3.31.60 \(\int \frac {\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [3060]

3.31.60.1 Optimal result

Integrand size = 53, antiderivative size = 470 \[ \int \frac {\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (4620 b c+3072 a c^4 x\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-5775 b-2688 a c^3 x\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-4096 c^5+2464 a c^2 x\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\sqrt {-b+a^2 x^2} \left (3072 c^4 \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}-2688 c^3 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+2464 c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}\right )}{6160 a c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}}-\frac {15 b \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{32 a c^{9/4}}+\frac {15 b \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{32 a c^{9/4}} \]

output

1/6160*((3072*a*c^4*x+4620*b*c)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)+(- 
2688*a*c^3*x-5775*b)*(a*x+(a^2*x^2-b)^(1/2))^(1/3)*(c+(a*x+(a^2*x^2-b)^(1/ 
2))^(1/3))^(3/4)+(-4096*c^5+2464*a*c^2*x)*(a*x+(a^2*x^2-b)^(1/2))^(2/3)*(c 
+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)+(a^2*x^2-b)^(1/2)*(3072*c^4*(c+(a*x+ 
(a^2*x^2-b)^(1/2))^(1/3))^(3/4)-2688*c^3*(a*x+(a^2*x^2-b)^(1/2))^(1/3)*(c+ 
(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)+2464*c^2*(a*x+(a^2*x^2-b)^(1/2))^(2/3 
)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)))/a/c^2/(a*x+(a^2*x^2-b)^(1/2))^ 
(2/3)-15/32*b*arctan((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1/4))/a/c^ 
(9/4)+15/32*b*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1/4))/a/c 
^(9/4)
 

3.31.60.2 Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\frac {2 \sqrt [4]{c} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4} \left (-1155 b \left (-4 c+5 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )+32 c^2 \left (96 c^2 \sqrt {-b+a^2 x^2}-84 c \sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-128 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+77 \sqrt {-b+a^2 x^2} \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+a x \left (96 c^2-84 c \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}+77 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}}-5775 b \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )+5775 b \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{12320 a c^{9/4}} \]

input

Integrate[(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

((2*c^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4)*(-1155*b*(-4*c + 
5*(a*x + Sqrt[-b + a^2*x^2])^(1/3)) + 32*c^2*(96*c^2*Sqrt[-b + a^2*x^2] - 
84*c*Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/3) - 128*c^3*(a*x + 
Sqrt[-b + a^2*x^2])^(2/3) + 77*Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2 
])^(2/3) + a*x*(96*c^2 - 84*c*(a*x + Sqrt[-b + a^2*x^2])^(1/3) + 77*(a*x + 
 Sqrt[-b + a^2*x^2])^(2/3)))))/(a*x + Sqrt[-b + a^2*x^2])^(2/3) - 5775*b*A 
rcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)] + 5775*b*ArcTa 
nh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(12320*a*c^(9/4) 
)
 

3.31.60.3 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.

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

\(\Big \downarrow \) 7299

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

input

Int[(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

$Aborted
 

3.31.60.3.1 Defintions of rubi rules used

rule 7299

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

3.31.60.4 Maple [F]

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

input

int((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((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)
 

3.31.60.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {5775 \, a c^{2} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {1}{4}} \log \left (3375 \, a^{3} c^{7} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {3}{4}} + 3375 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 5775 i \, a c^{2} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {1}{4}} \log \left (3375 i \, a^{3} c^{7} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {3}{4}} + 3375 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) + 5775 i \, a c^{2} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {1}{4}} \log \left (-3375 i \, a^{3} c^{7} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {3}{4}} + 3375 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 5775 \, a c^{2} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {1}{4}} \log \left (-3375 \, a^{3} c^{7} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {3}{4}} + 3375 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 4 \, {\left (4096 \, c^{5} - 2464 \, a c^{2} x - 2464 \, \sqrt {a^{2} x^{2} - b} c^{2} + 21 \, {\left (128 \, c^{3} + 275 \, a x - 275 \, \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {2}{3}} - 12 \, {\left (256 \, c^{4} + 385 \, a c x - 385 \, \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {3}{4}}}{24640 \, a c^{2}} \]

input

integrate((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

1/24640*(5775*a*c^2*(b^4/(a^4*c^9))^(1/4)*log(3375*a^3*c^7*(b^4/(a^4*c^9)) 
^(3/4) + 3375*b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 5775*I*a* 
c^2*(b^4/(a^4*c^9))^(1/4)*log(3375*I*a^3*c^7*(b^4/(a^4*c^9))^(3/4) + 3375* 
b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) + 5775*I*a*c^2*(b^4/(a^4* 
c^9))^(1/4)*log(-3375*I*a^3*c^7*(b^4/(a^4*c^9))^(3/4) + 3375*b^3*(c + (a*x 
 + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 5775*a*c^2*(b^4/(a^4*c^9))^(1/4)*log 
(-3375*a^3*c^7*(b^4/(a^4*c^9))^(3/4) + 3375*b^3*(c + (a*x + sqrt(a^2*x^2 - 
 b))^(1/3))^(1/4)) - 4*(4096*c^5 - 2464*a*c^2*x - 2464*sqrt(a^2*x^2 - b)*c 
^2 + 21*(128*c^3 + 275*a*x - 275*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - 
b))^(2/3) - 12*(256*c^4 + 385*a*c*x - 385*sqrt(a^2*x^2 - b)*c)*(a*x + sqrt 
(a^2*x^2 - b))^(1/3))*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(3/4))/(a*c^2)
 

3.31.60.6 Sympy [F]

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

input

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

3.31.60.7 Maxima [F]

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

input

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

3.31.60.8 Giac [F(-1)]

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

input

integrate((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
 

3.31.60.9 Mupad [F(-1)]

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

input

int((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((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)