Integrand size = 58, antiderivative size = 60 \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (b c+a d-(b+c+d) x+x^2\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{\sqrt {d} (a-x)}\right )}{\sqrt {d}} \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 32.00 (sec) , antiderivative size = 289, normalized size of antiderivative = 4.82 \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (b c+a d-(b+c+d) x+x^2\right )} \, dx=-\frac {2 i \sqrt {\frac {b-x}{a-x}} \sqrt {\frac {c-x}{a-x}} (-a+x)^{3/2} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right ),\frac {a-c}{a-b}\right )-\operatorname {EllipticPi}\left (-\frac {2 (a-c)}{-2 a+b+c+d-\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}},i \text {arcsinh}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right ),\frac {a-c}{a-b}\right )-\operatorname {EllipticPi}\left (-\frac {2 (a-c)}{-2 a+b+c+d+\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}},i \text {arcsinh}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right ),\frac {a-c}{a-b}\right )\right )}{\sqrt {a-b} \sqrt {(-a+x) (-b+x) (-c+x)}} \]
Integrate[(a*b + a*c - b*c - 2*a*x + x^2)/(Sqrt[(-a + x)*(-b + x)*(-c + x) ]*(b*c + a*d - (b + c + d)*x + x^2)),x]
((-2*I)*Sqrt[(b - x)/(a - x)]*Sqrt[(c - x)/(a - x)]*(-a + x)^(3/2)*(Ellipt icF[I*ArcSinh[Sqrt[a - b]/Sqrt[-a + x]], (a - c)/(a - b)] - EllipticPi[(-2 *(a - c))/(-2*a + b + c + d - Sqrt[b^2 - 2*b*c + c^2 - 4*a*d + 2*b*d + 2*c *d + d^2]), I*ArcSinh[Sqrt[a - b]/Sqrt[-a + x]], (a - c)/(a - b)] - Ellipt icPi[(-2*(a - c))/(-2*a + b + c + d + Sqrt[b^2 - 2*b*c + c^2 - 4*a*d + 2*b *d + 2*c*d + d^2]), I*ArcSinh[Sqrt[a - b]/Sqrt[-a + x]], (a - c)/(a - b)]) )/(Sqrt[a - b]*Sqrt[(-a + x)*(-b + x)*(-c + x)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 11.23 (sec) , antiderivative size = 454, normalized size of antiderivative = 7.57, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {7269, 25, 7279, 2009}
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.
\(\displaystyle \int \frac {a b+a c-2 a x-b c+x^2}{\sqrt {(x-a) (x-b) (x-c)} \left (a d-x (b+c+d)+b c+x^2\right )} \, dx\) |
\(\Big \downarrow \) 7269 |
\(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int -\frac {-x^2+2 a x+b c-a (b+c)}{\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \left (x^2-(b+c+d) x+b c+a d\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {-x^2+2 a x+b c-a (b+c)}{\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \left (x^2-(b+c+d) x+b c+a d\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle -\frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (-\frac {-2 b c+a (b+c-d)-(2 a-b-c-d) x}{\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \left (x^2+(-b-c-d) x+b c+a d\right )}-\frac {1}{\sqrt {x-a} \sqrt {x-b} \sqrt {x-c}}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \left (\frac {2 \sqrt {b-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \operatorname {EllipticPi}\left (\frac {2 (a-b)}{2 a-b-c-d-\sqrt {b^2-2 c b+2 d b+c^2+d^2-4 a d+2 c d}},\arcsin \left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right ),\frac {a-b}{a-c}\right )}{\sqrt {x-b} \sqrt {x-c}}+\frac {2 \sqrt {b-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \operatorname {EllipticPi}\left (\frac {2 (a-b)}{2 a-b-c-d+\sqrt {b^2-2 c b+2 d b+c^2+d^2-4 a d+2 c d}},\arcsin \left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right ),\frac {a-b}{a-c}\right )}{\sqrt {x-b} \sqrt {x-c}}-\frac {2 \sqrt {b-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right ),\frac {a-b}{a-c}\right )}{\sqrt {x-b} \sqrt {x-c}}\right )}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
Int[(a*b + a*c - b*c - 2*a*x + x^2)/(Sqrt[(-a + x)*(-b + x)*(-c + x)]*(b*c + a*d - (b + c + d)*x + x^2)),x]
-((Sqrt[-a + x]*Sqrt[-b + x]*Sqrt[-c + x]*((-2*Sqrt[-a + b]*Sqrt[-((b - x) /(a - b))]*Sqrt[-((c - x)/(a - c))]*EllipticF[ArcSin[Sqrt[-a + x]/Sqrt[-a + b]], (a - b)/(a - c)])/(Sqrt[-b + x]*Sqrt[-c + x]) + (2*Sqrt[-a + b]*Sqr t[-((b - x)/(a - b))]*Sqrt[-((c - x)/(a - c))]*EllipticPi[(2*(a - b))/(2*a - b - c - d - Sqrt[b^2 - 2*b*c + c^2 - 4*a*d + 2*b*d + 2*c*d + d^2]), Arc Sin[Sqrt[-a + x]/Sqrt[-a + b]], (a - b)/(a - c)])/(Sqrt[-b + x]*Sqrt[-c + x]) + (2*Sqrt[-a + b]*Sqrt[-((b - x)/(a - b))]*Sqrt[-((c - x)/(a - c))]*El lipticPi[(2*(a - b))/(2*a - b - c - d + Sqrt[b^2 - 2*b*c + c^2 - 4*a*d + 2 *b*d + 2*c*d + d^2]), ArcSin[Sqrt[-a + x]/Sqrt[-a + b]], (a - b)/(a - c)]) /(Sqrt[-b + x]*Sqrt[-c + x])))/Sqrt[-((a - x)*(b - x)*(c - x))])
3.8.75.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Simp[ a^IntPart[p]*((a*v^m*w^n*z^q)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[ p])*z^(q*FracPart[p]))) Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; FreeQ[{a , m, n, p, q}, x] && !IntegerQ[p] && !FreeQ[v, x] && !FreeQ[w, x] && !F reeQ[z, x]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 2.58 (sec) , antiderivative size = 8138, normalized size of antiderivative = 135.63
method | result | size |
default | \(\text {Expression too large to display}\) | \(8138\) |
elliptic | \(\text {Expression too large to display}\) | \(8296\) |
int((a*b+a*c-2*a*x-b*c+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/2)/(b*c+a*d-(b+c+d)* x+x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (49) = 98\).
