\(\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [2755]

Optimal result

Integrand size = 68, antiderivative size = 257 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=-\frac {4 \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}}{a c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}-\frac {4 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} a c^{4/3}}-\frac {4 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{3 a c^{4/3}}+\frac {2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{3 a c^{4/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

Defer[Int][1/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(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 [4]{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 = 0.58 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {2 \left (-\frac {6 \sqrt [3]{c} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}-2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )+\log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )\right )}{3 a c^{4/3}} \]

[In]

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

[Out]

(2*((-6*c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3))/(a*x + Sqrt[-b + a^2*x^2])^(1/4) - 2*Sqrt[3]*Arc
Tan[(1 + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/c^(1/3))/Sqrt[3]] - 2*Log[-c^(1/3) + (c + (a*x + Sqr
t[-b + a^2*x^2])^(1/4))^(1/3)] + 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)]))/(3*a*c^(4/3))

Maple [F]

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

[In]

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

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 758, normalized size of antiderivative = 2.95 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [4]{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 (3 \, \sqrt {\frac {1}{3}} b c \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \log \left (-6 \, \sqrt {\frac {1}{3}} {\left (a \left (-c\right )^{\frac {2}{3}} x - \sqrt {a^{2} x^{2} - b} \left (-c\right )^{\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 {\left (-c\right )^{\frac {1}{3}}}{c}} - 3 \, {\left (a \left (-c\right )^{\frac {2}{3}} x - \sqrt {\frac {1}{3}} {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} - \sqrt {a^{2} x^{2} - b} \left (-c\right )^{\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 \left (-c\right )^{\frac {1}{3}} c x - \sqrt {a^{2} x^{2} - b} \left (-c\right )^{\frac {1}{3}} c\right )} \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} - \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} + 2 \, b\right ) + b \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {2}{3}} - \left (-c\right )^{\frac {1}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {1}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) - 6 \, {\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}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}}{3 \, a b c^{2}}, -\frac {2 \, {\left (6 \, \sqrt {\frac {1}{3}} b c \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \arctan \left (-\sqrt {\frac {1}{3}} \left (-c\right )^{\frac {1}{3}} \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}} + 2 \, \sqrt {\frac {1}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}}\right ) - b \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {2}{3}} - \left (-c\right )^{\frac {1}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right ) + 2 \, b \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {1}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) + 6 \, {\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}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}}{3 \, a b c^{2}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

Maxima [F]

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

[In]

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

Giac [F(-1)]

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

[In]

integrate(1/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3),x, algorit
hm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 7.69 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=-\frac {3\,{\left (\frac {c}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}}+1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {4}{3};\ \frac {7}{3};\ -\frac {c}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}}\right )}{a\,{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/3}} \]

[In]

int(1/((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)*(a^2*x^2 - b)^(1/2)),x)

[Out]

-(3*(c/(a*x + (a^2*x^2 - b)^(1/2))^(1/4) + 1)^(1/3)*hypergeom([1/3, 4/3], 7/3, -c/(a*x + (a^2*x^2 - b)^(1/2))^
(1/4)))/(a*(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))