3.23.60 \(\int \frac {(-a+x) (-2 a+b+x)}{((-a+x) (-b+x)^2)^{3/4} (a+b^2 d-(1+2 b d) x+d x^2)} \, dx\) [2260]

3.23.60.1 Optimal result

Integrand size = 52, antiderivative size = 171 \[ \int \frac {(-a+x) (-2 a+b+x)}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=-\frac {4 \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}-2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{(b-x) (-a+x)}\right )+2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{(b-x) (-a+x)}\right ) \]

output

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

3.23.60.2 Mathematica [A] (verified)

Time = 13.97 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.76 \[ \int \frac {(-a+x) (-2 a+b+x)}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=\frac {2 (-b+x) \left (-2 a+2 x+\sqrt [4]{d} \sqrt {b-x} (-a+x)^{3/4} \arctan \left (\frac {\sqrt [4]{d} \sqrt {b-x}}{\sqrt [4]{-a+x}}\right )-\sqrt [4]{d} \sqrt {b-x} (-a+x)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt {b-x}}{\sqrt [4]{-a+x}}\right )\right )}{\left ((b-x)^2 (-a+x)\right )^{3/4}} \]

input

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

output

(2*(-b + x)*(-2*a + 2*x + d^(1/4)*Sqrt[b - x]*(-a + x)^(3/4)*ArcTan[(d^(1/ 
4)*Sqrt[b - x])/(-a + x)^(1/4)] - d^(1/4)*Sqrt[b - x]*(-a + x)^(3/4)*ArcTa 
nh[(d^(1/4)*Sqrt[b - x])/(-a + x)^(1/4)]))/((b - x)^2*(-a + x))^(3/4)
 

3.23.60.3 Rubi [F]

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.

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

\(\Big \downarrow \) 7270

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

input

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

output

$Aborted
 

3.23.60.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2153

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* 
x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ 
[m] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] && IGtQ[n, 0])
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7270

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p 
]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])))   Int[u*v 
^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !Free 
Q[v, x] &&  !FreeQ[w, x]
 

3.23.60.4 Maple [F]

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

input

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

output

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

3.23.60.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.23.60.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.23.60.7 Maxima [F]

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

input

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

output

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

3.23.60.8 Giac [F]

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

input

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

output

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

3.23.60.9 Mupad [F(-1)]

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

input

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

output

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