Integrand size = 58, antiderivative size = 119 \[ \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}\right )}{d^{3/4}} \] 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)/(b-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 )/(b-x))/d^(3/4)
Time = 15.34 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.61 \[ \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)^2}}{b-x}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{(b-x)^2 x (-a+x)}}{-b+x}\right )\right )}{d^{3/4}} \] Input:
Integrate[(-(a*b) + (-a + 2*b)*x)/((x*(-a + x)*(-b + x)^2)^(1/4)*(-b^2 + ( 2*b - a*d)*x + (-1 + d)*x^2)),x]
Output:
(2*(ArcTan[(d^(1/4)*(x*(-a + x)*(-b + x)^2)^(1/4))/(b - x)] + ArcTanh[(d^( 1/4)*((b - x)^2*x*(-a + x))^(1/4))/(-b + 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.
\(\displaystyle \int \frac {x (2 b-a)-a b}{\sqrt [4]{x (x-a) (x-b)^2} \left (x (2 b-a d)-b^2+(d-1) x^2\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {a b+(a-2 b) x}{\sqrt [4]{x} \left (b^2+(1-d) x^2-(2 b-a d) x\right ) \sqrt [4]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2}}dx}{\sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {\sqrt {x} (a b+(a-2 b) x)}{\left (b^2+(1-d) x^2-(2 b-a d) x\right ) \sqrt [4]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2}}d\sqrt [4]{x}}{\sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a-2 b) x^{3/2}}{\left (b^2+(1-d) x^2-(2 b-a d) x\right ) \sqrt [4]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2}}+\frac {a b \sqrt {x}}{\left (b^2+(1-d) x^2-(2 b-a d) x\right ) \sqrt [4]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {a \sqrt {x} (b+x)-2 b x^{3/2}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \left (b^2+(1-d) x^2-(2 b-a d) x\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int -\frac {2 b x^{3/2}-a \sqrt {x} (b+x)}{\sqrt [4]{a-x} \sqrt {b-x} \left (b^2+(1-d) x^2-(2 b-a d) x\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {2 b x^{3/2}-a \sqrt {x} (b+x)}{\sqrt [4]{a-x} \sqrt {b-x} \left (b^2+(1-d) x^2-(2 b-a d) x\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {a \left (1-\frac {2 b}{a}\right ) x^{3/2}}{\sqrt [4]{a-x} \sqrt {b-x} \left (-b^2-(1-d) x^2+(2 b-a d) x\right )}+\frac {a b \sqrt {x}}{\sqrt [4]{a-x} \sqrt {b-x} \left (-b^2-(1-d) x^2+(2 b-a d) x\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \left (-\frac {(a-2 b) \left (-\sqrt {d} \sqrt {a^2 d-4 a b+4 b^2}-a d+2 b\right ) \int \frac {\sqrt {x}}{\sqrt [4]{a-x} \sqrt {b-x} \left (2 b-a d-2 (1-d) x-\sqrt {d} \sqrt {d a^2-4 b a+4 b^2}\right )}d\sqrt [4]{x}}{\sqrt {d} \sqrt {a^2 d-4 a b+4 b^2}}-\frac {2 a b (1-d) \int \frac {\sqrt {x}}{\sqrt [4]{a-x} \sqrt {b-x} \left (2 b-a d-2 (1-d) x-\sqrt {d} \sqrt {d a^2-4 b a+4 b^2}\right )}d\sqrt [4]{x}}{\sqrt {d} \sqrt {a^2 d-4 a b+4 b^2}}+\frac {(a-2 b) \left (\sqrt {d} \sqrt {a^2 d-4 a b+4 b^2}-a d+2 b\right ) \int \frac {\sqrt {x}}{\sqrt [4]{a-x} \sqrt {b-x} \left (2 b-a d-2 (1-d) x+\sqrt {d} \sqrt {d a^2-4 b a+4 b^2}\right )}d\sqrt [4]{x}}{\sqrt {d} \sqrt {a^2 d-4 a b+4 b^2}}-\frac {2 a b (1-d) \int \frac {\sqrt {x}}{\sqrt [4]{a-x} \sqrt {b-x} \left (-2 b+a d+2 (1-d) x-\sqrt {d} \sqrt {d a^2-4 b a+4 b^2}\right )}d\sqrt [4]{x}}{\sqrt {d} \sqrt {a^2 d-4 a b+4 b^2}}\right )}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
Input:
Int[(-(a*b) + (-a + 2*b)*x)/((x*(-a + x)*(-b + x)^2)^(1/4)*(-b^2 + (2*b - a*d)*x + (-1 + d)*x^2)),x]
Output:
$Aborted
\[\int \frac {-a b +\left (-a +2 b \right ) x}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (-b^{2}+\left (-a d +2 b \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
Input:
int((-a*b+(-a+2*b)*x)/(x*(-a+x)*(-b+x)^2)^(1/4)/(-b^2+(-a*d+2*b)*x+(-1+d)* x^2),x)
Output:
int((-a*b+(-a+2*b)*x)/(x*(-a+x)*(-b+x)^2)^(1/4)/(-b^2+(-a*d+2*b)*x+(-1+d)* x^2),x)
Timed out. \[ \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((-a*b+(-a+2*b)*x)/(x*(-a+x)*(-b+x)^2)^(1/4)/(-b^2+(-a*d+2*b)*x+( -1+d)*x^2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((-a*b+(-a+2*b)*x)/(x*(-a+x)*(-b+x)**2)**(1/4)/(-b**2+(-a*d+2*b)* x+(-1+d)*x**2),x)
