\(\int \frac {x^3 (5 b+9 a x^4)}{\sqrt [4]{b x+a x^5} (1+b x^5+a x^9)} \, dx\) [2970]

Optimal result

Integrand size = 41, antiderivative size = 371 \[ \int \frac {x^3 \left (5 b+9 a x^4\right )}{\sqrt [4]{b x+a x^5} \left (1+b x^5+a x^9\right )} \, dx=-\sqrt {2} \arctan \left (\frac {-2^{2/3} x \sqrt [4]{b x+a x^5}+x^2 \sqrt [4]{b x+a x^5}}{2 \sqrt [6]{2}-\sqrt {2} x+2^{2/3} x \sqrt [4]{b x+a x^5}-x^2 \sqrt [4]{b x+a x^5}}\right )+\sqrt {2} \arctan \left (\frac {-2^{2/3} x \sqrt [4]{b x+a x^5}+x^2 \sqrt [4]{b x+a x^5}}{-2 \sqrt [6]{2}+\sqrt {2} x+2^{2/3} x \sqrt [4]{b x+a x^5}-x^2 \sqrt [4]{b x+a x^5}}\right )-\sqrt {2} \text {arctanh}\left (\frac {-2 2^{5/6} x \sqrt [4]{b x+a x^5}+4 \sqrt [6]{2} x^2 \sqrt [4]{b x+a x^5}-\sqrt {2} x^3 \sqrt [4]{b x+a x^5}}{-2 \sqrt [3]{2}+2\ 2^{2/3} x-x^2-2 \sqrt [3]{2} x^2 \sqrt {b x+a x^5}+2\ 2^{2/3} x^3 \sqrt {b x+a x^5}-x^4 \sqrt {b x+a x^5}}\right ) \] Output:

-2^(1/2)*arctan((-2^(2/3)*x*(a*x^5+b*x)^(1/4)+x^2*(a*x^5+b*x)^(1/4))/(2*2^ 
(1/6)-x*2^(1/2)+2^(2/3)*x*(a*x^5+b*x)^(1/4)-x^2*(a*x^5+b*x)^(1/4)))+2^(1/2 
)*arctan((-2^(2/3)*x*(a*x^5+b*x)^(1/4)+x^2*(a*x^5+b*x)^(1/4))/(-2*2^(1/6)+ 
x*2^(1/2)+2^(2/3)*x*(a*x^5+b*x)^(1/4)-x^2*(a*x^5+b*x)^(1/4)))-2^(1/2)*arct 
anh((-2*2^(5/6)*x*(a*x^5+b*x)^(1/4)+4*2^(1/6)*x^2*(a*x^5+b*x)^(1/4)-2^(1/2 
)*(a*x^5+b*x)^(1/4)*x^3)/(-2*2^(1/3)+2*2^(2/3)*x-x^2-2*2^(1/3)*x^2*(a*x^5+ 
b*x)^(1/2)+2*2^(2/3)*x^3*(a*x^5+b*x)^(1/2)-x^4*(a*x^5+b*x)^(1/2)))
                                                                                    
                                                                                    
 

Mathematica [F]

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

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

Output:

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

\(\displaystyle \int \frac {x^3 \left (9 a x^4+5 b\right )}{\sqrt [4]{a x^5+b x} \left (a x^9+b x^5+1\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^4+b} \int \frac {x^{11/4} \left (9 a x^4+5 b\right )}{\sqrt [4]{a x^4+b} \left (a x^9+b x^5+1\right )}dx}{\sqrt [4]{a x^5+b x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^4+b} \int \frac {x^{7/2} \left (9 a x^4+5 b\right )}{\sqrt [4]{a x^4+b} \left (a x^9+b x^5+1\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^5+b x}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^4+b} \int \left (\frac {9 a x^{15/2}}{\sqrt [4]{a x^4+b} \left (a x^9+b x^5+1\right )}+\frac {5 b x^{7/2}}{\sqrt [4]{a x^4+b} \left (a x^9+b x^5+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^5+b x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^4+b} \left (5 b \int \frac {x^{7/2}}{\sqrt [4]{a x^4+b} \left (a x^9+b x^5+1\right )}d\sqrt [4]{x}+9 a \int \frac {x^{15/2}}{\sqrt [4]{a x^4+b} \left (a x^9+b x^5+1\right )}d\sqrt [4]{x}\right )}{\sqrt [4]{a x^5+b x}}\)

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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