\(\int \frac {(-a+x) (-b+x) (-a b+x^2)}{x \sqrt {x (-a+x) (-b+x)} (a b-(a+b+d) x+x^2)} \, dx\) [874]

Optimal result

Integrand size = 54, antiderivative size = 67 \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{x \sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx=\frac {2 \sqrt {a b x+(-a-b) x^2+x^3}}{x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(-2*Sqrt[(a - x)*(b - x)*x])/x - (4*Sqrt[a]*Sqrt[(a - x)*(b - x)*x]*Sqrt[1 - x/a]*EllipticE[ArcSin[Sqrt[x]/Sqr
t[a]], a/b])/((a - x)*Sqrt[x]*Sqrt[1 - x/b]) - (2*Sqrt[a]*(a - b)*Sqrt[(a - x)*(b - x)*x]*Sqrt[1 - x/a]*Sqrt[1
- x/b]*EllipticF[ArcSin[Sqrt[x]/Sqrt[a]], a/b])/((a - x)*(b - x)*Sqrt[x]) + ((a + b + d + Sqrt[a^2 - 2*a*(b -
d) + (b + d)^2])*Sqrt[(a - x)*(b - x)*x]*Defer[Int][(Sqrt[a - x]*Sqrt[b - x])/(x^(3/2)*(-a - b - d - Sqrt[a^2
- 2*a*b + b^2 + 2*a*d + 2*b*d + d^2] + 2*x)), x])/(Sqrt[a - x]*Sqrt[b - x]*Sqrt[x]) + ((a + b + d - Sqrt[a^2
- 2*a*(b - d) + (b + d)^2])*Sqrt[(a - x)*(b - x)*x]*Defer[Int][(Sqrt[a - x]*Sqrt[b - x])/(x^(3/2)*(-a - b - d
+ Sqrt[a^2 - 2*a*b + b^2 + 2*a*d + 2*b*d + d^2] + 2*x)), x])/(Sqrt[a - x]*Sqrt[b - x]*Sqrt[x])

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {(a-x) (b-x) x} \left (-a b+x^2\right )}{x^2 \left (a b-(a+b+d) x+x^2\right )} \, dx \\ & = \frac {\sqrt {(a-x) (b-x) x} \int \frac {\sqrt {a-x} \sqrt {b-x} \left (-a b+x^2\right )}{x^{3/2} \left (a b-(a+b+d) x+x^2\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \\ & = \frac {\sqrt {(a-x) (b-x) x} \int \left (\frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2}}-\frac {\sqrt {a-x} \sqrt {b-x} (2 a b-(a+b+d) x)}{x^{3/2} \left (a b+(-a-b-d) x+x^2\right )}\right ) \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \\ & = \frac {\sqrt {(a-x) (b-x) x} \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2}} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\sqrt {(a-x) (b-x) x} \int \frac {\sqrt {a-x} \sqrt {b-x} (2 a b-(a+b+d) x)}{x^{3/2} \left (a b+(-a-b-d) x+x^2\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \\ & = -\frac {2 \sqrt {(a-x) (b-x) x}}{x}-\frac {\sqrt {(a-x) (b-x) x} \int \left (\frac {\left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}\right ) \sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )}+\frac {\left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}\right ) \sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )}\right ) \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}+\frac {\left (2 \sqrt {(a-x) (b-x) x}\right ) \int \frac {\frac {1}{2} (-a-b)+x}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \\ & = -\frac {2 \sqrt {(a-x) (b-x) x}}{x}-\frac {\left (2 \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {b-x}}{\sqrt {a-x} \sqrt {x}} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}+\frac {\left ((-a+b) \sqrt {(a-x) (b-x) x}\right ) \int \frac {1}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \\ & = -\frac {2 \sqrt {(a-x) (b-x) x}}{x}-\frac {\left (\left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (2 \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}}\right ) \int \frac {\sqrt {1-\frac {x}{b}}}{\sqrt {x} \sqrt {1-\frac {x}{a}}} \, dx}{(a-x) \sqrt {x} \sqrt {1-\frac {x}{b}}}+\frac {\left ((-a+b) \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}} \, dx}{(a-x) (b-x) \sqrt {x}} \\ & = -\frac {2 \sqrt {(a-x) (b-x) x}}{x}-\frac {4 \sqrt {a} \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{(a-x) \sqrt {x} \sqrt {1-\frac {x}{b}}}-\frac {2 \sqrt {a} (a-b) \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{(a-x) (b-x) \sqrt {x}}-\frac {\left (\left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.79 \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{x \sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx=\frac {2 \sqrt {x (-a+x) (-b+x)}}{x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {x (-a+x) (-b+x)}}\right ) \]