Output:
Timed out
\[ \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {a b + {\left (a - 2 \, b\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{4}} {\left ({\left (d - 1\right )} x^{2} - b^{2} - {\left (a d - 2 \, b\right )} x\right )}} \,d x } \] Input:
integrate((-a*b+(-a+2*b)*x)/(x*(-a+x)*(-b+x)^2)^(1/4)/(-b^2+(-a*d+2*b)*x+( -1+d)*x^2),x, algorithm="maxima")
Output:
-integrate((a*b + (a - 2*b)*x)/((-(a - x)*(b - x)^2*x)^(1/4)*((d - 1)*x^2 - b^2 - (a*d - 2*b)*x)), x)
\[ \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {a b + {\left (a - 2 \, b\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{4}} {\left ({\left (d - 1\right )} x^{2} - b^{2} - {\left (a d - 2 \, b\right )} x\right )}} \,d x } \] Input:
integrate((-a*b+(-a+2*b)*x)/(x*(-a+x)*(-b+x)^2)^(1/4)/(-b^2+(-a*d+2*b)*x+( -1+d)*x^2),x, algorithm="giac")
Output:
integrate(-(a*b + (a - 2*b)*x)/((-(a - x)*(b - x)^2*x)^(1/4)*((d - 1)*x^2 - b^2 - (a*d - 2*b)*x)), x)
Timed out. \[ \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx=-\int \frac {a\,b+x\,\left (a-2\,b\right )}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (x\,\left (2\,b-a\,d\right )-b^2+x^2\,\left (d-1\right )\right )} \,d x \] Input:
int(-(a*b + x*(a - 2*b))/((-x*(a - x)*(b - x)^2)^(1/4)*(x*(2*b - a*d) - b^ 2 + x^2*(d - 1))),x)
Output:
-int((a*b + x*(a - 2*b))/((-x*(a - x)*(b - x)^2)^(1/4)*(x*(2*b - a*d) - b^ 2 + x^2*(d - 1))), x)
\[ \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx=\left (\int \frac {x}{x^{\frac {5}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} a d +x^{\frac {1}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} b^{2}-2 x^{\frac {5}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} b -x^{\frac {9}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} d +x^{\frac {9}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}}}d x \right ) a -2 \left (\int \frac {x}{x^{\frac {5}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} a d +x^{\frac {1}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} b^{2}-2 x^{\frac {5}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} b -x^{\frac {9}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} d +x^{\frac {9}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}}}d x \right ) b +\left (\int \frac {1}{x^{\frac {5}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} a d +x^{\frac {1}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} b^{2}-2 x^{\frac {5}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} b -x^{\frac {9}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}} d +x^{\frac {9}{4}} \left (-a \,b^{2}+2 a b x -a \,x^{2}+b^{2} x -2 b \,x^{2}+x^{3}\right )^{\frac {1}{4}}}d x \right ) a b \] Input:
int((-a*b+(-a+2*b)*x)/(x*(-a+x)*(-b+x)^2)^(1/4)/(-b^2+(-a*d+2*b)*x+(-1+d)* x^2),x)
Output:
int(x/(x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)* *(1/4)*a*d*x + x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*b**2 - 2*x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*b*x - x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b** 2*x - 2*b*x**2 + x**3)**(1/4)*d*x**2 + x**(1/4)*( - a*b**2 + 2*a*b*x - a*x **2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*x**2),x)*a - 2*int(x/(x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*a*d*x + x**(1 /4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*b**2 - 2*x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1 /4)*b*x - x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x** 3)**(1/4)*d*x**2 + x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x **2 + x**3)**(1/4)*x**2),x)*b + int(1/(x**(1/4)*( - a*b**2 + 2*a*b*x - a*x **2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*a*d*x + x**(1/4)*( - a*b**2 + 2*a*b *x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*b**2 - 2*x**(1/4)*( - a*b** 2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*b*x - x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*d*x**2 + x** (1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*x** 2),x)*a*b