\(\int \frac {-3 b+a x^4}{(b-x^3+a x^4) \sqrt [4]{b x+a x^5}} \, dx\) [675]

Optimal result

Integrand size = 37, antiderivative size = 53 \[ \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx=-2 \arctan \left (\frac {\left (b x+a x^5\right )^{3/4}}{b+a x^4}\right )-2 \text {arctanh}\left (\frac {\left (b x+a x^5\right )^{3/4}}{b+a x^4}\right ) \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^4+b} \int \frac {\sqrt {x} \left (3 b-a x^4\right )}{\sqrt [4]{a x^4+b} \left (a x^4-x^3+b\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^5+b x}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^4+b} \int \left (\frac {\sqrt {x} \left (4 b-x^3\right )}{\sqrt [4]{a x^4+b} \left (a x^4-x^3+b\right )}-\frac {\sqrt {x}}{\sqrt [4]{a x^4+b}}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^5+b x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^4+b} \left (4 b \int \frac {\sqrt {x}}{\sqrt [4]{a x^4+b} \left (a x^4-x^3+b\right )}d\sqrt [4]{x}-\int \frac {x^{7/2}}{\sqrt [4]{a x^4+b} \left (a x^4-x^3+b\right )}d\sqrt [4]{x}-\frac {x^{3/4} \sqrt [4]{\frac {a x^4}{b}+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {1}{4},\frac {19}{16},-\frac {a x^4}{b}\right )}{3 \sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a x^5+b x}}\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\ln \left (\frac {{\left (x \left (a \,x^{4}+b \right )\right )}^{\frac {1}{4}}-x}{x}\right )-\ln \left (\frac {x +{\left (x \left (a \,x^{4}+b \right )\right )}^{\frac {1}{4}}}{x}\right )+2 \arctan \left (\frac {{\left (x \left (a \,x^{4}+b \right )\right )}^{\frac {1}{4}}}{x}\right )\)	\(60\)

Input:

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

Output:

ln(((x*(a*x^4+b))^(1/4)-x)/x)-ln((x+(x*(a*x^4+b))^(1/4))/x)+2*arctan((x*(a 
*x^4+b))^(1/4)/x)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx=\left (\int \frac {x^{4}}{x^{\frac {17}{4}} \left (a \,x^{4}+b \right )^{\frac {1}{4}} a +x^{\frac {1}{4}} \left (a \,x^{4}+b \right )^{\frac {1}{4}} b -x^{\frac {13}{4}} \left (a \,x^{4}+b \right )^{\frac {1}{4}}}d x \right ) a -3 \left (\int \frac {1}{x^{\frac {17}{4}} \left (a \,x^{4}+b \right )^{\frac {1}{4}} a +x^{\frac {1}{4}} \left (a \,x^{4}+b \right )^{\frac {1}{4}} b -x^{\frac {13}{4}} \left (a \,x^{4}+b \right )^{\frac {1}{4}}}d x \right ) b \] Input:

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

Output:

int(x**4/(x**(1/4)*(a*x**4 + b)**(1/4)*a*x**4 + x**(1/4)*(a*x**4 + b)**(1/ 
4)*b - x**(1/4)*(a*x**4 + b)**(1/4)*x**3),x)*a - 3*int(1/(x**(1/4)*(a*x**4 
 + b)**(1/4)*a*x**4 + x**(1/4)*(a*x**4 + b)**(1/4)*b - x**(1/4)*(a*x**4 + 
b)**(1/4)*x**3),x)*b