Integrand size = 82, antiderivative size = 76 \[ \int \frac {a b x-x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=-\frac {\arctan \left (\frac {x}{\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}} \] Output:
-arctan(x/d^(1/4)/(a*b*x+(-a-b)*x^2+x^3)^(1/2))/d^(1/4)+arctanh(x/d^(1/4)/ (a*b*x+(-a-b)*x^2+x^3)^(1/2))/d^(1/4)
Time = 10.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \frac {a b x-x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\frac {-\arctan \left (\frac {x}{\sqrt [4]{d} \sqrt {x (-a+x) (-b+x)}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{d} \sqrt {x (-a+x) (-b+x)}}\right )}{\sqrt [4]{d}} \] Input:
Integrate[(a*b*x - x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(a^2*b^2*d - 2*a*b*(a + b)*d*x + (-1 + a^2*d + 4*a*b*d + b^2*d)*x^2 - 2*(a + b)*d*x^3 + d*x^4)),x ]
Output:
(-ArcTan[x/(d^(1/4)*Sqrt[x*(-a + x)*(-b + x)])] + ArcTanh[x/(d^(1/4)*Sqrt[ x*(-a + x)*(-b + x)])])/d^(1/4)
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.
| \(\displaystyle \int \frac {a b x-x^3}{\sqrt {x (x-a) (x-b)} \left (x^2 \left (a^2 d+4 a b d+b^2 d-1\right )+a^2 b^2 d-2 d x^3 (a+b)-2 a b d x (a+b)+d x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x \left (a b-x^2\right )}{\sqrt {x (x-a) (x-b)} \left (x^2 \left (a^2 d+4 a b d+b^2 d-1\right )+a^2 b^2 d-2 d x^3 (a+b)-2 a b d x (a+b)+d x^4\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {\sqrt {x} \left (a b-x^2\right )}{\sqrt {x^2-(a+b) x+a b} \left (d x^4-2 (a+b) d x^3-\left (-d a^2-4 b d a-b^2 d+1\right ) x^2-2 a b (a+b) d x+a^2 b^2 d\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x \left (a b-x^2\right )}{\sqrt {x^2-(a+b) x+a b} \left (d x^4-2 (a+b) d x^3-\left (-d a^2-4 b d a-b^2 d+1\right ) x^2-2 a b (a+b) d x+a^2 b^2 d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {x^3}{\sqrt {x^2-(a+b) x+a b} \left (-d x^4+2 a \left (\frac {b}{a}+1\right ) d x^3+\left (1-\left (a^2+4 b a+b^2\right ) d\right ) x^2+2 a^2 b \left (\frac {b}{a}+1\right ) d x-a^2 b^2 d\right )}+\frac {a b x}{\sqrt {x^2-(a+b) x+a b} \left (d x^4-2 a \left (\frac {b}{a}+1\right ) d x^3-\left (1-\left (a^2+4 b a+b^2\right ) d\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) d x+a^2 b^2 d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \left (\int \frac {x^3}{\sqrt {x^2-(a+b) x+a b} \left (-d x^4+2 a \left (\frac {b}{a}+1\right ) d x^3+\left (1-\left (a^2+4 b a+b^2\right ) d\right ) x^2+2 a^2 b \left (\frac {b}{a}+1\right ) d x-a^2 b^2 d\right )}d\sqrt {x}+a b \int \frac {x}{\sqrt {x^2-(a+b) x+a b} \left (d x^4-2 a \left (\frac {b}{a}+1\right ) d x^3-\left (1-\left (a^2+4 b a+b^2\right ) d\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) d x+a^2 b^2 d\right )}d\sqrt {x}\right )}{\sqrt {x (a-x) (b-x)}}\) |
Input:
Int[(a*b*x - x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(a^2*b^2*d - 2*a*b*(a + b)*d* x + (-1 + a^2*d + 4*a*b*d + b^2*d)*x^2 - 2*(a + b)*d*x^3 + d*x^4)),x]
Output:
$Aborted
Time = 0.84 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(\frac {\left (\frac {1}{d}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}{-\left (\frac {1}{d}\right )^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}\right )+2 \arctan \left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \left (\frac {1}{d}\right )^{\frac {1}{4}}}\right )\right )}{2}\) | \(84\) |
| pseudoelliptic | \(\frac {\left (\frac {1}{d}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}{-\left (\frac {1}{d}\right )^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}\right )+2 \arctan \left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \left (\frac {1}{d}\right )^{\frac {1}{4}}}\right )\right )}{2}\) | \(84\) |
| elliptic | \(-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (-2 a d -2 b d \right ) \textit {\_Z}^{3}+\left (a^{2} d +4 a b d +b^{2} d -1\right ) \textit {\_Z}^{2}+\left (-2 a^{2} b d -2 a \,b^{2} d \right ) \textit {\_Z} +a^{2} b^{2} d \right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-a b \right ) \left (-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, \frac {-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b}{b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +3 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -4 \underline {\hspace {1.25 ex}}\alpha a d b -\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +a^{2} b d +a \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{b}\) | \(304\) |
Input:
int((a*b*x-x^3)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2*d-2*a*b*(a+b)*d*x+(a^2*d+ 4*a*b*d+b^2*d-1)*x^2-2*(a+b)*d*x^3+d*x^4),x,method=_RETURNVERBOSE)
Output:
1/2*(1/d)^(1/4)*(ln(((1/d)^(1/4)*x+(x*(a-x)*(b-x))^(1/2))/(-(1/d)^(1/4)*x+ (x*(a-x)*(b-x))^(1/2)))+2*arctan((x*(a-x)*(b-x))^(1/2)/x/(1/d)^(1/4)))
Result contains complex when optimal does not.
