Integrand size = 71, antiderivative size = 125 \[ \int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x+(1-d) x^2\right )} \, dx=-\frac {4 \left (a b x^2+(-a-b) x^3+x^4\right )^{3/4}}{x (-a+x) (-b+x)}+2 \sqrt [4]{d} \arctan \left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )+2 \sqrt [4]{d} \text {arctanh}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right ) \]
-4*(a*b*x^2+(-a-b)*x^3+x^4)^(3/4)/x/(-a+x)/(-b+x)+2*d^(1/4)*arctan(x/d^(1/ 4)/(a*b*x^2+(-a-b)*x^3+x^4)^(1/4))+2*d^(1/4)*arctanh(x/d^(1/4)/(a*b*x^2+(- a-b)*x^3+x^4)^(1/4))
Time = 67.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.37 \[ \int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x+(1-d) x^2\right )} \, dx=-\frac {2 x \left (2 \sqrt {\frac {x}{-a+x}}+\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}}}{\sqrt {\frac {x}{-a+x}}}\right )-\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}}}{\sqrt {\frac {x}{-a+x}}}\right )\right )}{\sqrt {\frac {x}{-a+x}} \sqrt [4]{x^2 (-a+x) (-b+x)}} \]
Integrate[(-2*a*b*x^2 + (a + b)*x^3)/((-a + x)*(-b + x)*(x^2*(-a + x)*(-b + x))^(1/4)*(-(a*b*d) + (a + b)*d*x + (1 - d)*x^2)),x]
(-2*x*(2*Sqrt[x/(-a + x)] + d^(1/4)*((b - x)/(a - x))^(1/4)*ArcTan[(d^(1/4 )*((b - x)/(a - x))^(1/4))/Sqrt[x/(-a + x)]] - d^(1/4)*((b - x)/(a - x))^( 1/4)*ArcTanh[(d^(1/4)*((b - x)/(a - x))^(1/4))/Sqrt[x/(-a + x)]]))/(Sqrt[x /(-a + 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.
| \(\displaystyle \int \frac {x^3 (a+b)-2 a b x^2}{(x-a) (x-b) \sqrt [4]{x^2 (x-a) (x-b)} \left (d x (a+b)-a b d+(1-d) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x^2 (x (a+b)-2 a b)}{(x-a) (x-b) \sqrt [4]{x^2 (x-a) (x-b)} \left (d x (a+b)-a b d+(1-d) x^2\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{-x (a+b)+a b+x^2} \int \frac {x^{3/2} (2 a b-(a+b) x)}{(a-x) (b-x) \sqrt [4]{x^2-(a+b) x+a b} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}dx}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{-x (a+b)+a b+x^2} \int \frac {x^2 (2 a b-(a+b) x)}{(a-x) (b-x) \sqrt [4]{x^2-(a+b) x+a b} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {x^2 (2 a b-(a+b) x)}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \left (-\frac {d a^2+2 b a+b^2 d}{(1-d)^2 (a-x)^{5/4} (b-x)^{5/4}}+\frac {(a+b) x}{(1-d) (a-x)^{5/4} (b-x)^{5/4}}+\frac {a b d \left (d a^2+2 b a+b^2 d\right )-d \left (d a^3+3 b a^2+3 b^2 a+b^3 d\right ) x}{(d-1)^2 (a-x)^{5/4} (b-x)^{5/4} \left ((d-1) x^2-(a+b) d x+a b d\right )}\right )d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a-x} \sqrt [4]{b-x} \int \frac {-b x^3-a \left (x^3-2 b x^2\right )}{(a-x)^{5/4} (b-x)^{5/4} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt [4]{x^2 (a-x) (b-x)}}\) |
Int[(-2*a*b*x^2 + (a + b)*x^3)/((-a + x)*(-b + x)*(x^2*(-a + x)*(-b + x))^ (1/4)*(-(a*b*d) + (a + b)*d*x + (1 - d)*x^2)),x]
3.19.35.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, 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 = 0.94 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.08