[In]

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

[Out]

(2*Sqrt[x*(-a + x)*(-b + x)])/x - 2*Sqrt[d]*ArcTanh[(Sqrt[d]*x)/Sqrt[x*(-a + x)*(-b + x)]]

Maple [A] (verified)

Time = 2.56 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \sqrt {d}}\right ) x +2 \sqrt {x \left (a -x \right ) \left (b -x \right )}}{x}\)	\(50\)
default	\(-\frac {2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \sqrt {d}}\right ) x -2 \sqrt {x \left (a -x \right ) \left (b -x \right )}}{x}\)	\(51\)
risch	\(\frac {2 \left (a -x \right ) \left (b -x \right )}{\sqrt {x \left (-a +x \right ) \left (-b +x \right )}}-2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \sqrt {d}}\right )\)	\(55\)
elliptic	\(\text {Expression too large to display}\)	\(3462\)

[In]

int((-a+x)*(-b+x)*(-a*b+x^2)/x/(x*(-a+x)*(-b+x))^(1/2)/(a*b-(a+b+d)*x+x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 722, normalized size of antiderivative = 10.78 \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{x \sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx=\frac {b\,d\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\frac {a}{2}-\frac {b}{2}+\frac {d}{2}-\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a+b+d-\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {a}{2}-\frac {b}{2}+\frac {d}{2}-\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}\right )}-\frac {2\,a\,b\,\left (\frac {\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )-\frac {\sqrt {\frac {b-x}{a-b}+1}\,\sqrt {\frac {b-x}{b}}}{\sqrt {1-\frac {b-x}{b}}}}{\frac {b}{a-b}+1}-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,b\,d\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,b\,\left (a\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )-\left (a-b\right )\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}+\frac {b\,d\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\frac {a}{2}-\frac {b}{2}+\frac {d}{2}+\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a+b+d+\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {a}{2}-\frac {b}{2}+\frac {d}{2}+\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}\right )} \]

[In]

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

[Out]

(b*d*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(-b/(a/2 - b/2 + d/2 - (2*a*d - 2*a*b + 2
*b*d + a^2 + b^2 + d^2)^(1/2)/2), asin(((b - x)/b)^(1/2)), -b/(a - b))*(a + b + d - (2*a*d - 2*a*b + 2*b*d + a
^2 + b^2 + d^2)^(1/2)))/((x^3 - x^2*(a + b) + a*b*x)^(1/2)*(a/2 - b/2 + d/2 - (2*a*d - 2*a*b + 2*b*d + a^2 + b
^2 + d^2)^(1/2)/2)) - (2*a*b*((ellipticE(asin(((b - x)/b)^(1/2)), -b/(a - b)) - (((b - x)/(a - b) + 1)^(1/2)*(
(b - x)/b)^(1/2))/(1 - (b - x)/b)^(1/2))/(b/(a - b) + 1) - ellipticF(asin(((b - x)/b)^(1/2)), -b/(a - b)))*(x/
b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2))/(x^3 - x^2*(a + b) + a*b*x)^(1/2) - (2*b*d*ellipticF(asin(
((b - x)/b)^(1/2)), -b/(a - b))*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2))/(x^3 - x^2*(a + b) + a*
b*x)^(1/2) - (2*b*(a*ellipticF(asin(((b - x)/b)^(1/2)), -b/(a - b)) - (a - b)*ellipticE(asin(((b - x)/b)^(1/2)
), -b/(a - b)))*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2))/(x^3 - x^2*(a + b) + a*b*x)^(1/2) + (b*
d*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(-b/(a/2 - b/2 + d/2 + (2*a*d - 2*a*b + 2*b*
d + a^2 + b^2 + d^2)^(1/2)/2), asin(((b - x)/b)^(1/2)), -b/(a - b))*(a + b + d + (2*a*d - 2*a*b + 2*b*d + a^2
+ b^2 + d^2)^(1/2)))/((x^3 - x^2*(a + b) + a*b*x)^(1/2)*(a/2 - b/2 + d/2 + (2*a*d - 2*a*b + 2*b*d + a^2 + b^2
+ d^2)^(1/2)/2))