Time = 12.24 (sec) , antiderivative size = 349, normalized size of antiderivative = 5.82 \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (b c+a d-(b+c+d) x+x^2\right )} \, dx=\left [\frac {\log \left (\frac {b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} - 2 \, {\left (b + c - 3 \, d\right )} x^{3} + x^{4} + {\left (b^{2} + 4 \, b c + c^{2} - 6 \, {\left (a + b + c\right )} d + d^{2}\right )} x^{2} - 4 \, \sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} {\left (b c - a d - {\left (b + c - d\right )} x + x^{2}\right )} \sqrt {d} - 2 \, {\left (b^{2} c + b c^{2} + a d^{2} - 3 \, {\left (a b + {\left (a + b\right )} c\right )} d\right )} x}{b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} - 2 \, {\left (b + c + d\right )} x^{3} + x^{4} + {\left (b^{2} + 4 \, b c + c^{2} + 2 \, {\left (a + b + c\right )} d + d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c + b c^{2} + a d^{2} + {\left (a b + {\left (a + b\right )} c\right )} d\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (-\frac {\sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} {\left (b c - a d - {\left (b + c - d\right )} x + x^{2}\right )} \sqrt {-d}}{2 \, {\left (a b c d + {\left (a + b + c\right )} d x^{2} - d x^{3} - {\left (a b + {\left (a + b\right )} c\right )} d x\right )}}\right )}{d}\right ] \]
integrate((a*b+a*c-2*a*x-b*c+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/2)/(b*c+a*d-(b +c+d)*x+x^2),x, algorithm="fricas")
[1/2*log((b^2*c^2 - 6*a*b*c*d + a^2*d^2 - 2*(b + c - 3*d)*x^3 + x^4 + (b^2 + 4*b*c + c^2 - 6*(a + b + c)*d + d^2)*x^2 - 4*sqrt(-a*b*c - (a + b + c)* x^2 + x^3 + (a*b + (a + b)*c)*x)*(b*c - a*d - (b + c - d)*x + x^2)*sqrt(d) - 2*(b^2*c + b*c^2 + a*d^2 - 3*(a*b + (a + b)*c)*d)*x)/(b^2*c^2 + 2*a*b*c *d + a^2*d^2 - 2*(b + c + d)*x^3 + x^4 + (b^2 + 4*b*c + c^2 + 2*(a + b + c )*d + d^2)*x^2 - 2*(b^2*c + b*c^2 + a*d^2 + (a*b + (a + b)*c)*d)*x))/sqrt( d), sqrt(-d)*arctan(-1/2*sqrt(-a*b*c - (a + b + c)*x^2 + x^3 + (a*b + (a + b)*c)*x)*(b*c - a*d - (b + c - d)*x + x^2)*sqrt(-d)/(a*b*c*d + (a + b + c )*d*x^2 - d*x^3 - (a*b + (a + b)*c)*d*x))/d]
Timed out. \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (b c+a d-(b+c+d) x+x^2\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (b c+a d-(b+c+d) x+x^2\right )} \, dx=\text {Exception raised: ValueError} \]
integrate((a*b+a*c-2*a*x-b*c+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/2)/(b*c+a*d-(b +c+d)*x+x^2),x, algorithm="maxima")
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((d+c+b)^2>0)', see `assume?` for more deta
\[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (b c+a d-(b+c+d) x+x^2\right )} \, dx=\int { \frac {a b + a c - b c - 2 \, a x + x^{2}}{\sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}} {\left (b c + a d - {\left (b + c + d\right )} x + x^{2}\right )}} \,d x } \]
integrate((a*b+a*c-2*a*x-b*c+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/2)/(b*c+a*d-(b +c+d)*x+x^2),x, algorithm="giac")
