Integrand size = 58, antiderivative size = 111 \[ \int \frac {(-a+x) (-b+x) (-2 a b+(a+b) x)}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=-\frac {4 \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}{x}-2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )+2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right ) \]
-4*(a*b*x^2+(-a-b)*x^3+x^4)^(1/4)/x-2*d^(1/4)*arctan(d^(1/4)*x/(a*b*x^2+(- a-b)*x^3+x^4)^(1/4))+2*d^(1/4)*arctanh(d^(1/4)*x/(a*b*x^2+(-a-b)*x^3+x^4)^ (1/4))
Time = 13.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.81 \[ \int \frac {(-a+x) (-b+x) (-2 a b+(a+b) x)}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=-\frac {4 \sqrt [4]{x^2 (-a+x) (-b+x)}}{x}-2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x^2 (-a+x) (-b+x)}}\right )+2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x^2 (-a+x) (-b+x)}}\right ) \]
Integrate[((-a + x)*(-b + x)*(-2*a*b + (a + b)*x))/((x^2*(-a + x)*(-b + x) )^(3/4)*(-(a*b) + (a + b)*x + (-1 + d)*x^2)),x]
(-4*(x^2*(-a + x)*(-b + x))^(1/4))/x - 2*d^(1/4)*ArcTan[(d^(1/4)*x)/(x^2*( -a + x)*(-b + x))^(1/4)] + 2*d^(1/4)*ArcTanh[(d^(1/4)*x)/(x^2*(-a + x)*(-b + 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.
| \(\displaystyle \int \frac {(x-a) (x-b) (x (a+b)-2 a b)}{\left (x^2 (x-a) (x-b)\right )^{3/4} \left (x (a+b)-a b+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{3/2} \left (-x (a+b)+a b+x^2\right )^{3/4} \int \frac {(a-x) (b-x) (2 a b-(a+b) x)}{x^{3/2} \left (x^2-(a+b) x+a b\right )^{3/4} \left ((1-d) x^2-(a+b) x+a b\right )}dx}{\left (x^2 (a-x) (b-x)\right )^{3/4}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 x^{3/2} \left (-x (a+b)+a b+x^2\right )^{3/4} \int \frac {(a-x) (b-x) (2 a b-(a+b) x)}{x \left (x^2-(a+b) x+a b\right )^{3/4} \left ((1-d) x^2-(a+b) x+a b\right )}d\sqrt {x}}{\left (x^2 (a-x) (b-x)\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 x^{3/2} \left (-x (a+b)+a b+x^2\right )^{3/4} \int \left (\frac {2 a b}{x \left (x^2-(a+b) x+a b\right )^{3/4}}+\frac {a+b}{(d-1) \left (x^2-(a+b) x+a b\right )^{3/4}}+\frac {d \left (a b (a+b)-\left (a^2+2 b d a+b^2\right ) x\right )}{(1-d) \left (x^2-(a+b) x+a b\right )^{3/4} \left ((1-d) x^2-(a+b) x+a b\right )}\right )d\sqrt {x}}{\left (x^2 (a-x) (b-x)\right )^{3/4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 x^{3/2} \left (-x (a+b)+a b+x^2\right )^{3/4} \left (-\frac {d \left ((a+b) \sqrt {a^2-2 a b (1-2 d)+b^2}+a^2+2 a b d+b^2\right ) \int \frac {1}{\left (-a-b+2 (1-d) x-\sqrt {a^2-2 b a+4 b d a+b^2}\right ) \left (x^2+(-a-b) x+a b\right )^{3/4}}d\sqrt {x}}{1-d}-\frac {d \left (-(a+b) \sqrt {a^2-2 a b (1-2 d)+b^2}+a^2+2 a b d+b^2\right ) \int \frac {1}{\left (-a-b+2 (1-d) x+\sqrt {a^2-2 b a+4 b d a+b^2}\right ) \left (x^2+(-a-b) x+a b\right )^{3/4}}d\sqrt {x}}{1-d}-\frac {\sqrt {x} (a+b) (b-x) \left (\frac {b (a-x)}{a (b-x)}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {(a-b) x}{a (b-x)}\right )}{b (1-d) \left (-x (a+b)+a b+x^2\right )^{3/4}}+\frac {8 a b \operatorname {Gamma}\left (\frac {3}{4}\right ) \left (1-\frac {x}{a}\right )^{3/4} \left (1-\frac {x}{b}\right )^{3/4} \sqrt [4]{1-\frac {2 x}{-\sqrt {(a-b)^2}+a+b}} \left (\left (1-\frac {4 x}{-\sqrt {(a-b)^2}+a+b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {3}{2},\frac {2 \sqrt {(a-b)^2} x}{a (2 b-x)+\left (\sqrt {(a-b)^2}-b\right ) x}\right )+\frac {8 x \sqrt {(a-b)^2} (a-x) (b-x) \operatorname {Hypergeometric2F1}\left (\frac {7}{4},2,\frac {5}{2},\frac {2 \sqrt {(a-b)^2} x}{a (2 b-x)+\left (\sqrt {(a-b)^2}-b\right ) x}\right )}{\left (-\sqrt {(a-b)^2}+a+b\right ) \left (\sqrt {(a-b)^2}+a+b-2 x\right ) \left (a (2 b-x)+x \left (\sqrt {(a-b)^2}-b\right )\right )}\right )}{\sqrt {x} \operatorname {Gamma}\left (-\frac {1}{4}\right ) \left (1-\frac {2 x}{\sqrt {(a-b)^2}+a+b}\right )^{3/4} \left (-x (a+b)+a b+x^2\right )^{3/4}}\right )}{\left (x^2 (a-x) (b-x)\right )^{3/4}}\) |
