\(\int \frac {x^4 (2 b+a x^6)}{\sqrt [4]{-b+a x^6} (-b-x^4+a x^6)^2} \, dx\) [1322]

Optimal result

Integrand size = 42, antiderivative size = 95 \[ \int \frac {x^4 \left (2 b+a x^6\right )}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )^2} \, dx=-\frac {x \left (-b+a x^6\right )^{3/4}}{2 \left (-b-x^4+a x^6\right )}-\frac {1}{4} \arctan \left (\frac {x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )-\frac {1}{4} \text {arctanh}\left (\frac {x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 8.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {x^4 \left (2 b+a x^6\right )}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )^2} \, dx=\frac {1}{4} \left (\frac {2 x \left (-b+a x^6\right )^{3/4}}{b+x^4-a x^6}+\arctan \left (\frac {x}{\sqrt [4]{-b+a x^6}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-b+a x^6}}\right )\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.38

method	result	size
pseudoelliptic	\(\frac {\left (-a \,x^{6}+x^{4}+b \right ) \ln \left (\frac {\left (a \,x^{6}-b \right )^{\frac {1}{4}}-x}{x}\right )+\left (a \,x^{6}-x^{4}-b \right ) \ln \left (\frac {\left (a \,x^{6}-b \right )^{\frac {1}{4}}+x}{x}\right )+\left (-2 a \,x^{6}+2 x^{4}+2 b \right ) \arctan \left (\frac {\left (a \,x^{6}-b \right )^{\frac {1}{4}}}{x}\right )-4 \left (a \,x^{6}-b \right )^{\frac {3}{4}} x}{8 a \,x^{6}-8 x^{4}-8 b}\)	\(131\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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