3.32.23 \(\int \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx\) [3123]

3.32.23.1 Optimal result

Integrand size = 29, antiderivative size = 719 \[ \int \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {\left (2835 b c^3-19683 a c^7 x+68040 a^2 c^3 x^2\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (4095 b c^2+6561 a c^6 x\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-5460 b c-4374 a c^5 x\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (9100 b+3402 a c^4 x\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-19683 c^7+68040 a c^3 x\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+6561 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}-4374 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+3402 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{73710 a^2 c^3 x+73710 a c^3 \sqrt {-b+a^2 x^2}}-\frac {20 b \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{81 \sqrt {3} a c^{11/3}}+\frac {20 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{243 a c^{11/3}}-\frac {10 b \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{243 a c^{11/3}} \]

output

((-19683*a*c^7*x+68040*a^2*c^3*x^2+2835*b*c^3)*(c+(a*x+(a^2*x^2-b)^(1/2))^ 
(1/4))^(1/3)+(6561*a*c^6*x+4095*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))^(1/3)+(-4374*a*c^5*x-5460*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))^(1/3)+(3402*a*c^4*x+9100 
*b)*(a*x+(a^2*x^2-b)^(1/2))^(3/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)+ 
(a^2*x^2-b)^(1/2)*((-19683*c^7+68040*a*c^3*x)*(c+(a*x+(a^2*x^2-b)^(1/2))^( 
1/4))^(1/3)+6561*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))^(1/3)-4374*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))^(1/3)+3402*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))^(1/3)))/(73710*a^2*c^3*x+73710*a*c^3*(a^2*x^2-b)^(1/2 
))-20/243*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^(11/3)+20/243*b*ln(-c^(1/3)+(c+(a*x+(a^2*x^2 
-b)^(1/2))^(1/4))^(1/3))/a/c^(11/3)-10/243*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 
^(11/3)
 

3.32.23.2 Mathematica [A] (verified)

Time = 1.50 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.62 \[ \int \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {\frac {3 c^{2/3} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \left (35 b \left (81 c^3+117 c^2 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-156 c \sqrt {a x+\sqrt {-b+a^2 x^2}}+260 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )+243 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (-81 c^4+280 a x+27 c^3 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-18 c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}+14 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )\right )}{a x+\sqrt {-b+a^2 x^2}}-18200 \sqrt {3} b \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+18200 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-9100 b \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{221130 a c^{11/3}} \]

input

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

output

((3*c^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)*(35*b*(81*c^3 + 1 
17*c^2*(a*x + Sqrt[-b + a^2*x^2])^(1/4) - 156*c*Sqrt[a*x + Sqrt[-b + a^2*x 
^2]] + 260*(a*x + Sqrt[-b + a^2*x^2])^(3/4)) + 243*c^3*(a*x + Sqrt[-b + a^ 
2*x^2])*(-81*c^4 + 280*a*x + 27*c^3*(a*x + Sqrt[-b + a^2*x^2])^(1/4) - 18* 
c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]] + 14*c*(a*x + Sqrt[-b + a^2*x^2])^(3/4) 
)))/(a*x + Sqrt[-b + a^2*x^2]) - 18200*Sqrt[3]*b*ArcTan[(1 + (2*(c + (a*x 
+ Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/c^(1/3))/Sqrt[3]] + 18200*b*Log[-c^(1/ 
3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)] - 9100*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)])/(221130*a*c^(11/3))
 

3.32.23.3 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.

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

\(\Big \downarrow \) 7299

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

input

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

output

$Aborted
 

3.32.23.3.1 Defintions of rubi rules used

rule 7299

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

3.32.23.4 Maple [F]

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

input

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

output

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

3.32.23.5 Fricas [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.55 \[ \int \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=-\frac {18200 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{6}} c \arctan \left (\frac {\sqrt {3} \sqrt {c^{2}} c + 2 \, \sqrt {3} {\left (c^{2}\right )}^{\frac {5}{6}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{3 \, c^{2}}\right ) + 9100 \, b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} c + {\left (c^{2}\right )}^{\frac {1}{3}} c + {\left (c^{2}\right )}^{\frac {2}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) - 18200 \, b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c - {\left (c^{2}\right )}^{\frac {2}{3}}\right ) + 3 \, {\left (19683 \, c^{9} - 70875 \, a c^{5} x + 2835 \, \sqrt {a^{2} x^{2} - b} c^{5} - 14 \, {\left (243 \, c^{6} + 650 \, a c^{2} x - 650 \, \sqrt {a^{2} x^{2} - b} c^{2}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} + 6 \, {\left (729 \, c^{7} + 910 \, a c^{3} x - 910 \, \sqrt {a^{2} x^{2} - b} c^{3}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 9 \, {\left (729 \, c^{8} + 455 \, a c^{4} x - 455 \, \sqrt {a^{2} x^{2} - b} c^{4}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{221130 \, a c^{5}} \]

input

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

output

-1/221130*(18200*sqrt(3)*b*(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) + 
 9100*b*(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)) - 1 
8200*b*(c^2)^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c - (c^ 
2)^(2/3)) + 3*(19683*c^9 - 70875*a*c^5*x + 2835*sqrt(a^2*x^2 - b)*c^5 - 14 
*(243*c^6 + 650*a*c^2*x - 650*sqrt(a^2*x^2 - b)*c^2)*(a*x + sqrt(a^2*x^2 - 
 b))^(3/4) + 6*(729*c^7 + 910*a*c^3*x - 910*sqrt(a^2*x^2 - b)*c^3)*sqrt(a* 
x + sqrt(a^2*x^2 - b)) - 9*(729*c^8 + 455*a*c^4*x - 455*sqrt(a^2*x^2 - b)* 
c^4)*(a*x + sqrt(a^2*x^2 - b))^(1/4))*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4) 
)^(1/3))/(a*c^5)
 

3.32.23.6 Sympy [F]

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

input

integrate((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)
 

3.32.23.7 Maxima [F]

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

input

integrate((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)
 

3.32.23.8 Giac [F]

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

input

integrate((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)
 

3.32.23.9 Mupad [F(-1)]

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

input

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

output

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