3.22.55 \(\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [2155]

3.22.55.1 Optimal result

Integrand size = 68, antiderivative size = 157 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=-\frac {3 \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{a c \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}-\frac {3 \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{2 a c^{5/4}}+\frac {3 \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{2 a c^{5/4}} \]

output

-3*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)/a/c/(a*x+(a^2*x^2-b)^(1/2))^(1/ 
3)-3/2*arctan((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1/4))/a/c^(5/4)+3 
/2*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1/4))/a/c^(5/4)
 

3.22.55.2 Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {3 \left (-\frac {2 \sqrt [4]{c} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}-\arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )\right )}{2 a c^{5/4}} \]

input

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

output

(3*((-2*c^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(a*x + Sqrt[ 
-b + a^2*x^2])^(1/3) - ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4) 
/c^(1/4)] + ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)]) 
)/(2*a*c^(5/4))
 

3.22.55.3 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.

\(\displaystyle \int \frac {1}{\sqrt {a^2 x^2-b} \sqrt [3]{\sqrt {a^2 x^2-b}+a x} \sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}} \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {1}{\sqrt {a^2 x^2-b} \sqrt [3]{\sqrt {a^2 x^2-b}+a x} \sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}dx\)

input

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

output

$Aborted
 

3.22.55.3.1 Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.22.55.4 Maple [F]

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

input

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

output

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

3.22.55.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.89 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {3 \, {\left (a b c \left (\frac {1}{a^{4} c^{5}}\right )^{\frac {1}{4}} \log \left (a^{3} c^{4} \left (\frac {1}{a^{4} c^{5}}\right )^{\frac {3}{4}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - i \, a b c \left (\frac {1}{a^{4} c^{5}}\right )^{\frac {1}{4}} \log \left (i \, a^{3} c^{4} \left (\frac {1}{a^{4} c^{5}}\right )^{\frac {3}{4}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) + i \, a b c \left (\frac {1}{a^{4} c^{5}}\right )^{\frac {1}{4}} \log \left (-i \, a^{3} c^{4} \left (\frac {1}{a^{4} c^{5}}\right )^{\frac {3}{4}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - a b c \left (\frac {1}{a^{4} c^{5}}\right )^{\frac {1}{4}} \log \left (-a^{3} c^{4} \left (\frac {1}{a^{4} c^{5}}\right )^{\frac {3}{4}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 4 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {2}{3}} {\left (a x - \sqrt {a^{2} x^{2} - b}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {3}{4}}\right )}}{4 \, a b c} \]

input

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

output

3/4*(a*b*c*(1/(a^4*c^5))^(1/4)*log(a^3*c^4*(1/(a^4*c^5))^(3/4) + (c + (a*x 
 + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - I*a*b*c*(1/(a^4*c^5))^(1/4)*log(I*a^ 
3*c^4*(1/(a^4*c^5))^(3/4) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) + 
 I*a*b*c*(1/(a^4*c^5))^(1/4)*log(-I*a^3*c^4*(1/(a^4*c^5))^(3/4) + (c + (a* 
x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - a*b*c*(1/(a^4*c^5))^(1/4)*log(-a^3* 
c^4*(1/(a^4*c^5))^(3/4) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 4 
*(a*x + sqrt(a^2*x^2 - b))^(2/3)*(a*x - sqrt(a^2*x^2 - b))*(c + (a*x + sqr 
t(a^2*x^2 - b))^(1/3))^(3/4))/(a*b*c)
 

3.22.55.6 Sympy [F]

\[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{\sqrt [4]{c + \sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}\, dx \]

input

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

output

Integral(1/((c + (a*x + sqrt(a**2*x**2 - b))**(1/3))**(1/4)*(a*x + sqrt(a* 
*2*x**2 - b))**(1/3)*sqrt(a**2*x**2 - b)), x)
 

3.22.55.7 Maxima [F]

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

input

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

output

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

3.22.55.8 Giac [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.22.55.9 Mupad [B] (verification not implemented)

Time = 6.61 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=-\frac {12\,{\left (\frac {c}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {5}{4};\ \frac {9}{4};\ -\frac {c}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}}\right )}{5\,a\,{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\right )}^{1/4}} \]

input

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

output

-(12*(c/(a*x + (a^2*x^2 - b)^(1/2))^(1/3) + 1)^(1/4)*hypergeom([1/4, 5/4], 
 9/4, -c/(a*x + (a^2*x^2 - b)^(1/2))^(1/3)))/(5*a*(a*x + (a^2*x^2 - b)^(1/ 
2))^(1/3)*(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/3))^(1/4))