\(\int \frac {-4 b+a x^3}{\sqrt [4]{-b+a x^3} (b-a x^3+x^4)} \, dx\) [625]

Optimal result

Integrand size = 36, antiderivative size = 49 \[ \int \frac {-4 b+a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{-b+a x^3}}{x}\right )+2 \text {arctanh}\left (\frac {x \left (-b+a x^3\right )^{3/4}}{b-a x^3}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {-4 b+a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{-b+a x^3}}{x}\right )+2 \text {arctanh}\left (\frac {x \left (-b+a x^3\right )^{3/4}}{b-a x^3}\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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