Time = 1.01 (sec) , antiderivative size = 787, normalized size of antiderivative = 10.36 \[ \int \frac {a b x-x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx =\text {Too large to display} \] Input:
integrate((a*b*x-x^3)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2*d-2*a*b*(a+b)*d*x+( a^2*d+4*a*b*d+b^2*d-1)*x^2-2*(a+b)*d*x^3+d*x^4),x, algorithm="fricas")
Output:
1/4*log((a^2*b^2*d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d*x + ((a ^2 + 4*a*b + b^2)*d + 1)*x^2 + 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d^(1/4)* x + (a*b*d - (a + b)*d*x + d*x^2)/d^(1/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2*d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d *x + ((a^2 + 4*a*b + b^2)*d - 1)*x^2))/d^(1/4) - 1/4*log((a^2*b^2*d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d*x + ((a^2 + 4*a*b + b^2)*d + 1)* x^2 - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d^(1/4)*x + (a*b*d - (a + b)*d*x + d*x^2)/d^(1/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2* d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d*x + ((a^2 + 4*a*b + b^2) *d - 1)*x^2))/d^(1/4) + 1/4*I*log((a^2*b^2*d - 2*(a + b)*d*x^3 + d*x^4 - 2 *(a^2*b + a*b^2)*d*x + ((a^2 + 4*a*b + b^2)*d + 1)*x^2 - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(I*d^(1/4)*x + (-I*a*b*d + I*(a + b)*d*x - I*d*x^2)/d^(1/ 4)) - 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2*d - 2*(a + b)* d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d*x + ((a^2 + 4*a*b + b^2)*d - 1)*x^2))/ d^(1/4) - 1/4*I*log((a^2*b^2*d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^ 2)*d*x + ((a^2 + 4*a*b + b^2)*d + 1)*x^2 - 2*sqrt(a*b*x - (a + b)*x^2 + x^ 3)*(-I*d^(1/4)*x + (I*a*b*d - I*(a + b)*d*x + I*d*x^2)/d^(1/4)) - 2*(a*b*d *x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2*d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d*x + ((a^2 + 4*a*b + b^2)*d - 1)*x^2))/d^(1/4)
Timed out. \[ \int \frac {a b x-x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((a*b*x-x**3)/(x*(-a+x)*(-b+x))**(1/2)/(a**2*b**2*d-2*a*b*(a+b)*d *x+(a**2*d+4*a*b*d+b**2*d-1)*x**2-2*(a+b)*d*x**3+d*x**4),x)
Output:
Timed out
\[ \int \frac {a b x-x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\int { \frac {a b x - x^{3}}{{\left (a^{2} b^{2} d - 2 \, {\left (a + b\right )} a b d x - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} + {\left (a^{2} d + 4 \, a b d + b^{2} d - 1\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \] Input:
integrate((a*b*x-x^3)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2*d-2*a*b*(a+b)*d*x+( a^2*d+4*a*b*d+b^2*d-1)*x^2-2*(a+b)*d*x^3+d*x^4),x, algorithm="maxima")
Output:
integrate((a*b*x - x^3)/((a^2*b^2*d - 2*(a + b)*a*b*d*x - 2*(a + b)*d*x^3 + d*x^4 + (a^2*d + 4*a*b*d + b^2*d - 1)*x^2)*sqrt((a - x)*(b - x)*x)), x)
\[ \int \frac {a b x-x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\int { \frac {a b x - x^{3}}{{\left (a^{2} b^{2} d - 2 \, {\left (a + b\right )} a b d x - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} + {\left (a^{2} d + 4 \, a b d + b^{2} d - 1\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \] Input:
