Integrand size = 54, antiderivative size = 81 \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a b x+(-a-b) x^2+x^3}}{x}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b x+(-a-b) x^2+x^3}}{x}\right )}{d^{3/4}} \]
2*arctan(d^(1/4)*(a*b*x+(-a-b)*x^2+x^3)^(1/4)/x)/d^(3/4)-2*arctanh(d^(1/4) *(a*b*x+(-a-b)*x^2+x^3)^(1/4)/x)/d^(3/4)
Time = 11.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.77 \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)}}{x}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)}}{x}\right )\right )}{d^{3/4}} \]
Integrate[(3*a*b - 2*(a + b)*x + x^2)/((x*(-a + x)*(-b + x))^(1/4)*(-(a*b* d) + (a + b)*d*x - d*x^2 + x^3)),x]
(2*(ArcTan[(d^(1/4)*(x*(-a + x)*(-b + x))^(1/4))/x] - ArcTanh[(d^(1/4)*(x* (-a + x)*(-b + x))^(1/4))/x]))/d^(3/4)
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.
| \(\displaystyle \int \frac {-2 x (a+b)+3 a b+x^2}{\sqrt [4]{x (x-a) (x-b)} \left (d x (a+b)-a b d-d x^2+x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{-x (a+b)+a b+x^2} \int -\frac {x^2-2 (a+b) x+3 a b}{\sqrt [4]{x} \sqrt [4]{x^2-(a+b) x+a b} \left (-x^3+d x^2-(a+b) d x+a b d\right )}dx}{\sqrt [4]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{-x (a+b)+a b+x^2} \int \frac {x^2-2 (a+b) x+3 a b}{\sqrt [4]{x} \sqrt [4]{x^2-(a+b) x+a b} \left (-x^3+d x^2-(a+b) d x+a b d\right )}dx}{\sqrt [4]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{-x (a+b)+a b+x^2} \int \frac {\sqrt {x} \left (x^2-2 (a+b) x+3 a b\right )}{\sqrt [4]{x^2-(a+b) x+a b} \left (-x^3+d x^2-(a+b) d x+a b d\right )}d\sqrt [4]{x}}{\sqrt [4]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{-x (a+b)+a b+x^2} \int \left (\frac {x^{5/2}}{\sqrt [4]{x^2-(a+b) x+a b} \left (-x^3+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}+\frac {2 (-a-b) x^{3/2}}{\sqrt [4]{x^2-(a+b) x+a b} \left (-x^3+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}+\frac {3 a b \sqrt {x}}{\sqrt [4]{x^2-(a+b) x+a b} \left (-x^3+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{-x (a+b)+a b+x^2} \left (3 a b \int \frac {\sqrt {x}}{\sqrt [4]{x^2-(a+b) x+a b} \left (-x^3+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}d\sqrt [4]{x}-2 (a+b) \int \frac {x^{3/2}}{\sqrt [4]{x^2-(a+b) x+a b} \left (-x^3+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}d\sqrt [4]{x}+\int \frac {x^{5/2}}{\sqrt [4]{x^2-(a+b) x+a b} \left (-x^3+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}d\sqrt [4]{x}\right )}{\sqrt [4]{x (a-x) (b-x)}}\) |
Int[(3*a*b - 2*(a + b)*x + x^2)/((x*(-a + x)*(-b + x))^(1/4)*(-(a*b*d) + ( a + b)*d*x - d*x^2 + x^3)),x]
3.11.73.3.1 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]
\[\int \frac {3 a b -2 \left (a +b \right ) x +x^{2}}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{4}} \left (-a b d +\left (a +b \right ) d x -d \,x^{2}+x^{3}\right )}d x\]
Timed out. \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\text {Timed out} \]
integrate((3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(1/4)/(-a*b*d+(a+b)*d*x- d*x^2+x^3),x, algorithm="fricas")
Timed out. \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\text {Timed out} \]
\[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\int { -\frac {3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}}{{\left (a b d - {\left (a + b\right )} d x + d x^{2} - x^{3}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{4}}} \,d x } \]
integrate((3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(1/4)/(-a*b*d+(a+b)*d*x- d*x^2+x^3),x, algorithm="maxima")
-integrate((3*a*b - 2*(a + b)*x + x^2)/((a*b*d - (a + b)*d*x + d*x^2 - x^3 )*((a - x)*(b - x)*x)^(1/4)), x)
\[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\int { -\frac {3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}}{{\left (a b d - {\left (a + b\right )} d x + d x^{2} - x^{3}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{4}}} \,d x } \]
integrate((3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(1/4)/(-a*b*d+(a+b)*d*x- d*x^2+x^3),x, algorithm="giac")
integrate(-(3*a*b - 2*(a + b)*x + x^2)/((a*b*d - (a + b)*d*x + d*x^2 - x^3 )*((a - x)*(b - x)*x)^(1/4)), x)
Timed out. \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=-\int \frac {3\,a\,b+x^2-2\,x\,\left (a+b\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/4}\,\left (-x^3+d\,x^2-d\,\left (a+b\right )\,x+a\,b\,d\right )} \,d x \]
int(-(3*a*b + x^2 - 2*x*(a + b))/((x*(a - x)*(b - x))^(1/4)*(d*x^2 - x^3 - d*x*(a + b) + a*b*d)),x)