Integrand size = 54, antiderivative size = 83 \[ \int \frac {a-3 b+2 x}{\sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a b+(-a-b) x+x^2}}{a-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b+(-a-b) x+x^2}}{a-x}\right )}{d^{3/4}} \]
-2*arctan(d^(1/4)*(a*b+(-a-b)*x+x^2)^(1/4)/(a-x))/d^(3/4)+2*arctanh(d^(1/4 )*(a*b+(-a-b)*x+x^2)^(1/4)/(a-x))/d^(3/4)
Time = 10.55 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.77 \[ \int \frac {a-3 b+2 x}{\sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} (-b+x)}{((-a+x) (-b+x))^{3/4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{(-a+x) (-b+x)}}{a-x}\right )\right )}{d^{3/4}} \]
Integrate[(a - 3*b + 2*x)/(((-a + x)*(-b + x))^(1/4)*(-a^3 + b*d - (-3*a^2 + d)*x - 3*a*x^2 + x^3)),x]
(2*(ArcTan[(d^(1/4)*(-b + x))/((-a + x)*(-b + x))^(3/4)] + ArcTanh[(d^(1/4 )*((-a + x)*(-b + x))^(1/4))/(a - x)]))/d^(3/4)
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.
\(\displaystyle \int \frac {a-3 b+2 x}{\sqrt [4]{(x-a) (x-b)} \left (-a^3-x \left (d-3 a^2\right )-3 a x^2+b d+x^3\right )} \, dx\) |
\(\Big \downarrow \) 2048 |
\(\displaystyle \int \frac {a-3 b+2 x}{\sqrt [4]{x (-a-b)+a b+x^2} \left (-a^3-x \left (d-3 a^2\right )-3 a x^2+b d+x^3\right )}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-a+3 b-2 x}{\sqrt [4]{-x (a+b)+a b+x^2} \left (a^3-x \left (3 a^2-d\right )+3 a x^2-b d-x^3\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 b \left (1-\frac {a}{3 b}\right )}{\sqrt [4]{-x (a+b)+a b+x^2} \left (a^3-x \left (3 a^2-d\right )+3 a x^2-b d-x^3\right )}+\frac {2 x}{\sqrt [4]{-x (a+b)+a b+x^2} \left (-a^3+x \left (3 a^2-d\right )-3 a x^2+b d+x^3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {x}{\sqrt [4]{x^2-(a+b) x+a b} \left (-a^3-3 x^2 a+x^3+b d+\left (3 a^2-d\right ) x\right )}dx-(a-3 b) \int \frac {1}{\sqrt [4]{x^2-(a+b) x+a b} \left (a^3+3 x^2 a-x^3-b d-\left (3 a^2-d\right ) x\right )}dx\) |
Int[(a - 3*b + 2*x)/(((-a + x)*(-b + x))^(1/4)*(-a^3 + b*d - (-3*a^2 + d)* x - 3*a*x^2 + x^3)),x]
3.12.10.3.1 Defintions of rubi rules used
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]
\[\int \frac {a -3 b +2 x}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{4}} \left (-a^{3}+d b -\left (-3 a^{2}+d \right ) x -3 a \,x^{2}+x^{3}\right )}d x\]
Timed out. \[ \int \frac {a-3 b+2 x}{\sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx=\text {Timed out} \]
integrate((a-3*b+2*x)/((-a+x)*(-b+x))^(1/4)/(-a^3+b*d-(-3*a^2+d)*x-3*a*x^2 +x^3),x, algorithm="fricas")
Timed out. \[ \int \frac {a-3 b+2 x}{\sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx=\text {Timed out} \]
\[ \int \frac {a-3 b+2 x}{\sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx=\int { -\frac {a - 3 \, b + 2 \, x}{{\left (a^{3} + 3 \, a x^{2} - x^{3} - b d - {\left (3 \, a^{2} - d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{4}}} \,d x } \]
integrate((a-3*b+2*x)/((-a+x)*(-b+x))^(1/4)/(-a^3+b*d-(-3*a^2+d)*x-3*a*x^2 +x^3),x, algorithm="maxima")
-integrate((a - 3*b + 2*x)/((a^3 + 3*a*x^2 - x^3 - b*d - (3*a^2 - d)*x)*(( a - x)*(b - x))^(1/4)), x)
\[ \int \frac {a-3 b+2 x}{\sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx=\int { -\frac {a - 3 \, b + 2 \, x}{{\left (a^{3} + 3 \, a x^{2} - x^{3} - b d - {\left (3 \, a^{2} - d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{4}}} \,d x } \]
integrate((a-3*b+2*x)/((-a+x)*(-b+x))^(1/4)/(-a^3+b*d-(-3*a^2+d)*x-3*a*x^2 +x^3),x, algorithm="giac")
integrate(-(a - 3*b + 2*x)/((a^3 + 3*a*x^2 - x^3 - b*d - (3*a^2 - d)*x)*(( a - x)*(b - x))^(1/4)), x)
Timed out. \[ \int \frac {a-3 b+2 x}{\sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx=\int -\frac {a-3\,b+2\,x}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{1/4}\,\left (3\,a\,x^2-b\,d+x\,\left (d-3\,a^2\right )+a^3-x^3\right )} \,d x \]
int(-(a - 3*b + 2*x)/(((a - x)*(b - x))^(1/4)*(3*a*x^2 - b*d + x*(d - 3*a^ 2) + a^3 - x^3)),x)