integrate((a*b*x-x^3)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2*d-2*a*b*(a+b)*d*x+( a^2*d+4*a*b*d+b^2*d-1)*x^2-2*(a+b)*d*x^3+d*x^4),x, algorithm="giac")
Output:
integrate((a*b*x - x^3)/((a^2*b^2*d - 2*(a + b)*a*b*d*x - 2*(a + b)*d*x^3 + d*x^4 + (a^2*d + 4*a*b*d + b^2*d - 1)*x^2)*sqrt((a - x)*(b - x)*x)), x)
Time = 14.46 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.30 \[ \int \frac {a b x-x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\frac {\ln \left (\frac {x+2\,d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}+\sqrt {d}\,x^2+a\,b\,\sqrt {d}-a\,\sqrt {d}\,x-b\,\sqrt {d}\,x}{x-\sqrt {d}\,x^2-a\,b\,\sqrt {d}+a\,\sqrt {d}\,x+b\,\sqrt {d}\,x}\right )}{2\,d^{1/4}}+\frac {\ln \left (\frac {x-\sqrt {d}\,x^2-a\,b\,\sqrt {d}+a\,\sqrt {d}\,x+b\,\sqrt {d}\,x-d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,2{}\mathrm {i}}{x+\sqrt {d}\,x^2+a\,b\,\sqrt {d}-a\,\sqrt {d}\,x-b\,\sqrt {d}\,x}\right )\,1{}\mathrm {i}}{2\,d^{1/4}} \] Input:
int(-(x^3 - a*b*x)/((x*(a - x)*(b - x))^(1/2)*(x^2*(a^2*d + b^2*d + 4*a*b* d - 1) + d*x^4 + a^2*b^2*d - 2*d*x^3*(a + b) - 2*a*b*d*x*(a + b))),x)
Output:
log((x + 2*d^(1/4)*(x*(a - x)*(b - x))^(1/2) + d^(1/2)*x^2 + a*b*d^(1/2) - a*d^(1/2)*x - b*d^(1/2)*x)/(x - d^(1/2)*x^2 - a*b*d^(1/2) + a*d^(1/2)*x + b*d^(1/2)*x))/(2*d^(1/4)) + (log((x - d^(1/4)*(x*(a - x)*(b - x))^(1/2)*2 i - d^(1/2)*x^2 - a*b*d^(1/2) + a*d^(1/2)*x + b*d^(1/2)*x)/(x + d^(1/2)*x^ 2 + a*b*d^(1/2) - a*d^(1/2)*x - b*d^(1/2)*x))*1i)/(2*d^(1/4))
\[ \int \frac {a b x-x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\text {too large to display} \] Input:
int((a*b*x-x^3)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2*d-2*a*b*(a+b)*d*x+(a^2*d+ 4*a*b*d+b^2*d-1)*x^2-2*(a+b)*d*x^3+d*x^4),x)
Output:
( - sqrt(x)*sqrt(b - x)*sqrt( - a + x)*i + sqrt(x)*sqrt(b - x)*sqrt(a - x) - int((sqrt(x)*sqrt(b - x)*sqrt( - a + x)*x**2)/(a**3*b**3*d - 3*a**3*b** 2*d*x + 3*a**3*b*d*x**2 - a**3*d*x**3 - 3*a**2*b**3*d*x + 9*a**2*b**2*d*x* *2 - 9*a**2*b*d*x**3 + 3*a**2*d*x**4 + 3*a*b**3*d*x**2 - 9*a*b**2*d*x**3 + 9*a*b*d*x**4 - a*b*x**2 - 3*a*d*x**5 + a*x**3 - b**3*d*x**3 + 3*b**2*d*x* *4 - 3*b*d*x**5 + b*x**3 + d*x**6 - x**4),x)*a**3*d*i - 9*int((sqrt(x)*sqr t(b - x)*sqrt( - a + x)*x**2)/(a**3*b**3*d - 3*a**3*b**2*d*x + 3*a**3*b*d* x**2 - a**3*d*x**3 - 3*a**2*b**3*d*x + 9*a**2*b**2*d*x**2 - 9*a**2*b*d*x** 3 + 3*a**2*d*x**4 + 3*a*b**3*d*x**2 - 9*a*b**2*d*x**3 + 9*a*b*d*x**4 - a*b *x**2 - 3*a*d*x**5 + a*x**3 - b**3*d*x**3 + 3*b**2*d*x**4 - 3*b*d*x**5 + b *x**3 + d*x**6 - x**4),x)*a**2*b*d*i - 9*int((sqrt(x)*sqrt(b - x)*sqrt( - a + x)*x**2)/(a**3*b**3*d - 3*a**3*b**2*d*x + 3*a**3*b*d*x**2 - a**3*d*x** 3 - 3*a**2*b**3*d*x + 9*a**2*b**2*d*x**2 - 9*a**2*b*d*x**3 + 3*a**2*d*x**4 + 3*a*b**3*d*x**2 - 9*a*b**2*d*x**3 + 9*a*b*d*x**4 - a*b*x**2 - 3*a*d*x** 5 + a*x**3 - b**3*d*x**3 + 3*b**2*d*x**4 - 3*b*d*x**5 + b*x**3 + d*x**6 - x**4),x)*a*b**2*d*i + int((sqrt(x)*sqrt(b - x)*sqrt( - a + x)*x**2)/(a**3* b**3*d - 3*a**3*b**2*d*x + 3*a**3*b*d*x**2 - a**3*d*x**3 - 3*a**2*b**3*d*x + 9*a**2*b**2*d*x**2 - 9*a**2*b*d*x**3 + 3*a**2*d*x**4 + 3*a*b**3*d*x**2 - 9*a*b**2*d*x**3 + 9*a*b*d*x**4 - a*b*x**2 - 3*a*d*x**5 + a*x**3 - b**3*d *x**3 + 3*b**2*d*x**4 - 3*b*d*x**5 + b*x**3 + d*x**6 - x**4),x)*a*i - i...