\(\int \frac {x^4 (-9+4 x^5)}{\sqrt {-x+x^6} (a-a x^5+x^9)} \, dx\) [339]

Optimal result

Integrand size = 35, antiderivative size = 28 \[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (a-a x^5+x^9\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {x^5}{\sqrt {a} \sqrt {-x+x^6}}\right )}{\sqrt {a}} \] Output:

-2*arctanh(x^5/a^(1/2)/(x^6-x)^(1/2))/a^(1/2)
 

Mathematica [F]

\[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (a-a x^5+x^9\right )} \, dx=\int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (a-a x^5+x^9\right )} \, dx \] Input:

Integrate[(x^4*(-9 + 4*x^5))/(Sqrt[-x + x^6]*(a - a*x^5 + x^9)),x]
 

Output:

Integrate[(x^4*(-9 + 4*x^5))/(Sqrt[-x + x^6]*(a - a*x^5 + x^9)), 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.

\(\displaystyle \int \frac {x^4 \left (4 x^5-9\right )}{\sqrt {x^6-x} \left (-a x^5+a+x^9\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt {x^5-1} \int -\frac {x^{7/2} \left (9-4 x^5\right )}{\sqrt {x^5-1} \left (x^9-a x^5+a\right )}dx}{\sqrt {x^6-x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {x} \sqrt {x^5-1} \int \frac {x^{7/2} \left (9-4 x^5\right )}{\sqrt {x^5-1} \left (x^9-a x^5+a\right )}dx}{\sqrt {x^6-x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \int \frac {x^4 \left (9-4 x^5\right )}{\sqrt {x^5-1} \left (x^9-a x^5+a\right )}d\sqrt {x}}{\sqrt {x^6-x}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \int \left (\frac {-4 a x^5+9 x^4+4 a}{\sqrt {x^5-1} \left (x^9-a x^5+a\right )}-\frac {4}{\sqrt {x^5-1}}\right )d\sqrt {x}}{\sqrt {x^6-x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \left (-4 a \int \frac {1}{\sqrt {x^5-1} \left (-x^9+a x^5-a\right )}d\sqrt {x}+4 a \int \frac {x^5}{\sqrt {x^5-1} \left (-x^9+a x^5-a\right )}d\sqrt {x}+9 \int \frac {x^4}{\sqrt {x^5-1} \left (x^9-a x^5+a\right )}d\sqrt {x}-\frac {4 \sqrt {x} \sqrt {1-x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{10},\frac {1}{2},\frac {11}{10},x^5\right )}{\sqrt {x^5-1}}\right )}{\sqrt {x^6-x}}\)

Input:

Int[(x^4*(-9 + 4*x^5))/(Sqrt[-x + x^6]*(a - a*x^5 + x^9)),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {x^{4} \left (4 x^{5}-9\right )}{\sqrt {x^{6}-x}\, \left (x^{9}-a \,x^{5}+a \right )}d x\]

Input:

int(x^4*(4*x^5-9)/(x^6-x)^(1/2)/(x^9-a*x^5+a),x)
 

Output:

int(x^4*(4*x^5-9)/(x^6-x)^(1/2)/(x^9-a*x^5+a),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (22) = 44\).

Time = 0.47 (sec) , antiderivative size = 160, normalized size of antiderivative = 5.71 \[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (a-a x^5+x^9\right )} \, dx=\left [\frac {\log \left (-\frac {x^{18} + 6 \, a x^{14} + a^{2} x^{10} - 6 \, a x^{9} - 2 \, a^{2} x^{5} - 4 \, {\left (x^{13} + a x^{9} - a x^{4}\right )} \sqrt {x^{6} - x} \sqrt {a} + a^{2}}{x^{18} - 2 \, a x^{14} + a^{2} x^{10} + 2 \, a x^{9} - 2 \, a^{2} x^{5} + a^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (x^{9} + a x^{5} - a\right )} \sqrt {x^{6} - x} \sqrt {-a}}{2 \, {\left (a x^{10} - a x^{5}\right )}}\right )}{a}\right ] \] Input:

integrate(x^4*(4*x^5-9)/(x^6-x)^(1/2)/(x^9-a*x^5+a),x, algorithm="fricas")
 

