3.27.41 \(\int \frac {(b^2-2 b x+x^2) (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3)}{(x (-a+x)^2 (-b+x)^3)^{3/4} (b d+(a^2-d) x-2 a x^2+x^3)} \, dx\) [2641]

3.27.41.1 Optimal result

Integrand size = 85, antiderivative size = 235 \[ \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6\right )^{3/4}}{(b-x)^2 x (-a+x)^2}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6\right )^{3/4}}{(b-x)^2 x (-a+x)^2}\right )}{d^{3/4}} \]

output

2*arctan(d^(1/4)*(-a^2*b^3*x+(3*a^2*b^2+2*a*b^3)*x^2+(-3*a^2*b-6*a*b^2-b^3 
)*x^3+(a^2+6*a*b+3*b^2)*x^4+(-2*a-3*b)*x^5+x^6)^(3/4)/(b-x)^2/x/(-a+x)^2)/ 
d^(3/4)-2*arctanh(d^(1/4)*(-a^2*b^3*x+(3*a^2*b^2+2*a*b^3)*x^2+(-3*a^2*b-6* 
a*b^2-b^3)*x^3+(a^2+6*a*b+3*b^2)*x^4+(-2*a-3*b)*x^5+x^6)^(3/4)/(b-x)^2/x/( 
-a+x)^2)/d^(3/4)
 

3.27.41.2 Mathematica [F]

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

input

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

output

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

3.27.41.3 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.

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

\(\Big \downarrow \) 2467

\(\displaystyle \frac {x^{3/4} \left (-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5\right )^{3/4} \int -\frac {\left (b^2-2 x b+x^2\right ) \left (-2 x^3+(2 a+3 b) x^2-4 a b x+a^2 b\right )}{x^{3/4} \left (x^3-2 a x^2+\left (a^2-d\right ) x+b d\right ) \left (x^5-(2 a+3 b) x^4+\left (a^2+6 b a+3 b^2\right ) x^3-b \left (3 a^2+6 b a+b^2\right ) x^2+a b^2 (3 a+2 b) x-a^2 b^3\right )^{3/4}}dx}{\left (x \left (-(a-x)^2\right ) (b-x)^3\right )^{3/4}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {x^{3/4} \left (-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5\right )^{3/4} \int \frac {\left (b^2-2 x b+x^2\right ) \left (-2 x^3+(2 a+3 b) x^2-4 a b x+a^2 b\right )}{x^{3/4} \left (x^3-2 a x^2+\left (a^2-d\right ) x+b d\right ) \left (x^5-(2 a+3 b) x^4+\left (a^2+6 b a+3 b^2\right ) x^3-b \left (3 a^2+6 b a+b^2\right ) x^2+a b^2 (3 a+2 b) x-a^2 b^3\right )^{3/4}}dx}{\left (x \left (-(a-x)^2\right ) (b-x)^3\right )^{3/4}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {4 x^{3/4} \left (-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5\right )^{3/4} \int \frac {\left (b^2-2 x b+x^2\right ) \left (-2 x^3+(2 a+3 b) x^2-4 a b x+a^2 b\right )}{\left (x^3-2 a x^2+\left (a^2-d\right ) x+b d\right ) \left (x^5-(2 a+3 b) x^4+\left (a^2+6 b a+3 b^2\right ) x^3-b \left (3 a^2+6 b a+b^2\right ) x^2+a b^2 (3 a+2 b) x-a^2 b^3\right )^{3/4}}d\sqrt [4]{x}}{\left (x \left (-(a-x)^2\right ) (b-x)^3\right )^{3/4}}\)

