\(\int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} (b-a x+x^3)} \, dx\) [619]

Optimal result

Integrand size = 34, antiderivative size = 49 \[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{-b x+a x^2}}{x}\right )-2 \text {arctanh}\left (\frac {\left (-b x+a x^2\right )^{3/4}}{-b+a x}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx \]

[In]

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

[Out]

(-8*a*x^(1/4)*(-b + a*x)^(1/4)*Defer[Subst][Defer[Int][x^6/((-b + a*x^4)^(1/4)*(-b + a*x^4 - x^12)), x], x, x^
(1/4)])/(-(b*x) + a*x^2)^(1/4) - (12*b*x^(1/4)*(-b + a*x)^(1/4)*Defer[Subst][Defer[Int][x^2/((-b + a*x^4)^(1/4
)*(b - a*x^4 + x^12)), x], x, x^(1/4)])/(-(b*x) + a*x^2)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x}\right ) \int \frac {-3 b+2 a x}{\sqrt [4]{x} \sqrt [4]{-b+a x} \left (b-a x+x^3\right )} \, dx}{\sqrt [4]{-b x+a x^2}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^2 \left (-3 b+2 a x^4\right )}{\sqrt [4]{-b+a x^4} \left (b-a x^4+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^2}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x}\right ) \text {Subst}\left (\int \left (-\frac {2 a x^6}{\sqrt [4]{-b+a x^4} \left (-b+a x^4-x^{12}\right )}-\frac {3 b x^2}{\sqrt [4]{-b+a x^4} \left (b-a x^4+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^2}} \\ & = -\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [4]{-b+a x^4} \left (-b+a x^4-x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^2}}-\frac {\left (12 b \sqrt [4]{x} \sqrt [4]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^4} \left (b-a x^4+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.85 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\frac {2 \sqrt [4]{x} \sqrt [4]{-b+a x} \left (\arctan \left (\frac {\sqrt [4]{-b+a x}}{x^{3/4}}\right )-\text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{-b+a x}}\right )\right )}{\sqrt [4]{x (-b+a x)}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {2 a x -3 b}{\left (a \,x^{2}-b x \right )^{\frac {1}{4}} \left (x^{3}-a x +b \right )}d x\]

[In]

int((2*a*x-3*b)/(a*x^2-b*x)^(1/4)/(x^3-a*x+b),x)

[Out]

int((2*a*x-3*b)/(a*x^2-b*x)^(1/4)/(x^3-a*x+b),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\text {Timed out} \]

[In]

integrate((2*a*x-3*b)/(a*x^2-b*x)^(1/4)/(x^3-a*x+b),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\text {Timed out} \]

[In]

integrate((2*a*x-3*b)/(a*x**2-b*x)**(1/4)/(x**3-a*x+b),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\int { \frac {2 \, a x - 3 \, b}{{\left (a x^{2} - b x\right )}^{\frac {1}{4}} {\left (x^{3} - a x + b\right )}} \,d x } \]

[In]

integrate((2*a*x-3*b)/(a*x^2-b*x)^(1/4)/(x^3-a*x+b),x, algorithm="maxima")

[Out]

integrate((2*a*x - 3*b)/((a*x^2 - b*x)^(1/4)*(x^3 - a*x + b)), x)

Giac [F]

\[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\int { \frac {2 \, a x - 3 \, b}{{\left (a x^{2} - b x\right )}^{\frac {1}{4}} {\left (x^{3} - a x + b\right )}} \,d x } \]

[In]

integrate((2*a*x-3*b)/(a*x^2-b*x)^(1/4)/(x^3-a*x+b),x, algorithm="giac")

[Out]

integrate((2*a*x - 3*b)/((a*x^2 - b*x)^(1/4)*(x^3 - a*x + b)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\int -\frac {3\,b-2\,a\,x}{{\left (a\,x^2-b\,x\right )}^{1/4}\,\left (x^3-a\,x+b\right )} \,d x \]

[In]

int(-(3*b - 2*a*x)/((a*x^2 - b*x)^(1/4)*(b - a*x + x^3)),x)

[Out]

int(-(3*b - 2*a*x)/((a*x^2 - b*x)^(1/4)*(b - a*x + x^3)), x)