\(\int \frac {x+3 x^5}{(-1+x^4) (-a-x+a x^4) \sqrt {-x+x^5}} \, dx\) [599]

Optimal result

Integrand size = 40, antiderivative size = 47 \[ \int \frac {x+3 x^5}{\left (-1+x^4\right ) \left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=\frac {2 \sqrt {-x+x^5}}{-1+x^4}-2 \sqrt {a} \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {-x+x^5}}\right ) \] Output:

2*(x^5-x)^(1/2)/(x^4-1)-2*a^(1/2)*arctanh(x/a^(1/2)/(x^5-x)^(1/2))
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2027

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^4-1} \left (-4 \int \frac {x}{\left (x^4-1\right )^{3/2} \left (a x^4-x-a\right )}d\sqrt {x}-\frac {3 \int \frac {x^2}{\left (x^4-1\right )^{3/2} \left (a x^4-x-a\right )}d\sqrt {x}}{a}-\frac {3 \sqrt {-\frac {(-1)^{3/4} \left (\sqrt [4]{-1} x+1\right )^2}{x}} \sqrt {\frac {i \left (1-x^4\right )}{x^2}} x^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {(-1)^{3/4} \left (i \sqrt {2} x^2-2 \sqrt [4]{-1} x+\sqrt {2}\right )}{x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{8 \sqrt {2+\sqrt {2}} a \left (\sqrt [4]{-1} x+1\right ) \sqrt {x^4-1}}-\frac {3 \sqrt {\frac {(-1)^{3/4} \left (1-\sqrt [4]{-1} x\right )^2}{x}} \sqrt {\frac {i \left (1-x^4\right )}{x^2}} x^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {(-1)^{3/4} \left (i \sqrt {2} x^2+2 \sqrt [4]{-1} x+\sqrt {2}\right )}{x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{8 \sqrt {2+\sqrt {2}} a \left (1-\sqrt [4]{-1} x\right ) \sqrt {x^4-1}}+\frac {3 x^{3/2}}{4 a \sqrt {x^4-1}}\right )}{\sqrt {x^5-x}}\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a}\, \sqrt {x^{5}-x}}{x}\right ) \sqrt {x^{5}-x}+2 x}{\sqrt {x^{5}-x}}\)	\(45\)

Input:

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

Output:

2*(-a^(1/2)*arctanh(a^(1/2)*(x^5-x)^(1/2)/x)*(x^5-x)^(1/2)+x)/(x^5-x)^(1/2 
)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 193, normalized size of antiderivative = 4.11 \[ \int \frac {x+3 x^5}{\left (-1+x^4\right ) \left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=\left [\frac {{\left (x^{4} - 1\right )} \sqrt {a} \log \left (\frac {a^{2} x^{8} - 2 \, a^{2} x^{4} + 6 \, a x^{5} - 4 \, {\left (a x^{4} - a + x\right )} \sqrt {x^{5} - x} \sqrt {a} + a^{2} - 6 \, a x + x^{2}}{a^{2} x^{8} - 2 \, a^{2} x^{4} - 2 \, a x^{5} + a^{2} + 2 \, a x + x^{2}}\right ) + 4 \, \sqrt {x^{5} - x}}{2 \, {\left (x^{4} - 1\right )}}, \frac {{\left (x^{4} - 1\right )} \sqrt {-a} \arctan \left (\frac {{\left (a x^{4} - a + x\right )} \sqrt {x^{5} - x} \sqrt {-a}}{2 \, {\left (a x^{5} - a x\right )}}\right ) + 2 \, \sqrt {x^{5} - x}}{x^{4} - 1}\right ] \] Input:

integrate((3*x^5+x)/(x^4-1)/(a*x^4-a-x)/(x^5-x)^(1/2),x, algorithm="fricas 
")
 

Output:

[1/2*((x^4 - 1)*sqrt(a)*log((a^2*x^8 - 2*a^2*x^4 + 6*a*x^5 - 4*(a*x^4 - a 
+ x)*sqrt(x^5 - x)*sqrt(a) + a^2 - 6*a*x + x^2)/(a^2*x^8 - 2*a^2*x^4 - 2*a 
*x^5 + a^2 + 2*a*x + x^2)) + 4*sqrt(x^5 - x))/(x^4 - 1), ((x^4 - 1)*sqrt(- 
a)*arctan(1/2*(a*x^4 - a + x)*sqrt(x^5 - x)*sqrt(-a)/(a*x^5 - a*x)) + 2*sq 
rt(x^5 - x))/(x^4 - 1)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x+3 x^5}{\left (-1+x^4\right ) \left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

integrate((3*x^5+x)/(x^4-1)/(a*x^4-a-x)/(x^5-x)^(1/2),x, algorithm="maxima 
")
 

Output:

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

Giac [F]

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

integrate((3*x^5+x)/(x^4-1)/(a*x^4-a-x)/(x^5-x)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 7.68 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.30 \[ \int \frac {x+3 x^5}{\left (-1+x^4\right ) \left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=\frac {2\,\sqrt {x^5-x}}{x^4-1}+\sqrt {a}\,\ln \left (\frac {a-x+2\,\sqrt {a}\,\sqrt {x^5-x}-a\,x^4}{-a\,x^4+x+a}\right ) \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x+3 x^5}{\left (-1+x^4\right ) \left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx=3 \left (\int \frac {\sqrt {x}\, \sqrt {x^{4}-1}\, x^{4}}{a \,x^{12}-3 a \,x^{8}-x^{9}+3 a \,x^{4}+2 x^{5}-a -x}d x \right )+\int \frac {\sqrt {x}\, \sqrt {x^{4}-1}}{a \,x^{12}-3 a \,x^{8}-x^{9}+3 a \,x^{4}+2 x^{5}-a -x}d x \] Input:

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

Output:

3*int((sqrt(x)*sqrt(x**4 - 1)*x**4)/(a*x**12 - 3*a*x**8 + 3*a*x**4 - a - x 
**9 + 2*x**5 - x),x) + int((sqrt(x)*sqrt(x**4 - 1))/(a*x**12 - 3*a*x**8 + 
3*a*x**4 - a - x**9 + 2*x**5 - x),x)