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (2 x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\frac {\left (2 \arctan \left (\frac {\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{x \left (\frac {1}{d}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{-x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}\right )\right ) \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{2}\right )}{\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}} \left (\frac {1}{d}\right )^{\frac {1}{4}}}\) | \(135\) |
int((-2*a*b*x^2+(a+b)*x^3)/(-a+x)/(-b+x)/(x^2*(-a+x)*(-b+x))^(1/4)/(-a*b*d +(a+b)*d*x+(1-d)*x^2),x,method=_RETURNVERBOSE)
-2*(2*x*(1/d)^(1/4)+1/2*(2*arctan((x^2*(a-x)*(b-x))^(1/4)/x/(1/d)^(1/4))-l n((x*(1/d)^(1/4)+(x^2*(a-x)*(b-x))^(1/4))/(-x*(1/d)^(1/4)+(x^2*(a-x)*(b-x) )^(1/4))))*(x^2*(a-x)*(b-x))^(1/4))/(x^2*(a-x)*(b-x))^(1/4)/(1/d)^(1/4)
Timed out. \[ \int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x+(1-d) x^2\right )} \, dx=\text {Timed out} \]
integrate((-2*a*b*x^2+(a+b)*x^3)/(-a+x)/(-b+x)/(x^2*(-a+x)*(-b+x))^(1/4)/( -a*b*d+(a+b)*d*x+(1-d)*x^2),x, algorithm="fricas")
Timed out. \[ \int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x+(1-d) x^2\right )} \, dx=\text {Timed out} \]
integrate((-2*a*b*x**2+(a+b)*x**3)/(-a+x)/(-b+x)/(x**2*(-a+x)*(-b+x))**(1/ 4)/(-a*b*d+(a+b)*d*x+(1-d)*x**2),x)
\[ \int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x+(1-d) x^2\right )} \, dx=\int { \frac {2 \, a b x^{2} - {\left (a + b\right )} x^{3}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{4}} {\left (a b d - {\left (a + b\right )} d x + {\left (d - 1\right )} x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}} \,d x } \]
integrate((-2*a*b*x^2+(a+b)*x^3)/(-a+x)/(-b+x)/(x^2*(-a+x)*(-b+x))^(1/4)/( -a*b*d+(a+b)*d*x+(1-d)*x^2),x, algorithm="maxima")
integrate((2*a*b*x^2 - (a + b)*x^3)/(((a - x)*(b - x)*x^2)^(1/4)*(a*b*d - (a + b)*d*x + (d - 1)*x^2)*(a - x)*(b - x)), x)
\[ \int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x+(1-d) x^2\right )} \, dx=\int { \frac {2 \, a b x^{2} - {\left (a + b\right )} x^{3}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{4}} {\left (a b d - {\left (a + b\right )} d x + {\left (d - 1\right )} x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}} \,d x } \]
integrate((-2*a*b*x^2+(a+b)*x^3)/(-a+x)/(-b+x)/(x^2*(-a+x)*(-b+x))^(1/4)/( -a*b*d+(a+b)*d*x+(1-d)*x^2),x, algorithm="giac")
integrate((2*a*b*x^2 - (a + b)*x^3)/(((a - x)*(b - x)*x^2)^(1/4)*(a*b*d - (a + b)*d*x + (d - 1)*x^2)*(a - x)*(b - x)), x)
Timed out. \[ \int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x+(1-d) x^2\right )} \, dx=\int -\frac {x^3\,\left (a+b\right )-2\,a\,b\,x^2}{\left (a-x\right )\,\left (b-x\right )\,\left (\left (d-1\right )\,x^2-d\,\left (a+b\right )\,x+a\,b\,d\right )\,{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/4}} \,d x \]
int(-(x^3*(a + b) - 2*a*b*x^2)/((a - x)*(b - x)*(x^2*(d - 1) - d*x*(a + b) + a*b*d)*(x^2*(a - x)*(b - x))^(1/4)),x)