\(\int \frac {3 b+a x^4}{(-b-x^3+a x^4) \sqrt [4]{-b x+a x^5}} \, dx\) [763]

Optimal result

Integrand size = 40, antiderivative size = 59 \[ \int \frac {3 b+a x^4}{\left (-b-x^3+a x^4\right ) \sqrt [4]{-b x+a x^5}} \, dx=-2 \arctan \left (\frac {\left (-b x+a x^5\right )^{3/4}}{-b+a x^4}\right )-2 \text {arctanh}\left (\frac {\left (-b x+a x^5\right )^{3/4}}{-b+a x^4}\right ) \] 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 [A] (verified)

Time = 11.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {3 b+a x^4}{\left (-b-x^3+a x^4\right ) \sqrt [4]{-b x+a x^5}} \, dx=-2 \arctan \left (\frac {\left (-b x+a x^5\right )^{3/4}}{-b+a x^4}\right )-2 \text {arctanh}\left (\frac {\left (-b x+a x^5\right )^{3/4}}{-b+a x^4}\right ) \] Input:

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

Output:

-2*ArcTan[(-(b*x) + a*x^5)^(3/4)/(-b + a*x^4)] - 2*ArcTanh[(-(b*x) + a*x^5 
)^(3/4)/(-b + a*x^4)]
 

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

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

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 = 0.53 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\ln \left (\frac {{\left (x \left (a \,x^{4}-b \right )\right )}^{\frac {1}{4}}-x}{x}\right )-\ln \left (\frac {x +{\left (x \left (a \,x^{4}-b \right )\right )}^{\frac {1}{4}}}{x}\right )+2 \arctan \left (\frac {{\left (x \left (a \,x^{4}-b \right )\right )}^{\frac {1}{4}}}{x}\right )\)	\(66\)

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

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

Output:

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