\(\int \frac {x^3 (-5 b+9 a x^4)}{\sqrt [4]{-b x+a x^5} (-2-b x^5+a x^9)} \, dx\) [751]

Optimal result

Integrand size = 43, antiderivative size = 58 \[ \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx=2^{3/4} \arctan \left (\frac {x \sqrt [4]{-b x+a x^5}}{\sqrt [4]{2}}\right )-2^{3/4} \text {arctanh}\left (\frac {x \sqrt [4]{-b x+a x^5}}{\sqrt [4]{2}}\right ) \] Output:

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

Mathematica [F]

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

Integrate[(x^3*(-5*b + 9*a*x^4))/((-(b*x) + a*x^5)^(1/4)*(-2 - b*x^5 + a*x 
^9)),x]
 

Output:

Integrate[(x^3*(-5*b + 9*a*x^4))/((-(b*x) + a*x^5)^(1/4)*(-2 - b*x^5 + a*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.

\(\displaystyle \int \frac {x^3 \left (9 a x^4-5 b\right )}{\sqrt [4]{a x^5-b x} \left (a x^9-b x^5-2\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^4-b} \int \frac {x^{11/4} \left (5 b-9 a x^4\right )}{\sqrt [4]{a x^4-b} \left (-a x^9+b x^5+2\right )}dx}{\sqrt [4]{a x^5-b x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^4-b} \int \frac {x^{7/2} \left (5 b-9 a x^4\right )}{\sqrt [4]{a x^4-b} \left (-a x^9+b x^5+2\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^5-b x}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^4-b} \int \left (\frac {9 a x^{15/2}}{\sqrt [4]{a x^4-b} \left (a x^9-b x^5-2\right )}+\frac {5 b x^{7/2}}{\sqrt [4]{a x^4-b} \left (-a x^9+b x^5+2\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^5-b x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^4-b} \left (5 b \int \frac {x^{7/2}}{\sqrt [4]{a x^4-b} \left (-a x^9+b x^5+2\right )}d\sqrt [4]{x}+9 a \int \frac {x^{15/2}}{\sqrt [4]{a x^4-b} \left (a x^9-b x^5-2\right )}d\sqrt [4]{x}\right )}{\sqrt [4]{a x^5-b x}}\)

Input:

Int[(x^3*(-5*b + 9*a*x^4))/((-(b*x) + a*x^5)^(1/4)*(-2 - b*x^5 + a*x^9)),x 
]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {x^{3} \left (9 a \,x^{4}-5 b \right )}{\left (a \,x^{5}-b x \right )^{\frac {1}{4}} \left (a \,x^{9}-b \,x^{5}-2\right )}d x\]

Input:

int(x^3*(9*a*x^4-5*b)/(a*x^5-b*x)^(1/4)/(a*x^9-b*x^5-2),x)
 

Output:

int(x^3*(9*a*x^4-5*b)/(a*x^5-b*x)^(1/4)/(a*x^9-b*x^5-2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx=\text {Timed out} \] Input:

integrate(x^3*(9*a*x^4-5*b)/(a*x^5-b*x)^(1/4)/(a*x^9-b*x^5-2),x, algorithm 
="fricas")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F]

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

integrate(x**3*(9*a*x**4-5*b)/(a*x**5-b*x)**(1/4)/(a*x**9-b*x**5-2),x)
 

Output:

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

Maxima [F]

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

integrate(x^3*(9*a*x^4-5*b)/(a*x^5-b*x)^(1/4)/(a*x^9-b*x^5-2),x, algorithm 
="maxima")
 

Output:

integrate((9*a*x^4 - 5*b)*x^3/((a*x^9 - b*x^5 - 2)*(a*x^5 - b*x)^(1/4)), x 
)
 

Giac [F]

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

integrate(x^3*(9*a*x^4-5*b)/(a*x^5-b*x)^(1/4)/(a*x^9-b*x^5-2),x, algorithm 
="giac")
 

Output:

integrate((9*a*x^4 - 5*b)*x^3/((a*x^9 - b*x^5 - 2)*(a*x^5 - b*x)^(1/4)), x 
)
 

Mupad [F(-1)]

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

int((x^3*(5*b - 9*a*x^4))/((a*x^5 - b*x)^(1/4)*(b*x^5 - a*x^9 + 2)),x)
 

Output:

int((x^3*(5*b - 9*a*x^4))/((a*x^5 - b*x)^(1/4)*(b*x^5 - a*x^9 + 2)), x)
 

Reduce [F]

\[ \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx=9 \left (\int \frac {x^{7}}{x^{\frac {37}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}} a -x^{\frac {21}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}} b -2 x^{\frac {1}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}}}d x \right ) a -5 \left (\int \frac {x^{3}}{x^{\frac {37}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}} a -x^{\frac {21}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}} b -2 x^{\frac {1}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}}}d x \right ) b \] Input:

int(x^3*(9*a*x^4-5*b)/(a*x^5-b*x)^(1/4)/(a*x^9-b*x^5-2),x)
 

Output:

9*int(x**7/(x**(1/4)*(a*x**4 - b)**(1/4)*a*x**9 - x**(1/4)*(a*x**4 - b)**( 
1/4)*b*x**5 - 2*x**(1/4)*(a*x**4 - b)**(1/4)),x)*a - 5*int(x**3/(x**(1/4)* 
(a*x**4 - b)**(1/4)*a*x**9 - x**(1/4)*(a*x**4 - b)**(1/4)*b*x**5 - 2*x**(1 
/4)*(a*x**4 - b)**(1/4)),x)*b