 \, x^{5} - x^{2} - 2\right )} x^{2}} \,d x } \]

input

integrate((x^5+x^2+4)*(2*x^5-2*x^4-x^2-2)^(1/4)/x^2/(2*x^5-x^2-2),x, algor 
ithm="maxima")
 

output

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

3.23.54.8 Giac [F]

\[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} {\left (x^{5} + x^{2} + 4\right )}}{{\left (2 \, x^{5} - x^{2} - 2\right )} x^{2}} \,d x } \]

input

integrate((x^5+x^2+4)*(2*x^5-2*x^4-x^2-2)^(1/4)/x^2/(2*x^5-x^2-2),x, algor 
ithm="giac")
 

output

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

3.23.54.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\int -\frac {\left (x^5+x^2+4\right )\,{\left (2\,x^5-2\,x^4-x^2-2\right )}^{1/4}}{x^2\,\left (-2\,x^5+x^2+2\right )} \,d x \]

input

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

output

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