Integrand size = 53, antiderivative size = 445 \[ \int \frac {1}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=-\frac {8 c \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{7 a}+\frac {3 b \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{8 a c \left (a x+\sqrt {-b+a^2 x^2}\right )^{4/3}}-\frac {13 b \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{32 a c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )}+\frac {117 b \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{256 a c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}}-\frac {585 b \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{1024 a c^4 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}+\frac {6 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{7 a}-\frac {585 b \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{2048 a c^{17/4}}+\frac {585 b \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{2048 a c^{17/4}} \]
-8/7*c*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)/a+3/8*b*(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))^(4/3)-13/32*b*(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))+117/256*b*(c+(a *x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)/a/c^3/(a*x+(a^2*x^2-b)^(1/2))^(2/3)-585 /1024*b*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)/a/c^4/(a*x+(a^2*x^2-b)^(1/ 2))^(1/3)+6/7*(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)/a-585/2048*b*arctan((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1 /4))/a/c^(17/4)+585/2048*b*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4) /c^(1/4))/a/c^(17/4)
Time = 1.42 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\frac {2 \sqrt [4]{c} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4} \left (2048 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{4/3} \left (-4 c+3 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )-7 b \left (-384 c^3+416 c^2 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-468 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+585 \left (a x+\sqrt {-b+a^2 x^2}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{4/3}}-4095 b \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )+4095 b \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{14336 a c^{17/4}} \]
Integrate[1/((a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^ 2])^(1/3))^(1/4)),x]
((2*c^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4)*(2048*c^4*(a*x + Sqrt[-b + a^2*x^2])^(4/3)*(-4*c + 3*(a*x + Sqrt[-b + a^2*x^2])^(1/3)) - 7* b*(-384*c^3 + 416*c^2*(a*x + Sqrt[-b + a^2*x^2])^(1/3) - 468*c*(a*x + Sqrt [-b + a^2*x^2])^(2/3) + 585*(a*x + Sqrt[-b + a^2*x^2]))))/(a*x + Sqrt[-b + a^2*x^2])^(4/3) - 4095*b*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1 /4)/c^(1/4)] + 4095*b*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4) /c^(1/4)])/(14336*a*c^(17/4))
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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {1}{\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}dx\) |
3.31.34.3.1 Defintions of rubi rules used
\[\int \frac {1}{\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )}^{\frac {1}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {4095 \, a b c^{4} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {1}{4}} \log \left (200201625 \, a^{3} c^{13} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {3}{4}} + 200201625 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 4095 i \, a b c^{4} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {1}{4}} \log \left (200201625 i \, a^{3} c^{13} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {3}{4}} + 200201625 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) + 4095 i \, a b c^{4} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {1}{4}} \log \left (-200201625 i \, a^{3} c^{13} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {3}{4}} + 200201625 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 4095 \, a b c^{4} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {1}{4}} \log \left (-200201625 \, a^{3} c^{13} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {3}{4}} + 200201625 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 4 \, {\left (8192 \, b c^{5} + 2912 \, a b c^{2} x - 2912 \, \sqrt {a^{2} x^{2} - b} b c^{2} - 21 \, {\left (256 \, a^{2} c^{3} x^{2} - 128 \, b c^{3} - 195 \, a b x - {\left (256 \, a c^{3} x - 195 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {2}{3}} - 12 \, {\left (512 \, b c^{4} + 273 \, a b c x - 273 \, \sqrt {a^{2} x^{2} - b} b c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {3}{4}}}{28672 \, a b c^{4}} \]
integrate(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),x, algorithm="fricas")
1/28672*(4095*a*b*c^4*(b^4/(a^4*c^17))^(1/4)*log(200201625*a^3*c^13*(b^4/( a^4*c^17))^(3/4) + 200201625*b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/ 4)) - 4095*I*a*b*c^4*(b^4/(a^4*c^17))^(1/4)*log(200201625*I*a^3*c^13*(b^4/ (a^4*c^17))^(3/4) + 200201625*b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1 /4)) + 4095*I*a*b*c^4*(b^4/(a^4*c^17))^(1/4)*log(-200201625*I*a^3*c^13*(b^ 4/(a^4*c^17))^(3/4) + 200201625*b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^ (1/4)) - 4095*a*b*c^4*(b^4/(a^4*c^17))^(1/4)*log(-200201625*a^3*c^13*(b^4/ (a^4*c^17))^(3/4) + 200201625*b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1 /4)) - 4*(8192*b*c^5 + 2912*a*b*c^2*x - 2912*sqrt(a^2*x^2 - b)*b*c^2 - 21* (256*a^2*c^3*x^2 - 128*b*c^3 - 195*a*b*x - (256*a*c^3*x - 195*b)*sqrt(a^2* x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(2/3) - 12*(512*b*c^4 + 273*a*b*c*x - 273*sqrt(a^2*x^2 - b)*b*c)*(a*x + sqrt(a^2*x^2 - b))^(1/3))*(c + (a*x + sq rt(a^2*x^2 - b))^(1/3))^(3/4))/(a*b*c^4)
\[ \int \frac {1}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{\sqrt [4]{c + \sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}}}\, dx \]
integrate(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),x)
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)), x)
\[ \int \frac {1}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {1}{{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}} \,d x } \]
integrate(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),x, algorithm="maxima")
integrate(1/((a*x + sqrt(a^2*x^2 - b))^(1/3)*(c + (a*x + sqrt(a^2*x^2 - b) )^(1/3))^(1/4)), x)
Timed out. \[ \int \frac {1}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Timed out} \]
integrate(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),x, algorithm="giac")
Timed out. \[ \int \frac {1}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\right )}^{1/4}} \,d x \]