\(\Big \downarrow \) 1380

\(\displaystyle -\frac {4 x^{3/4} \left (-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5\right )^{3/4} \int \frac {(b-x)^2 \left (-2 x^3+(2 a+3 b) x^2-4 a b x+a^2 b\right )}{\left (x^3-2 a x^2+\left (a^2-d\right ) x+b d\right ) \left (x^5-(2 a+3 b) x^4+\left (a^2+6 b a+3 b^2\right ) x^3-b \left (3 a^2+6 b a+b^2\right ) x^2+a b^2 (3 a+2 b) x-a^2 b^3\right )^{3/4}}d\sqrt [4]{x}}{\left (x \left (-(a-x)^2\right ) (b-x)^3\right )^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {4 x^{3/4} \left (-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5\right )^{3/4} \int \frac {(a-x) (b-x)^2 \left (2 x^2-3 b x+a b\right )}{\left ((a-x)^2 (x-b)^3\right )^{3/4} \left (b d+x \left (a^2-2 x a+x^2-d\right )\right )}d\sqrt [4]{x}}{\left (x \left (-(a-x)^2\right ) (b-x)^3\right )^{3/4}}\)

\(\Big \downarrow \) 2058

\(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/2} (x-b)^{9/4} \left (-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5\right )^{3/4} \int \frac {(b-x)^2 \left (2 x^2-3 b x+a b\right )}{\sqrt {a-x} (x-b)^{9/4} \left (b d+x \left (a^2-2 x a+x^2-d\right )\right )}d\sqrt [4]{x}}{\left (-(a-x)^2 (b-x)^3\right )^{3/4} \left (x \left (-(a-x)^2\right ) (b-x)^3\right )^{3/4}}\)

\(\Big \downarrow \) 281

\(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/2} (x-b)^{9/4} \left (-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5\right )^{3/4} \int \frac {2 x^2-3 b x+a b}{\sqrt {a-x} \sqrt [4]{x-b} \left (b d+x \left (a^2-2 x a+x^2-d\right )\right )}d\sqrt [4]{x}}{\left (-(a-x)^2 (b-x)^3\right )^{3/4} \left (x \left (-(a-x)^2\right ) (b-x)^3\right )^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/2} (x-b)^{9/4} \left (-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5\right )^{3/4} \int \left (\frac {2 x^2}{\sqrt {a-x} \sqrt [4]{x-b} \left (x^3-2 a x^2+a^2 \left (1-\frac {d}{a^2}\right ) x+b d\right )}+\frac {3 b x}{\sqrt {a-x} \sqrt [4]{x-b} \left (-x^3+2 a x^2-a^2 \left (1-\frac {d}{a^2}\right ) x-b d\right )}+\frac {a b}{\sqrt {a-x} \sqrt [4]{x-b} \left (x^3-2 a x^2+a^2 \left (1-\frac {d}{a^2}\right ) x+b d\right )}\right )d\sqrt [4]{x}}{\left (-(a-x)^2 (b-x)^3\right )^{3/4} \left (x \left (-(a-x)^2\right ) (b-x)^3\right )^{3/4}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/2} (x-b)^{9/4} \left (-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5\right )^{3/4} \left (3 b \int \frac {x}{\sqrt {a-x} \sqrt [4]{x-b} \left (-x^3+2 a x^2-a^2 \left (1-\frac {d}{a^2}\right ) x-b d\right )}d\sqrt [4]{x}+a b \int \frac {1}{\sqrt {a-x} \sqrt [4]{x-b} \left (x^3-2 a x^2+a^2 \left (1-\frac {d}{a^2}\right ) x+b d\right )}d\sqrt [4]{x}+2 \int \frac {x^2}{\sqrt {a-x} \sqrt [4]{x-b} \left (x^3-2 a x^2+a^2 \left (1-\frac {d}{a^2}\right ) x+b d\right )}d\sqrt [4]{x}\right )}{\left (-(a-x)^2 (b-x)^3\right )^{3/4} \left (x \left (-(a-x)^2\right ) (b-x)^3\right )^{3/4}}\)

input

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

output

$Aborted
 

3.27.41.3.1 Defintions of rubi rules used

rule 25

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

rule 281

Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ 
Symbol] :> Simp[(b/d)^p   Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, 
 c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] &&  !(IntegerQ[q] & 
& SimplerQ[a + b*x^n, c + d*x^n])
 

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 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 7293

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

3.27.41.4 Maple [F]

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

input

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

output

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

3.27.41.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.27.41.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.27.41.7 Maxima [F]

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

input

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

output

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

3.27.41.8 Giac [F]

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

input

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

output

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

3.27.41.9 Mupad [F(-1)]

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

input

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

output

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