3.19.36 \(\int \frac {(a^2-2 a x+x^2) (a b^2-2 (2 a-b) b x+(3 a-2 b) x^2)}{(x (-a+x) (-b+x)^2)^{3/4} (a^3 d+(b^2-3 a^2 d) x+(-2 b+3 a d) x^2+(1-d) x^3)} \, dx\) [1836]

3.19.36.1 Optimal result

Integrand size = 99, antiderivative size = 125 \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (a b^2-2 (2 a-b) b x+(3 a-2 b) x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a^3 d+\left (b^2-3 a^2 d\right ) x+(-2 b+3 a d) x^2+(1-d) x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{3/4}}{x (-b+x)^2}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{3/4}}{x (-b+x)^2}\right )}{d^{3/4}} \]

output

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

3.19.36.2 Mathematica [F]

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

input

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

output

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

3.19.36.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 2058

\(\displaystyle \frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {(a-x)^{5/4} \left (a b^2-2 (2 a-b) x b+(3 a-2 b) x^2\right )}{(x-b)^{3/2} \left (d a^3+(1-d) x^3-(2 b-3 a d) x^2+\left (b^2-3 a^2 d\right ) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\)

\(\Big \downarrow \) 1387

\(\displaystyle \frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {(a-x)^{5/4} ((3 a-2 b) x-a b)}{\sqrt {x-b} \left (d a^3+(1-d) x^3-(2 b-3 a d) x^2+\left (b^2-3 a^2 d\right ) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (\frac {a b (a-x)^{5/4}}{\sqrt {x-b} \left (-d a^3-(1-d) x^3+2 b \left (1-\frac {3 a d}{2 b}\right ) x^2-b^2 \left (1-\frac {3 a^2 d}{b^2}\right ) x\right )}+\frac {(3 a-2 b) x (a-x)^{5/4}}{\sqrt {x-b} \left (d a^3+(1-d) x^3-2 b \left (1-\frac {3 a d}{2 b}\right ) x^2+b^2 \left (1-\frac {3 a^2 d}{b^2}\right ) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \left (a b \int \frac {(a-x)^{5/4}}{\sqrt {x-b} \left (-d a^3-(1-d) x^3+2 b \left (1-\frac {3 a d}{2 b}\right ) x^2-b^2 \left (1-\frac {3 a^2 d}{b^2}\right ) x\right )}d\sqrt [4]{x}+(3 a-2 b) \int \frac {(a-x)^{5/4} x}{\sqrt {x-b} \left (d a^3+(1-d) x^3-2 b \left (1-\frac {3 a d}{2 b}\right ) x^2+b^2 \left (1-\frac {3 a^2 d}{b^2}\right ) x\right )}d\sqrt [4]{x}\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\)

input

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

output

$Aborted
 

3.19.36.3.1 Defintions of rubi rules used

rule 1380

Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

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] 
]
 

3.19.36.4 Maple [F]

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

input

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

output

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

3.19.36.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.19.36.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.19.36.7 Maxima [F]

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

input

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

output

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

3.19.36.8 Giac [F]

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

input

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

output

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

3.19.36.9 Mupad [F(-1)]

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

input

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

output

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