3.19.50 \(\int \frac {-((2 a-b) b^2)+(4 a-b) b x-(2 a+b) x^2+x^3}{((-a+x) (-b+x)^2)^{3/4} (a+b^2 d-(1+2 b d) x+d x^2)} \, dx\) [1850]

3.19.50.1 Optimal result

Integrand size = 77, antiderivative size = 127 \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{(b-x) (-a+x)}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{(b-x) (-a+x)}\right )}{d^{3/4}} \]

output

-2*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))/d^(3/4)+2*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))/d^(3/4)
 

3.19.50.2 Mathematica [A] (verified)

Time = 2.53 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.22 \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=\frac {\sqrt {2} \sqrt [4]{(b-x)^2 (-a+x)} \left (\arctan \left (\frac {-b \sqrt {d}+\sqrt {a-x}+\sqrt {d} x}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x}}{\sqrt {a-x}+\sqrt {d} (b-x)}\right )\right )}{d^{3/4} \sqrt [4]{a-x} \sqrt {b-x}} \]

input

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

output

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

3.19.50.3 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.

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

\(\Big \downarrow \) 7270

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2019

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

\(\Big \downarrow \) 2004

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 7239

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

input

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

output

$Aborted
 

3.19.50.3.1 Defintions of rubi rules used

rule 25

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

rule 2004

Int[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) 
, x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b 
, c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
 

rule 2019

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px 
, Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && 
 EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
 

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]
 

3.19.50.4 Maple [F]

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

input

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

output

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

3.19.50.5 Fricas [F(-1)]

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

input

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

3.19.50.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.19.50.7 Maxima [F]

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

input

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

3.19.50.8 Giac [F]

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

input

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

3.19.50.9 Mupad [F(-1)]

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

input

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

output

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