Int[((-a + x)*(-b + x)*(-2*a*b + (a + b)*x))/((x^2*(-a + x)*(-b + x))^(3/4 )*(-(a*b) + (a + b)*x + (-1 + d)*x^2)),x]
3.17.33.3.1 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[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 1.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95
| method | result | size |
| pseudoelliptic | \(\frac {x \left (\ln \left (\frac {d^{\frac {1}{4}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{-d^{\frac {1}{4}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{x \,d^{\frac {1}{4}}}\right )\right ) d^{\frac {1}{4}}-4 \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{x}\) | \(105\) |
int((-a+x)*(-b+x)*(-2*a*b+(a+b)*x)/(x^2*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x +(-1+d)*x^2),x,method=_RETURNVERBOSE)
(x*(ln((d^(1/4)*x+(x^2*(a-x)*(b-x))^(1/4))/(-d^(1/4)*x+(x^2*(a-x)*(b-x))^( 1/4)))+2*arctan((x^2*(a-x)*(b-x))^(1/4)/x/d^(1/4)))*d^(1/4)-4*(x^2*(a-x)*( b-x))^(1/4))/x
Timed out. \[ \int \frac {(-a+x) (-b+x) (-2 a b+(a+b) x)}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
integrate((-a+x)*(-b+x)*(-2*a*b+(a+b)*x)/(x^2*(-a+x)*(-b+x))^(3/4)/(-a*b+( a+b)*x+(-1+d)*x^2),x, algorithm="fricas")
Timed out. \[ \int \frac {(-a+x) (-b+x) (-2 a b+(a+b) x)}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
integrate((-a+x)*(-b+x)*(-2*a*b+(a+b)*x)/(x**2*(-a+x)*(-b+x))**(3/4)/(-a*b +(a+b)*x+(-1+d)*x**2),x)
\[ \int \frac {(-a+x) (-b+x) (-2 a b+(a+b) x)}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int { -\frac {{\left (2 \, a b - {\left (a + b\right )} x\right )} {\left (a - x\right )} {\left (b - x\right )}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {3}{4}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}} \,d x } \]
integrate((-a+x)*(-b+x)*(-2*a*b+(a+b)*x)/(x^2*(-a+x)*(-b+x))^(3/4)/(-a*b+( a+b)*x+(-1+d)*x^2),x, algorithm="maxima")
-integrate((2*a*b - (a + b)*x)*(a - x)*(b - x)/(((a - x)*(b - x)*x^2)^(3/4 )*((d - 1)*x^2 - a*b + (a + b)*x)), x)
\[ \int \frac {(-a+x) (-b+x) (-2 a b+(a+b) x)}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int { -\frac {{\left (2 \, a b - {\left (a + b\right )} x\right )} {\left (a - x\right )} {\left (b - x\right )}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {3}{4}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}} \,d x } \]
integrate((-a+x)*(-b+x)*(-2*a*b+(a+b)*x)/(x^2*(-a+x)*(-b+x))^(3/4)/(-a*b+( a+b)*x+(-1+d)*x^2),x, algorithm="giac")
integrate(-(2*a*b - (a + b)*x)*(a - x)*(b - x)/(((a - x)*(b - x)*x^2)^(3/4 )*((d - 1)*x^2 - a*b + (a + b)*x)), x)
Timed out. \[ \int \frac {(-a+x) (-b+x) (-2 a b+(a+b) x)}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int -\frac {\left (2\,a\,b-x\,\left (a+b\right )\right )\,\left (a-x\right )\,\left (b-x\right )}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{3/4}\,\left (\left (d-1\right )\,x^2+\left (a+b\right )\,x-a\,b\right )} \,d x \]
int(-((2*a*b - x*(a + b))*(a - x)*(b - x))/((x^2*(a - x)*(b - x))^(3/4)*(x *(a + b) - a*b + x^2*(d - 1))),x)