3.9.40 \(\int \frac {\sqrt {-1+x^2+x^5} (2+3 x^5)}{1+x^4-2 x^5+x^{10}} \, dx\) [840]

3.9.40.1 Optimal result

Integrand size = 35, antiderivative size = 63 \[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=-\sqrt {1+i} \arctan \left (\frac {\sqrt {-1-i} x}{\sqrt {-1+x^2+x^5}}\right )-\sqrt {1-i} \arctan \left (\frac {\sqrt {-1+i} x}{\sqrt {-1+x^2+x^5}}\right ) \]

output

-(1+I)^(1/2)*arctan((-1-I)^(1/2)*x/(x^5+x^2-1)^(1/2))-(1-I)^(1/2)*arctan(( 
-1+I)^(1/2)*x/(x^5+x^2-1)^(1/2))
 

3.9.40.2 Mathematica [A] (verified)

Time = 1.52 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=-\sqrt {1+i} \arctan \left (\frac {\sqrt {-1-i} x}{\sqrt {-1+x^2+x^5}}\right )-\sqrt {1-i} \arctan \left (\frac {\sqrt {-1+i} x}{\sqrt {-1+x^2+x^5}}\right ) \]

input

Integrate[(Sqrt[-1 + x^2 + x^5]*(2 + 3*x^5))/(1 + x^4 - 2*x^5 + x^10),x]
 

output

-(Sqrt[1 + I]*ArcTan[(Sqrt[-1 - I]*x)/Sqrt[-1 + x^2 + x^5]]) - Sqrt[1 - I] 
*ArcTan[(Sqrt[-1 + I]*x)/Sqrt[-1 + x^2 + x^5]]
 

3.9.40.3 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.

\(\displaystyle \int \frac {\sqrt {x^5+x^2-1} \left (3 x^5+2\right )}{x^{10}-2 x^5+x^4+1} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \sqrt {x^5+x^2-1} x^5}{x^{10}-2 x^5+x^4+1}+\frac {2 \sqrt {x^5+x^2-1}}{x^{10}-2 x^5+x^4+1}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \int \frac {\sqrt {x^5+x^2-1}}{x^{10}-2 x^5+x^4+1}dx+3 \int \frac {x^5 \sqrt {x^5+x^2-1}}{x^{10}-2 x^5+x^4+1}dx\)

input

Int[(Sqrt[-1 + x^2 + x^5]*(2 + 3*x^5))/(1 + x^4 - 2*x^5 + x^10),x]
 

output

$Aborted
 

3.9.40.3.1 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] 
]
 

3.9.40.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(269\) vs. \(2(51)=102\).

Time = 14.96 (sec) , antiderivative size = 270, normalized size of antiderivative = 4.29

method	result	size
pseudoelliptic	\(\frac {-\sqrt {2}\, \ln \left (\frac {x^{5}+\sqrt {2}\, x^{2}+\sqrt {x^{5}+x^{2}-1}\, \sqrt {2+2 \sqrt {2}}\, x +x^{2}-1}{x^{2}}\right )+\sqrt {2}\, \ln \left (\frac {x^{5}+\sqrt {2}\, x^{2}-\sqrt {x^{5}+x^{2}-1}\, \sqrt {2+2 \sqrt {2}}\, x +x^{2}-1}{x^{2}}\right )-2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x -2 \sqrt {x^{5}+x^{2}-1}}{x \sqrt {-2+2 \sqrt {2}}}\right )+\ln \left (\frac {x^{5}+\sqrt {2}\, x^{2}+\sqrt {x^{5}+x^{2}-1}\, \sqrt {2+2 \sqrt {2}}\, x +x^{2}-1}{x^{2}}\right )+2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x +2 \sqrt {x^{5}+x^{2}-1}}{x \sqrt {-2+2 \sqrt {2}}}\right )-\ln \left (\frac {x^{5}+\sqrt {2}\, x^{2}-\sqrt {x^{5}+x^{2}-1}\, \sqrt {2+2 \sqrt {2}}\, x +x^{2}-1}{x^{2}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}\)	\(270\)
trager	\(\operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3} x^{5}+8 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{5} x^{2}-\operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) x^{5}+2 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3} x^{2}+4 \sqrt {x^{5}+x^{2}-1}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x +2 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3}-\operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) x^{2}+\sqrt {x^{5}+x^{2}-1}\, x +\operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )}{x^{5}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{2}+x^{2}-1}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{5}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{4} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{2}-4 \sqrt {x^{5}+x^{2}-1}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x -\operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right )-\sqrt {x^{5}+x^{2}-1}\, x}{-x^{5}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{2}+x^{2}+1}\right )}{2}\)	\(486\)

input

int((x^5+x^2-1)^(1/2)*(3*x^5+2)/(x^10-2*x^5+x^4+1),x,method=_RETURNVERBOSE 
)
 

output

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

3.9.40.5 Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (43) = 86\).

