\(\int \frac {x (8 b+5 a x^3)}{\sqrt [4]{b+a x^3} (-b-a x^3+x^8)} \, dx\) [469]

Optimal result

Integrand size = 38, antiderivative size = 37 \[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b+a x^3}}{x^2}\right )-2 \text {arctanh}\left (\frac {x^2}{\sqrt [4]{b+a x^3}}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=\int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx \]

[In]

Int[(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(-b - a*x^3 + x^8)),x]

[Out]

-8*b*Defer[Int][x/((b + a*x^3)^(1/4)*(b + a*x^3 - x^8)), x] - 5*a*Defer[Int][x^4/((b + a*x^3)^(1/4)*(b + a*x^3
- x^8)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {8 b x}{\sqrt [4]{b+a x^3} \left (b+a x^3-x^8\right )}-\frac {5 a x^4}{\sqrt [4]{b+a x^3} \left (b+a x^3-x^8\right )}\right ) \, dx \\ & = -\left ((5 a) \int \frac {x^4}{\sqrt [4]{b+a x^3} \left (b+a x^3-x^8\right )} \, dx\right )-(8 b) \int \frac {x}{\sqrt [4]{b+a x^3} \left (b+a x^3-x^8\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b+a x^3}}{x^2}\right )-2 \text {arctanh}\left (\frac {x^2}{\sqrt [4]{b+a x^3}}\right ) \]

[In]

Integrate[(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(-b - a*x^3 + x^8)),x]

[Out]

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

Maple [F]

\[\int \frac {x \left (5 a \,x^{3}+8 b \right )}{\left (a \,x^{3}+b \right )^{\frac {1}{4}} \left (x^{8}-a \,x^{3}-b \right )}d x\]

[In]

int(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x)

[Out]

int(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=\int \frac {x \left (5 a x^{3} + 8 b\right )}{\sqrt [4]{a x^{3} + b} \left (- a x^{3} - b + x^{8}\right )}\, dx \]

[In]

integrate(x*(5*a*x**3+8*b)/(a*x**3+b)**(1/4)/(x**8-a*x**3-b),x)

[Out]

Integral(x*(5*a*x**3 + 8*b)/((a*x**3 + b)**(1/4)*(-a*x**3 - b + x**8)), x)

Maxima [F]

\[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=\int { \frac {{\left (5 \, a x^{3} + 8 \, b\right )} x}{{\left (x^{8} - a x^{3} - b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x, algorithm="maxima")

[Out]

integrate((5*a*x^3 + 8*b)*x/((x^8 - a*x^3 - b)*(a*x^3 + b)^(1/4)), x)

Giac [F]

\[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=\int { \frac {{\left (5 \, a x^{3} + 8 \, b\right )} x}{{\left (x^{8} - a x^{3} - b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x, algorithm="giac")

[Out]

integrate((5*a*x^3 + 8*b)*x/((x^8 - a*x^3 - b)*(a*x^3 + b)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=\int -\frac {x\,\left (5\,a\,x^3+8\,b\right )}{{\left (a\,x^3+b\right )}^{1/4}\,\left (-x^8+a\,x^3+b\right )} \,d x \]

[In]

int(-(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(b + a*x^3 - x^8)),x)

[Out]

int(-(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(b + a*x^3 - x^8)), x)