\(\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [2900]

Optimal result

Integrand size = 68, antiderivative size = 319 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {-6 c \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+8 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}}{3 a c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}}+\frac {8 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} a c^{7/3}}+\frac {8 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{9 a c^{7/3}}-\frac {4 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{9 a c^{7/3}} \]

[Out]

1/3*(-6*c*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)+8*(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))/a/c^2/(a*x+(a^2*x^2-b)^(1/2))^(1/2)+8/9*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^(7/3)+8/9*ln(-c^(1/3)+(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3))/a/c^(7/3)-4/
9*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^(7/3
)

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\frac {6 \sqrt [3]{c} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3} \left (-3 c+4 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}+8 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+8 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-4 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{9 a c^{7/3}} \]

[In]

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

[Out]

((6*c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)*(-3*c + 4*(a*x + Sqrt[-b + a^2*x^2])^(1/4)))/Sqrt[a*x
+ Sqrt[-b + a^2*x^2]] + 8*Sqrt[3]*ArcTan[(1 + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/c^(1/3))/Sqrt[
3]] + 8*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)] - 4*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqr
t[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(9*a*c^(7/3))

Maple [F]

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

[In]

int(1/(a^2*x^2-b)^(1/2)/(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),x)

[Out]

int(1/(a^2*x^2-b)^(1/2)/(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),x)

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.41 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\left [\frac {2 \, {\left (6 \, \sqrt {\frac {1}{3}} b c \sqrt {-\frac {1}{c^{\frac {2}{3}}}} \log \left (6 \, \sqrt {\frac {1}{3}} {\left (a c^{\frac {2}{3}} x - \sqrt {a^{2} x^{2} - b} c^{\frac {2}{3}}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - 3 \, {\left (a c^{\frac {2}{3}} x + \sqrt {\frac {1}{3}} {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - \sqrt {a^{2} x^{2} - b} c^{\frac {2}{3}}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} + 3 \, {\left (a c x - \sqrt {\frac {1}{3}} {\left (a c^{\frac {4}{3}} x - \sqrt {a^{2} x^{2} - b} c^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} + 2 \, b\right ) - 2 \, b c^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c^{\frac {1}{3}} + c^{\frac {2}{3}}\right ) + 4 \, b c^{\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^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} - 3 \, {\left (a c^{2} x - \sqrt {a^{2} x^{2} - b} c^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}}{9 \, a b c^{3}}, \frac {2 \, {\left (12 \, \sqrt {\frac {1}{3}} b c^{\frac {2}{3}} \arctan \left (\sqrt {\frac {1}{3}} + \frac {2 \, \sqrt {\frac {1}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right ) - 2 \, b c^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c^{\frac {1}{3}} + c^{\frac {2}{3}}\right ) + 4 \, b c^{\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^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} - 3 \, {\left (a c^{2} x - \sqrt {a^{2} x^{2} - b} c^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}}{9 \, a b c^{3}}\right ] \]

[In]

integrate(1/(a^2*x^2-b)^(1/2)/(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),x, algorit
hm="fricas")

[Out]

[2/9*(6*sqrt(1/3)*b*c*sqrt(-1/c^(2/3))*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(-1/c^(2/3)) - 3*(a*c^(2/3)*x + sqrt(1/3)*(
a*c*x - sqrt(a^2*x^2 - b)*c)*sqrt(-1/c^(2/3)) - 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^(4/3)*x - sqrt(a^2*x^2 - b)*c^(4/3))*sqrt
(-1/c^(2/3)) - sqrt(a^2*x^2 - b)*c)*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 2*b) - 2*b*c^(2/3)*log((c + (a*x + sqrt(
a^2*x^2 - b))^(1/4))^(2/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c^(1/3) + c^(2/3)) + 4*b*c^(2/3)*log(
(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) - c^(1/3)) + 3*(4*(a*c*x - sqrt(a^2*x^2 - b)*c)*(a*x + sqrt(a^2*x^
2 - b))^(3/4) - 3*(a*c^2*x - sqrt(a^2*x^2 - b)*c^2)*sqrt(a*x + sqrt(a^2*x^2 - b)))*(c + (a*x + sqrt(a^2*x^2 -
b))^(1/4))^(2/3))/(a*b*c^3), 2/9*(12*sqrt(1/3)*b*c^(2/3)*arctan(sqrt(1/3) + 2*sqrt(1/3)*(c + (a*x + sqrt(a^2*x
^2 - b))^(1/4))^(1/3)/c^(1/3)) - 2*b*c^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3) + (c + (a*x + sqr
t(a^2*x^2 - b))^(1/4))^(1/3)*c^(1/3) + c^(2/3)) + 4*b*c^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)
- c^(1/3)) + 3*(4*(a*c*x - sqrt(a^2*x^2 - b)*c)*(a*x + sqrt(a^2*x^2 - b))^(3/4) - 3*(a*c^2*x - sqrt(a^2*x^2 -
b)*c^2)*sqrt(a*x + sqrt(a^2*x^2 - b)))*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3))/(a*b*c^3)]

Sympy [F]

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

[In]

integrate(1/(a**2*x**2-b)**(1/2)/(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
),x)

[Out]

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

Maxima [F]

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

[In]

integrate(1/(a^2*x^2-b)^(1/2)/(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),x, algorit
hm="maxima")

[Out]

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

Giac [F(-1)]

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

[In]

integrate(1/(a^2*x^2-b)^(1/2)/(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),x, algorit
hm="giac")

[Out]

Timed out

Mupad [F(-1)]

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

[In]

int(1/((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)*(a^2*x^2 - b)^(1/2)),x)

[Out]

int(1/((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)*(a^2*x^2 - b)^(1/2)), x
)