\(\int \frac {(2 b+a x^6) (-b-x^4+a x^6)}{x^4 \sqrt [4]{-b+a x^6} (-b-2 x^4+a x^6)} \, dx\) [1323]

Optimal result

Integrand size = 56, antiderivative size = 95 \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\frac {\arctan \left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}} \]

[Out]

2/3*(a*x^6-b)^(3/4)/x^3+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]

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

[In]

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

[Out]

(2*(-b + a*x^6)^(3/4))/(3*x^3) + (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 {2 b}{x^4 \sqrt [4]{-b+a x^6}}+\frac {a x^2}{\sqrt [4]{-b+a x^6}}+\frac {-3 b-2 x^4}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}}\right ) \, dx \\ & = a \int \frac {x^2}{\sqrt [4]{-b+a x^6}} \, dx+(2 b) \int \frac {1}{x^4 \sqrt [4]{-b+a x^6}} \, dx+\int \frac {1}{\sqrt [4]{-b+a x^6}} \, dx+\int \frac {-3 b-2 x^4}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}} \, dx \\ & = \frac {1}{3} a \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^2}} \, dx,x,x^3\right )+\frac {1}{3} (2 b) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{-b+a x^2}} \, dx,x,x^3\right )+\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 {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\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-\frac {1}{3} a \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^2}} \, dx,x,x^3\right )-(3 b) \int \frac {1}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}} \, dx+\frac {\left (2 \sqrt {\frac {a x^6}{b}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3} \\ & = \frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\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-\frac {\left (2 \sqrt {\frac {a x^6}{b}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3}+\frac {\left (2 \sqrt {b} \sqrt {\frac {a x^6}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3}-\frac {\left (2 \sqrt {b} \sqrt {\frac {a x^6}{b}}\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {b}}}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3} \\ & = \frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\frac {2 a x^3 \sqrt [4]{-b+a x^6}}{3 \left (\sqrt {b}+\sqrt {-b+a x^6}\right )}-\frac {2 \sqrt [4]{b} \sqrt {\frac {a x^6}{\left (\sqrt {b}+\sqrt {-b+a x^6}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^6}\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 x^3}+\frac {\sqrt [4]{b} \sqrt {\frac {a x^6}{\left (\sqrt {b}+\sqrt {-b+a x^6}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^6}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 x^3}+\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-\frac {\left (2 \sqrt {b} \sqrt {\frac {a x^6}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3}+\frac {\left (2 \sqrt {b} \sqrt {\frac {a x^6}{b}}\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {b}}}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3} \\ & = \frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\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 = 8.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-b+a x^6}}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-b+a x^6}}\right )}{\sqrt [4]{2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.41 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(\frac {6 \arctan \left (\frac {\left (a \,x^{6}-b \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right ) 2^{\frac {3}{4}} x^{3}-3 \ln \left (\frac {-2^{\frac {1}{4}} x -\left (a \,x^{6}-b \right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (a \,x^{6}-b \right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} x^{3}+8 \left (a \,x^{6}-b \right )^{\frac {3}{4}}}{12 x^{3}}\)	\(99\)

[In]

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

[Out]

1/12*(6*arctan(1/2*(a*x^6-b)^(1/4)/x*2^(3/4))*2^(3/4)*x^3-3*ln((-2^(1/4)*x-(a*x^6-b)^(1/4))/(2^(1/4)*x-(a*x^6-
b)^(1/4)))*2^(3/4)*x^3+8*(a*x^6-b)^(3/4))/x^3

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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