3.25.11 \(\int \frac {1}{\sqrt {-b+a^2 x^2} (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}})^{2/3}} \, dx\) [2411]

3.25.11.1 Optimal result

Integrand size = 45, antiderivative size = 193 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx=-\frac {4 \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{a c^{2/3}}+\frac {4 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{a c^{2/3}}-\frac {2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{a c^{2/3}} \]

output

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

3.25.11.2 Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx=-\frac {2 \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )+\log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )\right )}{a c^{2/3}} \]

input

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

output

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

3.25.11.3 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.

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

\(\Big \downarrow \) 7299

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

input

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

output

$Aborted
 

3.25.11.3.1 Defintions of rubi rules used

rule 7299

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

3.25.11.4 Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(\frac {\frac {4 \ln \left (-c^{\frac {1}{3}}+{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right )}{c^{\frac {2}{3}}}-\frac {2 \ln \left (c^{\frac {2}{3}}+c^{\frac {1}{3}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}+{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}{c^{\frac {2}{3}}}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{c^{\frac {1}{3}}}+1\right )}{3}\right )}{c^{\frac {2}{3}}}}{a}\)	\(145\)
default	\(\frac {\frac {4 \ln \left (-c^{\frac {1}{3}}+{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right )}{c^{\frac {2}{3}}}-\frac {2 \ln \left (c^{\frac {2}{3}}+c^{\frac {1}{3}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}+{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}{c^{\frac {2}{3}}}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{c^{\frac {1}{3}}}+1\right )}{3}\right )}{c^{\frac {2}{3}}}}{a}\)	\(145\)

input

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

output

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

3.25.11.5 Fricas [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {3} {\left (c^{2}\right )}^{\frac {1}{6}} c \arctan \left (\frac {\sqrt {3} \sqrt {c^{2}} c + 2 \, \sqrt {3} {\left (c^{2}\right )}^{\frac {5}{6}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{3 \, c^{2}}\right ) + {\left (c^{2}\right )}^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} c + {\left (c^{2}\right )}^{\frac {1}{3}} c + {\left (c^{2}\right )}^{\frac {2}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) - 2 \, {\left (c^{2}\right )}^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c - {\left (c^{2}\right )}^{\frac {2}{3}}\right )\right )}}{a c^{2}} \]

input

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

output

-2*(2*sqrt(3)*(c^2)^(1/6)*c*arctan(1/3*(sqrt(3)*sqrt(c^2)*c + 2*sqrt(3)*(c 
^2)^(5/6)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3))/c^2) + (c^2)^(2/3)* 
log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3)*c + (c^2)^(1/3)*c + (c^2)^ 
(2/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)) - 2*(c^2)^(2/3)*log((c 
+ (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c - (c^2)^(2/3)))/(a*c^2)
 

3.25.11.6 Sympy [F]

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

input

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

output

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

3.25.11.7 Maxima [F]

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

input

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

output

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

3.25.11.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.25.11.9 Mupad [F(-1)]

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

input

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

output

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