\(\int \frac {2 b+a x^6}{\sqrt [4]{-b+a x^6} (-b-2 x^4+a x^6)} \, dx\) [995]

Optimal result

Integrand size = 39, antiderivative size = 75 \[ \int \frac {2 b+a x^6}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}} \]

[Out]

1/2*arctan(2^(1/4)*x*(a*x^6-b)^(3/4)/(-a*x^6+b))*2^(3/4)+1/2*arctanh(2^(1/4)*x*(a*x^6-b)^(3/4)/(-a*x^6+b))*2^(
3/4)

Rubi [F]

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

[In]

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

[Out]

(x*(1 - (a*x^6)/b)^(1/4)*Hypergeometric2F1[1/6, 1/4, 7/6, (a*x^6)/b])/(-b + a*x^6)^(1/4) - 3*b*Defer[Int][1/((
b + 2*x^4 - a*x^6)*(-b + a*x^6)^(1/4)), x] + 2*Defer[Int][x^4/((-b + a*x^6)^(1/4)*(-b - 2*x^4 + a*x^6)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt [4]{-b+a x^6}}+\frac {3 b+2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )}\right ) \, dx \\ & = \int \frac {1}{\sqrt [4]{-b+a x^6}} \, dx+\int \frac {3 b+2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx \\ & = \frac {\sqrt [4]{1-\frac {a x^6}{b}} \int \frac {1}{\sqrt [4]{1-\frac {a x^6}{b}}} \, dx}{\sqrt [4]{-b+a x^6}}+\int \left (-\frac {3 b}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}}+\frac {2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )}\right ) \, dx \\ & = \frac {x \sqrt [4]{1-\frac {a x^6}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},\frac {a x^6}{b}\right )}{\sqrt [4]{-b+a x^6}}+2 \int \frac {x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx-(3 b) \int \frac {1}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.39 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.67 \[ \int \frac {2 b+a x^6}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-b+a x^6}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-b+a x^6}}\right )}{\sqrt [4]{2}} \]

[In]

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

[Out]

-((ArcTan[(2^(1/4)*x)/(-b + a*x^6)^(1/4)] + ArcTanh[(2^(1/4)*x)/(-b + a*x^6)^(1/4)])/2^(1/4))

Maple [A] (verified)

Time = 2.93 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(\frac {2^{\frac {3}{4}} \left (2 \arctan \left (\frac {\left (a \,x^{6}-b \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right )-\ln \left (\frac {-2^{\frac {1}{4}} x -\left (a \,x^{6}-b \right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (a \,x^{6}-b \right )^{\frac {1}{4}}}\right )\right )}{4}\)	\(74\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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