\(\int \frac {\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx\) [2742]

Optimal result

Integrand size = 68, antiderivative size = 254 \[ \int \frac {\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=-\frac {4 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{a \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}-\frac {4 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} a c^{2/3}}+\frac {4 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{3 a c^{2/3}}-\frac {2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{3 a c^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {2 \left (-\frac {6 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}-\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{2/3}}+\frac {2 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{c^{2/3}}-\frac {\log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{c^{2/3}}\right )}{3 a} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{6}} c \arctan \left (\frac {\sqrt {3} \sqrt {c^{2}} c + 2 \, \sqrt {3} {\left (c^{2}\right )}^{\frac {5}{6}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{3 \, c^{2}}\right ) + b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} c + {\left (c^{2}\right )}^{\frac {1}{3}} c + {\left (c^{2}\right )}^{\frac {2}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) - 2 \, b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c - {\left (c^{2}\right )}^{\frac {2}{3}}\right ) + 6 \, {\left (a c^{2} x - \sqrt {a^{2} x^{2} - b} c^{2}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right )}}{3 \, a b c^{2}} \]

[In]

integrate((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x, algorithm
="fricas")

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((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]

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

[In]

integrate((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x, algorithm
="maxima")

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 7.46 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=-\frac {6\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ -\frac {c}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}}\right )}{a\,{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,{\left (\frac {c}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}}+1\right )}^{1/3}} \]

[In]

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

[Out]

-(6*(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(1/3)*hypergeom([-1/3, 2/3], 5/3, -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)^(1/3))