\(\int \frac {(-3 b+a x^2) (b-a x^2+x^3)}{x^3 (-b+a x^2+x^3) \sqrt [4]{-b x+a x^3}} \, dx\) [1859]

Optimal result

Integrand size = 52, antiderivative size = 128 \[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\frac {4 \left (-b x+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b x+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b x+a x^3}}{x^2+\sqrt {-b x+a x^3}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 12.71 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\frac {4 \left (-b x+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-x^2+\sqrt {-b x+a x^3}}{\sqrt {2} x \sqrt [4]{-b x+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b x+a x^3}}{x^2+\sqrt {-b x+a x^3}}\right ) \] Input:

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

Output:

(4*(-(b*x) + a*x^3)^(3/4))/(3*x^3) - 2*Sqrt[2]*ArcTan[(-x^2 + Sqrt[-(b*x) 
+ a*x^3])/(Sqrt[2]*x*(-(b*x) + a*x^3)^(1/4))] + 2*Sqrt[2]*ArcTanh[(Sqrt[2] 
*x*(-(b*x) + a*x^3)^(1/4))/(x^2 + Sqrt[-(b*x) + a*x^3])]
 

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.

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

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^2-b} \int \frac {\left (3 b-a x^2\right ) \left (x^3-a x^2+b\right )}{x^{13/4} \sqrt [4]{a x^2-b} \left (-x^3-a x^2+b\right )}dx}{\sqrt [4]{a x^3-b x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^2-b} \int \frac {\left (3 b-a x^2\right ) \left (x^3-a x^2+b\right )}{x^{5/2} \sqrt [4]{a x^2-b} \left (-x^3-a x^2+b\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^3-b x}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^2-b} \int \left (-\frac {a}{\sqrt {x} \sqrt [4]{a x^2-b}}+\frac {3 b}{x^{5/2} \sqrt [4]{a x^2-b}}+\frac {2 \sqrt {x} \left (3 b-a x^2\right )}{\sqrt [4]{a x^2-b} \left (-x^3-a x^2+b\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^3-b x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^2-b} \left (6 b \int \frac {\sqrt {x}}{\sqrt [4]{a x^2-b} \left (-x^3-a x^2+b\right )}d\sqrt [4]{x}+2 a \int \frac {x^{5/2}}{\sqrt [4]{a x^2-b} \left (x^3+a x^2-b\right )}d\sqrt [4]{x}+\frac {a \sqrt [4]{1-\frac {a x^2}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{8},\frac {1}{4},\frac {7}{8},\frac {a x^2}{b}\right )}{\sqrt [4]{x} \sqrt [4]{a x^2-b}}-\frac {b \sqrt [4]{1-\frac {a x^2}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {9}{8},\frac {1}{4},-\frac {1}{8},\frac {a x^2}{b}\right )}{3 x^{9/4} \sqrt [4]{a x^2-b}}\right )}{\sqrt [4]{a x^3-b x}}\)

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\left (\int \frac {x^{2}}{x^{\frac {9}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} a -x^{\frac {1}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} b +x^{\frac {13}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}}}d x \right ) a -\left (\int \frac {x}{x^{\frac {9}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} a -x^{\frac {1}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} b +x^{\frac {13}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}}}d x \right ) a^{2}-3 \left (\int \frac {1}{x^{\frac {21}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} a -x^{\frac {13}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} b +x^{\frac {25}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}}}d x \right ) b^{2}+4 \left (\int \frac {1}{x^{\frac {13}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} a -x^{\frac {5}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} b +x^{\frac {17}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}}}d x \right ) a b -3 \left (\int \frac {1}{x^{\frac {9}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} a -x^{\frac {1}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} b +x^{\frac {13}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}}}d x \right ) b \] Input:

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

Output:

int(x**2/(x**(1/4)*(a*x**2 - b)**(1/4)*a*x**2 - x**(1/4)*(a*x**2 - b)**(1/ 
4)*b + x**(1/4)*(a*x**2 - b)**(1/4)*x**3),x)*a - int(x/(x**(1/4)*(a*x**2 - 
 b)**(1/4)*a*x**2 - x**(1/4)*(a*x**2 - b)**(1/4)*b + x**(1/4)*(a*x**2 - b) 
**(1/4)*x**3),x)*a**2 - 3*int(1/(x**(1/4)*(a*x**2 - b)**(1/4)*a*x**5 - x** 
(1/4)*(a*x**2 - b)**(1/4)*b*x**3 + x**(1/4)*(a*x**2 - b)**(1/4)*x**6),x)*b 
**2 + 4*int(1/(x**(1/4)*(a*x**2 - b)**(1/4)*a*x**3 - x**(1/4)*(a*x**2 - b) 
**(1/4)*b*x + x**(1/4)*(a*x**2 - b)**(1/4)*x**4),x)*a*b - 3*int(1/(x**(1/4 
)*(a*x**2 - b)**(1/4)*a*x**2 - x**(1/4)*(a*x**2 - b)**(1/4)*b + x**(1/4)*( 
a*x**2 - b)**(1/4)*x**3),x)*b