3.18.63 \(\int \frac {a b^2-2 (2 a-b) b x+(3 a-2 b) x^2}{\sqrt [4]{x (-a+x) (-b+x)^2} (a^3+(-3 a^2+b^2 d) x+(3 a-2 b d) x^2+(-1+d) x^3)} \, dx\) [1763]

3.18.63.1 Optimal result

Integrand size = 85, antiderivative size = 119 \[ \int \frac {a b^2-2 (2 a-b) b x+(3 a-2 b) x^2}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (a^3+\left (-3 a^2+b^2 d\right ) x+(3 a-2 b d) x^2+(-1+d) x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{a-x}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{a-x}\right )}{d^{3/4}} \]

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)/(a-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 
)/(a-x))/d^(3/4)
 

3.18.63.2 Mathematica [A] (verified)

Time = 10.76 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.61 \[ \int \frac {a b^2-2 (2 a-b) b x+(3 a-2 b) x^2}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (a^3+\left (-3 a^2+b^2 d\right ) x+(3 a-2 b d) x^2+(-1+d) x^3\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)^2}}{a-x}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{(b-x)^2 x (-a+x)}}{-a+x}\right )\right )}{d^{3/4}} \]

input

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

output

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

3.18.63.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 1387

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.18.63.3.1 Defintions of rubi rules used

rule 1387

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)* 
(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^ 
p, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - 
b*d*e + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
 

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 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.18.63.4 Maple [F]

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

input

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

output

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

3.18.63.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.18.63.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.18.63.7 Maxima [F]

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

input

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

output

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

3.18.63.8 Giac [F]

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

input

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

output

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

3.18.63.9 Mupad [F(-1)]

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

input

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

output

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