Output:

[1/2*log(-(x^18 + 6*a*x^14 + a^2*x^10 - 6*a*x^9 - 2*a^2*x^5 - 4*(x^13 + a* 
x^9 - a*x^4)*sqrt(x^6 - x)*sqrt(a) + a^2)/(x^18 - 2*a*x^14 + a^2*x^10 + 2* 
a*x^9 - 2*a^2*x^5 + a^2))/sqrt(a), sqrt(-a)*arctan(1/2*(x^9 + a*x^5 - a)*s 
qrt(x^6 - x)*sqrt(-a)/(a*x^10 - a*x^5))/a]
 

Sympy [F]

\[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (a-a x^5+x^9\right )} \, dx=\int \frac {x^{4} \cdot \left (4 x^{5} - 9\right )}{\sqrt {x \left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (- a x^{5} + a + x^{9}\right )}\, dx \] Input:

integrate(x**4*(4*x**5-9)/(x**6-x)**(1/2)/(x**9-a*x**5+a),x)
 

Output:

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

Maxima [F]

\[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (a-a x^5+x^9\right )} \, dx=\int { \frac {{\left (4 \, x^{5} - 9\right )} x^{4}}{{\left (x^{9} - a x^{5} + a\right )} \sqrt {x^{6} - x}} \,d x } \] Input:

integrate(x^4*(4*x^5-9)/(x^6-x)^(1/2)/(x^9-a*x^5+a),x, algorithm="maxima")
 

Output:

integrate((4*x^5 - 9)*x^4/((x^9 - a*x^5 + a)*sqrt(x^6 - x)), x)
 

Giac [F]

\[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (a-a x^5+x^9\right )} \, dx=\int { \frac {{\left (4 \, x^{5} - 9\right )} x^{4}}{{\left (x^{9} - a x^{5} + a\right )} \sqrt {x^{6} - x}} \,d x } \] Input:

integrate(x^4*(4*x^5-9)/(x^6-x)^(1/2)/(x^9-a*x^5+a),x, algorithm="giac")
 

Output:

integrate((4*x^5 - 9)*x^4/((x^9 - a*x^5 + a)*sqrt(x^6 - x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (a-a x^5+x^9\right )} \, dx=\int \frac {x^4\,\left (4\,x^5-9\right )}{\sqrt {x^6-x}\,\left (x^9-a\,x^5+a\right )} \,d x \] Input:

int((x^4*(4*x^5 - 9))/((x^6 - x)^(1/2)*(a - a*x^5 + x^9)),x)
 

Output:

int((x^4*(4*x^5 - 9))/((x^6 - x)^(1/2)*(a - a*x^5 + x^9)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (a-a x^5+x^9\right )} \, dx=-4 \left (\int \frac {\sqrt {x}\, \sqrt {x^{5}-1}\, x^{8}}{-x^{14}+a \,x^{10}+x^{9}-2 a \,x^{5}+a}d x \right )+9 \left (\int \frac {\sqrt {x}\, \sqrt {x^{5}-1}\, x^{3}}{-x^{14}+a \,x^{10}+x^{9}-2 a \,x^{5}+a}d x \right ) \] Input:

int(x^4*(4*x^5-9)/(x^6-x)^(1/2)/(x^9-a*x^5+a),x)
 

Output:

 - 4*int((sqrt(x)*sqrt(x**5 - 1)*x**8)/(a*x**10 - 2*a*x**5 + a - x**14 + x 
**9),x) + 9*int((sqrt(x)*sqrt(x**5 - 1)*x**3)/(a*x**10 - 2*a*x**5 + a - x* 
*14 + x**9),x)