\(\int \frac {-b+a x^8}{(b+a x^8) \sqrt [4]{b-c x^4+a x^8}} \, dx\) [2130]

Optimal result

Integrand size = 36, antiderivative size = 154 \[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b-c x^4+a x^8}}{-\sqrt {c} x^2+\sqrt {b-c x^4+a x^8}}\right )}{2 \sqrt {2} \sqrt [4]{c}}-\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {b-c x^4+a x^8}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{b-c x^4+a x^8}}\right )}{2 \sqrt {2} \sqrt [4]{c}} \]

[Out]

-1/4*arctan(2^(1/2)*c^(1/4)*x*(a*x^8-c*x^4+b)^(1/4)/(-c^(1/2)*x^2+(a*x^8-c*x^4+b)^(1/2)))*2^(1/2)/c^(1/4)-1/4*
arctanh((1/2*c^(1/4)*x^2*2^(1/2)+1/2*(a*x^8-c*x^4+b)^(1/2)*2^(1/2)/c^(1/4))/x/(a*x^8-c*x^4+b)^(1/4))*2^(1/2)/c
^(1/4)

Rubi [F]

\[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx \]

[In]

Int[(-b + a*x^8)/((b + a*x^8)*(b - c*x^4 + a*x^8)^(1/4)),x]

[Out]

(x*(1 - (2*a*x^4)/(c - Sqrt[-4*a*b + c^2]))^(1/4)*(1 - (2*a*x^4)/(c + Sqrt[-4*a*b + c^2]))^(1/4)*AppellF1[1/4,
1/4, 1/4, 5/4, (2*a*x^4)/(c + Sqrt[-4*a*b + c^2]), (2*a*x^4)/(c - Sqrt[-4*a*b + c^2])])/(b - c*x^4 + a*x^8)^(
1/4) - Sqrt[b]*Defer[Int][1/((Sqrt[b] - Sqrt[-a]*x^4)*(b - c*x^4 + a*x^8)^(1/4)), x] - Sqrt[b]*Defer[Int][1/((
Sqrt[b] + Sqrt[-a]*x^4)*(b - c*x^4 + a*x^8)^(1/4)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt [4]{b-c x^4+a x^8}}-\frac {2 b}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}}\right ) \, dx \\ & = -\left ((2 b) \int \frac {1}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx\right )+\int \frac {1}{\sqrt [4]{b-c x^4+a x^8}} \, dx \\ & = -\left ((2 b) \int \left (\frac {1}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^4\right ) \sqrt [4]{b-c x^4+a x^8}}+\frac {1}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^4\right ) \sqrt [4]{b-c x^4+a x^8}}\right ) \, dx\right )+\frac {\left (\sqrt [4]{1+\frac {2 a x^4}{-c-\sqrt {-4 a b+c^2}}} \sqrt [4]{1+\frac {2 a x^4}{-c+\sqrt {-4 a b+c^2}}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {2 a x^4}{-c-\sqrt {-4 a b+c^2}}} \sqrt [4]{1+\frac {2 a x^4}{-c+\sqrt {-4 a b+c^2}}}} \, dx}{\sqrt [4]{b-c x^4+a x^8}} \\ & = \frac {x \sqrt [4]{1-\frac {2 a x^4}{c-\sqrt {-4 a b+c^2}}} \sqrt [4]{1-\frac {2 a x^4}{c+\sqrt {-4 a b+c^2}}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {2 a x^4}{c+\sqrt {-4 a b+c^2}},\frac {2 a x^4}{c-\sqrt {-4 a b+c^2}}\right )}{\sqrt [4]{b-c x^4+a x^8}}-\sqrt {b} \int \frac {1}{\left (\sqrt {b}-\sqrt {-a} x^4\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx-\sqrt {b} \int \frac {1}{\left (\sqrt {b}+\sqrt {-a} x^4\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.78 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.87 \[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b-c x^4+a x^8}}{-\sqrt {c} x^2+\sqrt {b-c x^4+a x^8}}\right )+\text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {b-c x^4+a x^8}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b-c x^4+a x^8}}\right )}{2 \sqrt {2} \sqrt [4]{c}} \]

[In]

Integrate[(-b + a*x^8)/((b + a*x^8)*(b - c*x^4 + a*x^8)^(1/4)),x]

[Out]

-1/2*(ArcTan[(Sqrt[2]*c^(1/4)*x*(b - c*x^4 + a*x^8)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[b - c*x^4 + a*x^8])] + ArcTa
nh[(Sqrt[c]*x^2 + Sqrt[b - c*x^4 + a*x^8])/(Sqrt[2]*c^(1/4)*x*(b - c*x^4 + a*x^8)^(1/4))])/(Sqrt[2]*c^(1/4))

Maple [A] (verified)

Time = 2.20 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.16

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{8}-c \,x^{4}+b}}{\left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{8}-c \,x^{4}+b}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )\right )}{8 c^{\frac {1}{4}}}\)	\(178\)

[In]

int((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x,method=_RETURNVERBOSE)

[Out]

1/8/c^(1/4)*2^(1/2)*(ln((-(a*x^8-c*x^4+b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)*x^2+(a*x^8-c*x^4+b)^(1/2))/((a*x^8-c
*x^4+b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)*x^2+(a*x^8-c*x^4+b)^(1/2)))+2*arctan((2^(1/2)*(a*x^8-c*x^4+b)^(1/4)+c^
(1/4)*x)/c^(1/4)/x)+2*arctan((2^(1/2)*(a*x^8-c*x^4+b)^(1/4)-c^(1/4)*x)/c^(1/4)/x))

Fricas [F(-1)]

Timed out. \[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\text {Timed out} \]

[In]

integrate((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\int \frac {a x^{8} - b}{\left (a x^{8} + b\right ) \sqrt [4]{a x^{8} + b - c x^{4}}}\, dx \]

[In]

integrate((a*x**8-b)/(a*x**8+b)/(a*x**8-c*x**4+b)**(1/4),x)

[Out]

Integral((a*x**8 - b)/((a*x**8 + b)*(a*x**8 + b - c*x**4)**(1/4)), x)

Maxima [F]

\[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\int { \frac {a x^{8} - b}{{\left (a x^{8} - c x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{8} + b\right )}} \,d x } \]

[In]

integrate((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^8 - b)/((a*x^8 - c*x^4 + b)^(1/4)*(a*x^8 + b)), x)

Giac [F]

\[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\int { \frac {a x^{8} - b}{{\left (a x^{8} - c x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{8} + b\right )}} \,d x } \]

[In]

integrate((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x, algorithm="giac")

[Out]

integrate((a*x^8 - b)/((a*x^8 - c*x^4 + b)^(1/4)*(a*x^8 + b)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\int -\frac {b-a\,x^8}{\left (a\,x^8+b\right )\,{\left (a\,x^8-c\,x^4+b\right )}^{1/4}} \,d x \]

[In]

int(-(b - a*x^8)/((b + a*x^8)*(b + a*x^8 - c*x^4)^(1/4)),x)

[Out]

int(-(b - a*x^8)/((b + a*x^8)*(b + a*x^8 - c*x^4)^(1/4)), x)