\(\int \frac {\sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\) [2039]

Optimal result

Integrand size = 56, antiderivative size = 145 \[ \int \frac {\sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {2 b \sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{a}-\frac {b \arctan \left (\sqrt [4]{2} \sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt [4]{2} a}-\frac {b \text {arctanh}\left (\sqrt [4]{2} \sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt [4]{2} a} \] Output:

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

Mathematica [A] (verified)

Time = 18.22 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=-\frac {b \sqrt [4]{x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \left (-2 \sqrt [4]{2} \sqrt [4]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}+\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}}{\sqrt [4]{a}}\right )+\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}}{\sqrt [4]{a}}\right )\right )}{\sqrt [4]{2} a \sqrt [4]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}} \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral((x*(a*x + b*sqrt(a**2*x**2/b**2 - a/b**2)))**(1/4)/sqrt(a*(a*x**2 
 - 1)/b**2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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