Time = 0.29 (sec) , antiderivative size = 301, normalized size of antiderivative = 4.78 \[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=\frac {1}{4} \, \sqrt {-i - 1} \log \left (\frac {x^{10} + \left (2 i + 2\right ) \, x^{7} - 2 \, x^{5} + \left (2 i - 1\right ) \, x^{4} - 2 \, \sqrt {-i - 1} {\left (i \, x^{6} - x^{3} - i \, x\right )} \sqrt {x^{5} + x^{2} - 1} - \left (2 i + 2\right ) \, x^{2} + 1}{x^{10} - 2 \, x^{5} + x^{4} + 1}\right ) - \frac {1}{4} \, \sqrt {-i - 1} \log \left (\frac {x^{10} + \left (2 i + 2\right ) \, x^{7} - 2 \, x^{5} + \left (2 i - 1\right ) \, x^{4} - 2 \, \sqrt {-i - 1} {\left (-i \, x^{6} + x^{3} + i \, x\right )} \sqrt {x^{5} + x^{2} - 1} - \left (2 i + 2\right ) \, x^{2} + 1}{x^{10} - 2 \, x^{5} + x^{4} + 1}\right ) - \frac {1}{4} \, \sqrt {i - 1} \log \left (\frac {x^{10} - \left (2 i - 2\right ) \, x^{7} - 2 \, x^{5} - \left (2 i + 1\right ) \, x^{4} - 2 \, \sqrt {i - 1} {\left (i \, x^{6} + x^{3} - i \, x\right )} \sqrt {x^{5} + x^{2} - 1} + \left (2 i - 2\right ) \, x^{2} + 1}{x^{10} - 2 \, x^{5} + x^{4} + 1}\right ) + \frac {1}{4} \, \sqrt {i - 1} \log \left (\frac {x^{10} - \left (2 i - 2\right ) \, x^{7} - 2 \, x^{5} - \left (2 i + 1\right ) \, x^{4} - 2 \, \sqrt {i - 1} {\left (-i \, x^{6} - x^{3} + i \, x\right )} \sqrt {x^{5} + x^{2} - 1} + \left (2 i - 2\right ) \, x^{2} + 1}{x^{10} - 2 \, x^{5} + x^{4} + 1}\right ) \]

input

integrate((x^5+x^2-1)^(1/2)*(3*x^5+2)/(x^10-2*x^5+x^4+1),x, algorithm="fri 
cas")
 

output

1/4*sqrt(-I - 1)*log((x^10 + (2*I + 2)*x^7 - 2*x^5 + (2*I - 1)*x^4 - 2*sqr 
t(-I - 1)*(I*x^6 - x^3 - I*x)*sqrt(x^5 + x^2 - 1) - (2*I + 2)*x^2 + 1)/(x^ 
10 - 2*x^5 + x^4 + 1)) - 1/4*sqrt(-I - 1)*log((x^10 + (2*I + 2)*x^7 - 2*x^ 
5 + (2*I - 1)*x^4 - 2*sqrt(-I - 1)*(-I*x^6 + x^3 + I*x)*sqrt(x^5 + x^2 - 1 
) - (2*I + 2)*x^2 + 1)/(x^10 - 2*x^5 + x^4 + 1)) - 1/4*sqrt(I - 1)*log((x^ 
10 - (2*I - 2)*x^7 - 2*x^5 - (2*I + 1)*x^4 - 2*sqrt(I - 1)*(I*x^6 + x^3 - 
I*x)*sqrt(x^5 + x^2 - 1) + (2*I - 2)*x^2 + 1)/(x^10 - 2*x^5 + x^4 + 1)) + 
1/4*sqrt(I - 1)*log((x^10 - (2*I - 2)*x^7 - 2*x^5 - (2*I + 1)*x^4 - 2*sqrt 
(I - 1)*(-I*x^6 - x^3 + I*x)*sqrt(x^5 + x^2 - 1) + (2*I - 2)*x^2 + 1)/(x^1 
0 - 2*x^5 + x^4 + 1))
 

3.9.40.6 Sympy [F]

\[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=\int \frac {\left (3 x^{5} + 2\right ) \sqrt {x^{5} + x^{2} - 1}}{x^{10} - 2 x^{5} + x^{4} + 1}\, dx \]

input

integrate((x**5+x**2-1)**(1/2)*(3*x**5+2)/(x**10-2*x**5+x**4+1),x)
 

output

Integral((3*x**5 + 2)*sqrt(x**5 + x**2 - 1)/(x**10 - 2*x**5 + x**4 + 1), x 
)
 

3.9.40.7 Maxima [F]

\[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=\int { \frac {{\left (3 \, x^{5} + 2\right )} \sqrt {x^{5} + x^{2} - 1}}{x^{10} - 2 \, x^{5} + x^{4} + 1} \,d x } \]

input

integrate((x^5+x^2-1)^(1/2)*(3*x^5+2)/(x^10-2*x^5+x^4+1),x, algorithm="max 
ima")
 

output

integrate((3*x^5 + 2)*sqrt(x^5 + x^2 - 1)/(x^10 - 2*x^5 + x^4 + 1), x)
 

3.9.40.8 Giac [F]

\[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=\int { \frac {{\left (3 \, x^{5} + 2\right )} \sqrt {x^{5} + x^{2} - 1}}{x^{10} - 2 \, x^{5} + x^{4} + 1} \,d x } \]

input

integrate((x^5+x^2-1)^(1/2)*(3*x^5+2)/(x^10-2*x^5+x^4+1),x, algorithm="gia 
c")
 

output

integrate((3*x^5 + 2)*sqrt(x^5 + x^2 - 1)/(x^10 - 2*x^5 + x^4 + 1), x)
 

3.9.40.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=\int \frac {\left (3\,x^5+2\right )\,\sqrt {x^5+x^2-1}}{x^{10}-2\,x^5+x^4+1} \,d x \]

input

int(((3*x^5 + 2)*(x^2 + x^5 - 1)^(1/2))/(x^4 - 2*x^5 + x^10 + 1),x)
 

output

int(((3*x^5 + 2)*(x^2 + x^5 - 1)^(1/2))/(x^4 - 2*x^5 + x^10 + 1), x)