\(\int \frac {-x+3 x^5}{(1+x^4) (a-x+a x^4) \sqrt {x+x^5}} \, dx\) [626]

Optimal result

Integrand size = 38, antiderivative size = 49 \[ \int \frac {-x+3 x^5}{\left (1+x^4\right ) \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\frac {2 \sqrt {x+x^5}}{1+x^4}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} \left (1+x^4\right )}\right ) \] Output:

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

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

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

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 = 0.40 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a}\, \sqrt {x \left (x^{4}+1\right )}}{x}\right ) \sqrt {x \left (x^{4}+1\right )}+2 x}{\sqrt {x \left (x^{4}+1\right )}}\)	\(45\)

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.67 \[ \int \frac {-x+3 x^5}{\left (1+x^4\right ) \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\left [\frac {{\left (x^{4} + 1\right )} \sqrt {a} \log \left (\frac {a^{2} x^{8} + 2 \, a^{2} x^{4} + 6 \, a x^{5} - 4 \, {\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{8} + 2 \, a^{2} x^{4} - 2 \, a x^{5} + a^{2} - 2 \, a x + x^{2}}\right ) + 4 \, \sqrt {x^{5} + x}}{2 \, {\left (x^{4} + 1\right )}}, \frac {{\left (x^{4} + 1\right )} \sqrt {-a} \arctan \left (\frac {{\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \sqrt {-a}}{2 \, {\left (a x^{5} + a x\right )}}\right ) + 2 \, \sqrt {x^{5} + x}}{x^{4} + 1}\right ] \] 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 = 8.64 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.12 \[ \int \frac {-x+3 x^5}{\left (1+x^4\right ) \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\frac {2\,\sqrt {x^5+x}}{x^4+1}+\sqrt {a}\,\ln \left (\frac {a+x-2\,\sqrt {a}\,\sqrt {x^5+x}+a\,x^4}{a\,x^4-x+a}\right ) \] Input:

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

Output:

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

Reduce [F]

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