Integrand size = 55, antiderivative size = 40 \[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a b x+(-a-b) x^2+x^3}}{x^2}\right )}{\sqrt {d}} \]
Time = 12.76 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x (-a+x) (-b+x)}}{x^2}\right )}{\sqrt {d}} \]
Integrate[(x*(3*a*b - 2*(a + b)*x + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(-(a* b*d) + (a + b)*d*x - d*x^2 + x^3)),x]
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.
| \(\displaystyle \int \frac {x \left (-2 x (a+b)+3 a b+x^2\right )}{\sqrt {x (x-a) (x-b)} \left (d x (a+b)-a b d-d x^2+x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int -\frac {\sqrt {x} \left (x^2-2 (a+b) x+3 a b\right )}{\sqrt {x^2-(a+b) x+a b} \left (-x^3+d x^2-(a+b) d x+a b d\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {\sqrt {x} \left (x^2-2 (a+b) x+3 a b\right )}{\sqrt {x^2-(a+b) x+a b} \left (-x^3+d x^2-(a+b) d x+a b d\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x \left (x^2-2 (a+b) x+3 a b\right )}{\sqrt {x^2-(a+b) x+a b} \left (-x^3+d x^2-(a+b) d x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {-\left ((2 a+2 b-d) x^2\right )+(3 a b-d b-a d) x+a b d}{\sqrt {x^2-(a+b) x+a b} \left (-x^3+d x^2-(a+b) d x+a b d\right )}-\frac {1}{\sqrt {x^2-(a+b) x+a b}}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \left (a b d \int \frac {1}{\sqrt {x^2-(a+b) x+a b} \left (-x^3+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}d\sqrt {x}+(3 a b-a d-b d) \int \frac {x}{\sqrt {x^2-(a+b) x+a b} \left (-x^3+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}d\sqrt {x}-(2 a+2 b-d) \int \frac {x^2}{\sqrt {x^2-(a+b) x+a b} \left (-x^3+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}d\sqrt {x}-\frac {\left (\sqrt {a} \sqrt {b}+x\right ) \sqrt {\frac {-x (a+b)+a b+x^2}{\left (\sqrt {a} \sqrt {b}+x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-x (a+b)+a b+x^2}}\right )}{\sqrt {x (a-x) (b-x)}}\) |
Int[(x*(3*a*b - 2*(a + b)*x + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(-(a*b*d) + (a + b)*d*x - d*x^2 + x^3)),x]
3.6.18.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.82 (sec) , antiderivative size = 291, normalized size of antiderivative = 7.28
| method | result | size |
| default | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}-d \,\textit {\_Z}^{2}+\left (a d +d b \right ) \textit {\_Z} -a b d \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b +\underline {\hspace {1.25 ex}}\alpha ^{2} d +3 \underline {\hspace {1.25 ex}}\alpha a b -\underline {\hspace {1.25 ex}}\alpha a d -\underline {\hspace {1.25 ex}}\alpha b d +a b d \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +a d +b^{2}\right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +a d +b^{2}}{b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha d -a d -d b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{b^{2}}\) | \(291\) |
| elliptic | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}-d \,\textit {\_Z}^{2}+\left (a d +d b \right ) \textit {\_Z} -a b d \right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d -a b d \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +a d +b^{2}\right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +a d +b^{2}}{b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha d -a d -d b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{b^{2}}\) | \(291\) |
int(x*(3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a*b*d+(a+b)*d*x-d*x^ 2+x^3),x,method=_RETURNVERBOSE)
-2*b*(-(-b+x)/b)^(1/2)*((-a+x)/(-a+b))^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^ 2+x^3)^(1/2)*EllipticF((-(-b+x)/b)^(1/2),(b/(-a+b))^(1/2))+2/b^2*sum((-2*_ alpha^2*a-2*_alpha^2*b+_alpha^2*d+3*_alpha*a*b-_alpha*a*d-_alpha*b*d+a*b*d )/(-3*_alpha^2+2*_alpha*d-a*d-b*d)*(_alpha^2+_alpha*b-_alpha*d+a*d+b^2)*(- (-b+x)/b)^(1/2)*((-a+x)/(-a+b))^(1/2)*(x/b)^(1/2)/(x*(a*b-a*x-b*x+x^2))^(1 /2)*EllipticPi((-(-b+x)/b)^(1/2),(_alpha^2+_alpha*b-_alpha*d+a*d+b^2)/b^2, (b/(-a+b))^(1/2)),_alpha=RootOf(_Z^3-d*_Z^2+(a*d+b*d)*_Z-a*b*d))
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (31) = 62\).
