Integrand size = 34, antiderivative size = 49 \[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{-b x+a x^2}}{x}\right )-2 \text {arctanh}\left (\frac {\left (-b x+a x^2\right )^{3/4}}{-b+a x}\right ) \]
Time = 1.85 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\frac {2 \sqrt [4]{x} \sqrt [4]{-b+a x} \left (\arctan \left (\frac {\sqrt [4]{-b+a x}}{x^{3/4}}\right )-\text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{-b+a x}}\right )\right )}{\sqrt [4]{x (-b+a x)}} \]
(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)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {2 a x-3 b}{\sqrt [4]{a x^2-b x} \left (-a x+b+x^3\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x-b} \int -\frac {3 b-2 a x}{\sqrt [4]{x} \sqrt [4]{a x-b} \left (x^3-a x+b\right )}dx}{\sqrt [4]{a x^2-b x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{a x-b} \int \frac {3 b-2 a x}{\sqrt [4]{x} \sqrt [4]{a x-b} \left (x^3-a x+b\right )}dx}{\sqrt [4]{a x^2-b x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x-b} \int \frac {\sqrt {x} (3 b-2 a x)}{\sqrt [4]{a x-b} \left (x^3-a x+b\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^2-b x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x-b} \int \left (\frac {2 a x^{3/2}}{\sqrt [4]{a x-b} \left (-x^3+a x-b\right )}+\frac {3 b \sqrt {x}}{\sqrt [4]{a x-b} \left (x^3-a x+b\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^2-b x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x-b} \left (3 b \int \frac {\sqrt {x}}{\sqrt [4]{a x-b} \left (x^3-a x+b\right )}d\sqrt [4]{x}+2 a \int \frac {x^{3/2}}{\sqrt [4]{a x-b} \left (-x^3+a x-b\right )}d\sqrt [4]{x}\right )}{\sqrt [4]{a x^2-b x}}\) |
3.7.19.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
\[\int \frac {2 a x -3 b}{\left (a \,x^{2}-b x \right )^{\frac {1}{4}} \left (x^{3}-a x +b \right )}d x\]
Timed out. \[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\text {Timed out} \]
\[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\int { \frac {2 \, a x - 3 \, b}{{\left (a x^{2} - b x\right )}^{\frac {1}{4}} {\left (x^{3} - a x + b\right )}} \,d x } \]
\[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\int { \frac {2 \, a x - 3 \, b}{{\left (a x^{2} - b x\right )}^{\frac {1}{4}} {\left (x^{3} - a x + b\right )}} \,d x } \]
Timed out. \[ \int \frac {-3 b+2 a x}{\sqrt [4]{-b x+a x^2} \left (b-a x+x^3\right )} \, dx=\int -\frac {3\,b-2\,a\,x}{{\left (a\,x^2-b\,x\right )}^{1/4}\,\left (x^3-a\,x+b\right )} \,d x \]