3.12.11 \(\int \frac {(a-3 b+2 x) (a^2-2 a x+x^2)}{((-a+x) (-b+x))^{3/4} (-b+a^3 d+(1-3 a^2 d) x+3 a d x^2-d x^3)} \, dx\) [1111]

3.12.11.1 Optimal result

Integrand size = 69, antiderivative size = 83 \[ \int \frac {(a-3 b+2 x) \left (a^2-2 a x+x^2\right )}{((-a+x) (-b+x))^{3/4} \left (-b+a^3 d+\left (1-3 a^2 d\right ) x+3 a d x^2-d x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (a b+(-a-b) x+x^2\right )^{3/4}}{b-x}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (a b+(-a-b) x+x^2\right )^{3/4}}{b-x}\right )}{d^{3/4}} \]

output

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

3.12.11.2 Mathematica [A] (verified)

Time = 11.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.77 \[ \int \frac {(a-3 b+2 x) \left (a^2-2 a x+x^2\right )}{((-a+x) (-b+x))^{3/4} \left (-b+a^3 d+\left (1-3 a^2 d\right ) x+3 a d x^2-d x^3\right )} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} (-a+x)}{\sqrt [4]{(-a+x) (-b+x)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} ((-a+x) (-b+x))^{3/4}}{b-x}\right )\right )}{d^{3/4}} \]

input

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

output

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

3.12.11.3 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.

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

\(\Big \downarrow \) 2048

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.12.11.3.1 Defintions of rubi rules used

rule 2009

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

rule 2048

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) 
, x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F 
reeQ[{a, b, c, d, e, n, p}, 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] 
]
 

3.12.11.4 Maple [F]

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

input

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

output

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

3.12.11.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.12.11.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.12.11.7 Maxima [F]

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

input

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

output

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

3.12.11.8 Giac [F]

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

input

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

output

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

3.12.11.9 Mupad [F(-1)]

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

input

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

output

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