Time = 0.47 (sec) , antiderivative size = 312, normalized size of antiderivative = 7.80 \[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\left [\frac {\log \left (\frac {a^{2} b^{2} d^{2} + 6 \, d x^{5} + x^{6} + {\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} x^{2} - {\left (6 \, {\left (a + b\right )} d - d^{2}\right )} x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d^{2} x + 2 \, {\left (3 \, a b d - {\left (a + b\right )} d^{2}\right )} x^{3} - 4 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3} + x^{4}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d}}{a^{2} b^{2} d^{2} - 2 \, d x^{5} + x^{6} + {\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} x^{2} + {\left (2 \, {\left (a + b\right )} d + d^{2}\right )} x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d^{2} x - 2 \, {\left (a b d + {\left (a + b\right )} d^{2}\right )} x^{3}}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {{\left (a b d - {\left (a + b\right )} d x + d x^{2} + x^{3}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a b d x^{2} - {\left (a + b\right )} d x^{3} + d x^{4}\right )}}\right )}{d}\right ] \]
integrate(x*(3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a*b*d+(a+b)*d* x-d*x^2+x^3),x, algorithm="fricas")
[1/2*log((a^2*b^2*d^2 + 6*d*x^5 + x^6 + (a^2 + 4*a*b + b^2)*d^2*x^2 - (6*( a + b)*d - d^2)*x^4 - 2*(a^2*b + a*b^2)*d^2*x + 2*(3*a*b*d - (a + b)*d^2)* x^3 - 4*(a*b*d*x - (a + b)*d*x^2 + d*x^3 + x^4)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(d))/(a^2*b^2*d^2 - 2*d*x^5 + x^6 + (a^2 + 4*a*b + b^2)*d^2*x^2 + (2*(a + b)*d + d^2)*x^4 - 2*(a^2*b + a*b^2)*d^2*x - 2*(a*b*d + (a + b)*d ^2)*x^3))/sqrt(d), sqrt(-d)*arctan(1/2*(a*b*d - (a + b)*d*x + d*x^2 + x^3) *sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(-d)/(a*b*d*x^2 - (a + b)*d*x^3 + d*x ^4))/d]
Timed out. \[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\text {Timed out} \]
integrate(x*(3*a*b-2*(a+b)*x+x**2)/(x*(-a+x)*(-b+x))**(1/2)/(-a*b*d+(a+b)* d*x-d*x**2+x**3),x)
\[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\int { -\frac {{\left (3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}\right )} x}{{\left (a b d - {\left (a + b\right )} d x + d x^{2} - x^{3}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate(x*(3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a*b*d+(a+b)*d* x-d*x^2+x^3),x, algorithm="maxima")
-integrate((3*a*b - 2*(a + b)*x + x^2)*x/((a*b*d - (a + b)*d*x + d*x^2 - x ^3)*sqrt((a - x)*(b - x)*x)), x)
\[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\int { -\frac {{\left (3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}\right )} x}{{\left (a b d - {\left (a + b\right )} d x + d x^{2} - x^{3}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate(x*(3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a*b*d+(a+b)*d* x-d*x^2+x^3),x, algorithm="giac")
integrate(-(3*a*b - 2*(a + b)*x + x^2)*x/((a*b*d - (a + b)*d*x + d*x^2 - x ^3)*sqrt((a - x)*(b - x)*x)), x)
Time = 5.28 (sec) , antiderivative size = 457, normalized size of antiderivative = 11.42 \[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\left (\sum _{k=1}^3\left (-\frac {2\,b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )-b};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (2\,a\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2+2\,b\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2-d\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2-3\,a\,b\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )+a\,d\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )+b\,d\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )-a\,b\,d\right )}{\left (\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )-b\right )\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (3\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2-2\,d\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )+a\,d+b\,d\right )}\right )\right )-\frac {2\,b\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \]
int(-(x*(3*a*b + x^2 - 2*x*(a + b)))/((x*(a - x)*(b - x))^(1/2)*(d*x^2 - x ^3 - d*x*(a + b) + a*b*d)),x)
symsum(-(2*b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipti cPi(-b/(root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k) - b), asin(((b - x)/ b)^(1/2)), -b/(a - b))*(2*a*root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k)^ 2 + 2*b*root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k)^2 - d*root(z^3 - d*z ^2 + d*z*(a + b) - a*b*d, z, k)^2 - 3*a*b*root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k) + a*d*root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k) + b*d*ro ot(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k) - a*b*d))/((root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k) - b)*(x*(a - x)*(b - x))^(1/2)*(a*d + b*d + 3* root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k)^2 - 2*d*root(z^3 - d*z^2 + d *z*(a + b) - a*b*d, z, k))), k, 1, 3) - (2*b*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)