3.18.61 \(\int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} (-b^2+(2 b-a d) x+(-1+d) x^2)} \, dx\) [1761]

3.18.61.1 Optimal result

Integrand size = 58, antiderivative size = 119 \[ \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}\right )}{d^{3/4}} \]

output

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

3.18.61.2 Mathematica [A] (verified)

Time = 15.36 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.61 \[ \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)^2}}{b-x}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{(b-x)^2 x (-a+x)}}{-b+x}\right )\right )}{d^{3/4}} \]

input

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

output

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

3.18.61.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.18.61.3.1 Defintions of rubi rules used

rule 25

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2035

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]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 2467

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]
 

rule 7239

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

rule 7279

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]
 

3.18.61.4 Maple [F]

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

input

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

output

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

3.18.61.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.18.61.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.18.61.7 Maxima [F]

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

input

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

output

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

3.18.61.8 Giac [F]

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

input

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

output

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

3.18.61.9 Mupad [F(-1)]

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

input

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

output

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