3.32.21 \(\int \frac {1}{\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [3121]

3.32.21.1 Optimal result

Integrand size = 29, antiderivative size = 699 \[ \int \frac {1}{\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (8910 b c^3-19683 a c^7 x\right ) \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (-9900 b c^2+13122 a c^6 x\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (11550 b c-10935 a c^5 x\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (-15400 b+9720 a c^4 x\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\sqrt {-b+a^2 x^2} \left (-19683 c^7 \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+13122 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}-10935 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+9720 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{17820 a^2 c^4 x+17820 a c^4 \sqrt {-b+a^2 x^2}}-\frac {70 b \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{81 \sqrt {3} a c^{13/3}}-\frac {70 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{243 a c^{13/3}}+\frac {35 b \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{243 a c^{13/3}} \]

output

((-19683*a*c^7*x+8910*b*c^3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)+(1312 
2*a*c^6*x-9900*b*c^2)*(a*x+(a^2*x^2-b)^(1/2))^(1/4)*(c+(a*x+(a^2*x^2-b)^(1 
/2))^(1/4))^(2/3)+(-10935*a*c^5*x+11550*b*c)*(a*x+(a^2*x^2-b)^(1/2))^(1/2) 
*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)+(9720*a*c^4*x-15400*b)*(a*x+(a^2* 
x^2-b)^(1/2))^(3/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)+(a^2*x^2-b)^(1 
/2)*(-19683*c^7*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)+13122*c^6*(a*x+(a^ 
2*x^2-b)^(1/2))^(1/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)-10935*c^5*(a 
*x+(a^2*x^2-b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)+9720*c 
^4*(a*x+(a^2*x^2-b)^(1/2))^(3/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3))) 
/(17820*a^2*c^4*x+17820*a*c^4*(a^2*x^2-b)^(1/2))-70/243*b*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))*3^(1/2)/a/ 
c^(13/3)-70/243*b*ln(-c^(1/3)+(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3))/a/c 
^(13/3)+35/243*b*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^(13/3)
 

3.32.21.2 Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\frac {3 \sqrt [3]{c} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3} \left (243 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (-81 c^3+54 c^2 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-45 c \sqrt {a x+\sqrt {-b+a^2 x^2}}+40 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )-110 b \left (-81 c^3+90 c^2 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-105 c \sqrt {a x+\sqrt {-b+a^2 x^2}}+140 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )\right )}{a x+\sqrt {-b+a^2 x^2}}-15400 \sqrt {3} b \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )-15400 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )+7700 b \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{53460 a c^{13/3}} \]

input

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

output

((3*c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)*(243*c^4*(a*x + S 
qrt[-b + a^2*x^2])*(-81*c^3 + 54*c^2*(a*x + Sqrt[-b + a^2*x^2])^(1/4) - 45 
*c*Sqrt[a*x + Sqrt[-b + a^2*x^2]] + 40*(a*x + Sqrt[-b + a^2*x^2])^(3/4)) - 
 110*b*(-81*c^3 + 90*c^2*(a*x + Sqrt[-b + a^2*x^2])^(1/4) - 105*c*Sqrt[a*x 
 + Sqrt[-b + a^2*x^2]] + 140*(a*x + Sqrt[-b + a^2*x^2])^(3/4))))/(a*x + Sq 
rt[-b + a^2*x^2]) - 15400*Sqrt[3]*b*ArcTan[(1 + (2*(c + (a*x + Sqrt[-b + a 
^2*x^2])^(1/4))^(1/3))/c^(1/3))/Sqrt[3]] - 15400*b*Log[-c^(1/3) + (c + (a* 
x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)] + 7700*b*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)])/(53460*a*c^(13/3))
 

3.32.21.3 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.

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

\(\Big \downarrow \) 7299

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

input

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

output

$Aborted
 

3.32.21.3.1 Defintions of rubi rules used

rule 7299

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

3.32.21.4 Maple [F]

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

input

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

output

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

3.32.21.5 Fricas [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 1039, normalized size of antiderivative = 1.49 \[ \int \frac {1}{\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Too large to display} \]

input

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

output

[1/53460*(23100*sqrt(1/3)*b*c*sqrt((-c)^(1/3)/c)*log(-6*sqrt(1/3)*(a*(-c)^ 
(2/3)*x - sqrt(a^2*x^2 - b)*(-c)^(2/3))*(a*x + sqrt(a^2*x^2 - b))^(3/4)*(c 
 + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3)*sqrt((-c)^(1/3)/c) - 3*(a*(-c)^( 
2/3)*x - sqrt(1/3)*(a*c*x - sqrt(a^2*x^2 - b)*c)*sqrt((-c)^(1/3)/c) - sqrt 
(a^2*x^2 - b)*(-c)^(2/3))*(a*x + sqrt(a^2*x^2 - b))^(3/4)*(c + (a*x + sqrt 
(a^2*x^2 - b))^(1/4))^(1/3) + 3*(a*c*x - sqrt(1/3)*(a*(-c)^(1/3)*c*x - sqr 
t(a^2*x^2 - b)*(-c)^(1/3)*c)*sqrt((-c)^(1/3)/c) - sqrt(a^2*x^2 - b)*c)*(a* 
x + sqrt(a^2*x^2 - b))^(3/4) + 2*b) + 7700*b*(-c)^(2/3)*log((-c)^(2/3) - ( 
-c)^(1/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) + (c + (a*x + sqrt(a 
^2*x^2 - b))^(1/4))^(2/3)) - 15400*b*(-c)^(2/3)*log((-c)^(1/3) + (c + (a*x 
 + sqrt(a^2*x^2 - b))^(1/4))^(1/3)) - 3*(19683*c^8 - 8910*a*c^4*x + 8910*s 
qrt(a^2*x^2 - b)*c^4 - 40*(243*c^5 - 385*a*c*x + 385*sqrt(a^2*x^2 - b)*c)* 
(a*x + sqrt(a^2*x^2 - b))^(3/4) + 15*(729*c^6 - 770*a*c^2*x + 770*sqrt(a^2 
*x^2 - b)*c^2)*sqrt(a*x + sqrt(a^2*x^2 - b)) - 18*(729*c^7 - 550*a*c^3*x + 
 550*sqrt(a^2*x^2 - b)*c^3)*(a*x + sqrt(a^2*x^2 - b))^(1/4))*(c + (a*x + s 
qrt(a^2*x^2 - b))^(1/4))^(2/3))/(a*c^5), -1/53460*(46200*sqrt(1/3)*b*c*sqr 
t(-(-c)^(1/3)/c)*arctan(-sqrt(1/3)*(-c)^(1/3)*sqrt(-(-c)^(1/3)/c) + 2*sqrt 
(1/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*sqrt(-(-c)^(1/3)/c)) - 7 
700*b*(-c)^(2/3)*log((-c)^(2/3) - (-c)^(1/3)*(c + (a*x + sqrt(a^2*x^2 - b) 
)^(1/4))^(1/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3)) + 15400*b...
 

3.32.21.6 Sympy [F]

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

input

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

output

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

3.32.21.7 Maxima [F]

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

input

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

output

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

3.32.21.8 Giac [F]

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

input

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

output

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

3.32.21.9 Mupad [F(-1)]

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

input

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

output

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