\(\int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx\) [2839]

Optimal result

Integrand size = 45, antiderivative size = 289 \[ \int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\frac {24 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{a}+\frac {4 \sqrt {3} \sqrt [6]{c} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{a}-\frac {4 \sqrt {3} \sqrt [6]{c} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{a}-\frac {8 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}\right )}{a}-\frac {4 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt [6]{c}+\frac {\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}\right )}{a} \]

[Out]

24*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)/a-4*3^(1/2)*c^(1/6)*arctan(-1/3*3^(1/2)+2/3*(c+(a*x+(a^2*x^2-b)^(1/
2))^(1/4))^(1/6)*3^(1/2)/c^(1/6))/a-4*3^(1/2)*c^(1/6)*arctan(1/3*3^(1/2)+2/3*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))
^(1/6)*3^(1/2)/c^(1/6))/a-8*c^(1/6)*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)/c^(1/6))/a-4*c^(1/6)*arcta
nh((c^(1/6)+(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/c^(1/6))/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6))/a

Rubi [F]

\[ \int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx \]

[In]

Int[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)/Sqrt[-b + a^2*x^2],x]

[Out]

Defer[Int][(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)/Sqrt[-b + a^2*x^2], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\frac {4 \left (6 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\sqrt {3} \sqrt [6]{c} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt {3}}\right )-\sqrt {3} \sqrt [6]{c} \arctan \left (\frac {1+\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt {3}}\right )-2 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}\right )-\sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}\right )\right )}{a} \]

[In]

Integrate[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)/Sqrt[-b + a^2*x^2],x]

[Out]

(4*(6*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6) + Sqrt[3]*c^(1/6)*ArcTan[(1 - (2*(c + (a*x + Sqrt[-b + a^2*
x^2])^(1/4))^(1/6))/c^(1/6))/Sqrt[3]] - Sqrt[3]*c^(1/6)*ArcTan[(1 + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^
(1/6))/c^(1/6))/Sqrt[3]] - 2*c^(1/6)*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)/c^(1/6)] - c^(1/6)*A
rcTanh[(c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(c^(1/6)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4)
)^(1/6))]))/a

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {24 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+24 \left (-\frac {\ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}+c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{3}}\right )}{12 c^{\frac {5}{6}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{6 c^{\frac {5}{6}}}-\frac {\ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{6 c^{\frac {5}{6}}}+\frac {\ln \left (-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{6 c^{\frac {5}{6}}}+\frac {\ln \left (c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}-c^{\frac {1}{3}}\right )}{12 c^{\frac {5}{6}}}-\frac {\sqrt {3}\, \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{6 c^{\frac {5}{6}}}\right ) c}{a}\)	\(320\)
default	\(\frac {24 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+24 \left (-\frac {\ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}+c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{3}}\right )}{12 c^{\frac {5}{6}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{6 c^{\frac {5}{6}}}-\frac {\ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{6 c^{\frac {5}{6}}}+\frac {\ln \left (-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{6 c^{\frac {5}{6}}}+\frac {\ln \left (c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}-c^{\frac {1}{3}}\right )}{12 c^{\frac {5}{6}}}-\frac {\sqrt {3}\, \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{6 c^{\frac {5}{6}}}\right ) c}{a}\)	\(320\)

[In]

int((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)/(a^2*x^2-b)^(1/2),x,method=_RETURNVERBOSE)

[Out]

4/a*(6*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)+6*(-1/12/c^(5/6)*ln((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)+c^(
1/6)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)+c^(1/3))-1/6/c^(5/6)*3^(1/2)*arctan(1/3*3^(1/2)+2/3*(c+(a*x+(a^2*
x^2-b)^(1/2))^(1/4))^(1/6)*3^(1/2)/c^(1/6))-1/6/c^(5/6)*ln((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)+c^(1/6))+1/
6/c^(5/6)*ln(-(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)+c^(1/6))+1/12/c^(5/6)*ln(c^(1/6)*(c+(a*x+(a^2*x^2-b)^(1/
2))^(1/4))^(1/6)-(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)-c^(1/3))-1/6/c^(5/6)*3^(1/2)*arctan(-1/3*3^(1/2)+2/3*
(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)*3^(1/2)/c^(1/6)))*c)

Fricas [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=-\frac {2 \, {\left ({\left (\sqrt {-3} a + a\right )} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \log \left (4 \, {\left (\sqrt {-3} a + a\right )} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + 8 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - {\left (\sqrt {-3} a + a\right )} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \log \left (-4 \, {\left (\sqrt {-3} a + a\right )} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + 8 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) + {\left (\sqrt {-3} a - a\right )} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \log \left (4 \, {\left (\sqrt {-3} a - a\right )} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + 8 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - {\left (\sqrt {-3} a - a\right )} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \log \left (-4 \, {\left (\sqrt {-3} a - a\right )} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + 8 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) + 2 \, a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \log \left (4 \, a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + 4 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 2 \, a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \log \left (-4 \, a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + 4 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 12 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right )}}{a} \]

[In]

integrate((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)/(a^2*x^2-b)^(1/2),x, algorithm="fricas")

[Out]

-2*((sqrt(-3)*a + a)*(c/a^6)^(1/6)*log(4*(sqrt(-3)*a + a)*(c/a^6)^(1/6) + 8*(c + (a*x + sqrt(a^2*x^2 - b))^(1/
4))^(1/6)) - (sqrt(-3)*a + a)*(c/a^6)^(1/6)*log(-4*(sqrt(-3)*a + a)*(c/a^6)^(1/6) + 8*(c + (a*x + sqrt(a^2*x^2
- b))^(1/4))^(1/6)) + (sqrt(-3)*a - a)*(c/a^6)^(1/6)*log(4*(sqrt(-3)*a - a)*(c/a^6)^(1/6) + 8*(c + (a*x + sqr
t(a^2*x^2 - b))^(1/4))^(1/6)) - (sqrt(-3)*a - a)*(c/a^6)^(1/6)*log(-4*(sqrt(-3)*a - a)*(c/a^6)^(1/6) + 8*(c +
(a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)) + 2*a*(c/a^6)^(1/6)*log(4*a*(c/a^6)^(1/6) + 4*(c + (a*x + sqrt(a^2*x^2
- b))^(1/4))^(1/6)) - 2*a*(c/a^6)^(1/6)*log(-4*a*(c/a^6)^(1/6) + 4*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6
)) - 12*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6))/a

Sympy [F]

\[ \int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\int \frac {\sqrt [6]{c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}}}{\sqrt {a^{2} x^{2} - b}}\, dx \]

[In]

integrate((c+(a*x+(a**2*x**2-b)**(1/2))**(1/4))**(1/6)/(a**2*x**2-b)**(1/2),x)

[Out]

Integral((c + (a*x + sqrt(a**2*x**2 - b))**(1/4))**(1/6)/sqrt(a**2*x**2 - b), x)

Maxima [F]

\[ \int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\int { \frac {{\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}}{\sqrt {a^{2} x^{2} - b}} \,d x } \]

[In]

integrate((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)/(a^2*x^2-b)^(1/2),x, algorithm="maxima")

[Out]

integrate((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)/sqrt(a^2*x^2 - b), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\text {Timed out} \]

[In]

integrate((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)/(a^2*x^2-b)^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\int \frac {{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/6}}{\sqrt {a^2\,x^2-b}} \,d x \]

[In]

int((c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(1/6)/(a^2*x^2 - b)^(1/2),x)

[Out]

int((c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(1/6)/(a^2*x^2 - b)^(1/2), x)