integrate((a*b + a*c - b*c - 2*a*x + x^2)/(sqrt(-(a - x)*(b - x)*(c - x))* (b*c + a*d - (b + c + d)*x + x^2)), x)
Time = 5.73 (sec) , antiderivative size = 711, normalized size of antiderivative = 11.85 \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (b c+a d-(b+c+d) x+x^2\right )} \, dx=\frac {2\,\left (a-c\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}}{\sqrt {x^3+\left (-a-b-c\right )\,x^2+\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c}}+\frac {2\,\left (a-c\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}\,\Pi \left (\frac {a-c}{\frac {b}{2}-\frac {c}{2}+\frac {d}{2}-\frac {\sqrt {b^2-2\,b\,c+2\,b\,d+c^2+2\,c\,d+d^2-4\,a\,d}}{2}};\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\left (a\,b+a\,c-a\,d-2\,b\,c+\left (b-2\,a+c+d\right )\,\left (\frac {b}{2}+\frac {c}{2}+\frac {d}{2}-\frac {\sqrt {b^2-2\,b\,c+2\,b\,d+c^2+2\,c\,d+d^2-4\,a\,d}}{2}\right )\right )}{\sqrt {x^3+\left (-a-b-c\right )\,x^2+\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c}\,\left (\frac {b}{2}-\frac {c}{2}+\frac {d}{2}-\frac {\sqrt {b^2-2\,b\,c+2\,b\,d+c^2+2\,c\,d+d^2-4\,a\,d}}{2}\right )\,\sqrt {b^2-2\,b\,c+2\,b\,d+c^2+2\,c\,d+d^2-4\,a\,d}}-\frac {2\,\left (a-c\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}\,\Pi \left (\frac {a-c}{\frac {b}{2}-\frac {c}{2}+\frac {d}{2}+\frac {\sqrt {b^2-2\,b\,c+2\,b\,d+c^2+2\,c\,d+d^2-4\,a\,d}}{2}};\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\left (a\,b+a\,c-a\,d-2\,b\,c+\left (b-2\,a+c+d\right )\,\left (\frac {b}{2}+\frac {c}{2}+\frac {d}{2}+\frac {\sqrt {b^2-2\,b\,c+2\,b\,d+c^2+2\,c\,d+d^2-4\,a\,d}}{2}\right )\right )}{\sqrt {x^3+\left (-a-b-c\right )\,x^2+\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c}\,\left (\frac {b}{2}-\frac {c}{2}+\frac {d}{2}+\frac {\sqrt {b^2-2\,b\,c+2\,b\,d+c^2+2\,c\,d+d^2-4\,a\,d}}{2}\right )\,\sqrt {b^2-2\,b\,c+2\,b\,d+c^2+2\,c\,d+d^2-4\,a\,d}} \]
int((a*b + a*c - b*c - 2*a*x + x^2)/((-(a - x)*(b - x)*(c - x))^(1/2)*(a*d + b*c - x*(b + c + d) + x^2)),x)
(2*(a - c)*ellipticF(asin((-(c - x)/(a - c))^(1/2)), (a - c)/(b - c))*((a - x)/(a - c))^(1/2)*(-(c - x)/(a - c))^(1/2)*((b - x)/(b - c))^(1/2))/(x*( a*b + a*c + b*c) - x^2*(a + b + c) + x^3 - a*b*c)^(1/2) + (2*(a - c)*((a - x)/(a - c))^(1/2)*(-(c - x)/(a - c))^(1/2)*((b - x)/(b - c))^(1/2)*ellipt icPi((a - c)/(b/2 - c/2 + d/2 - (2*b*d - 2*b*c - 4*a*d + 2*c*d + b^2 + c^2 + d^2)^(1/2)/2), asin((-(c - x)/(a - c))^(1/2)), (a - c)/(b - c))*(a*b + a*c - a*d - 2*b*c + (b - 2*a + c + d)*(b/2 + c/2 + d/2 - (2*b*d - 2*b*c - 4*a*d + 2*c*d + b^2 + c^2 + d^2)^(1/2)/2)))/((x*(a*b + a*c + b*c) - x^2*(a + b + c) + x^3 - a*b*c)^(1/2)*(b/2 - c/2 + d/2 - (2*b*d - 2*b*c - 4*a*d + 2*c*d + b^2 + c^2 + d^2)^(1/2)/2)*(2*b*d - 2*b*c - 4*a*d + 2*c*d + b^2 + c^2 + d^2)^(1/2)) - (2*(a - c)*((a - x)/(a - c))^(1/2)*(-(c - x)/(a - c))^ (1/2)*((b - x)/(b - c))^(1/2)*ellipticPi((a - c)/(b/2 - c/2 + d/2 + (2*b*d - 2*b*c - 4*a*d + 2*c*d + b^2 + c^2 + d^2)^(1/2)/2), asin((-(c - x)/(a - c))^(1/2)), (a - c)/(b - c))*(a*b + a*c - a*d - 2*b*c + (b - 2*a + c + d)* (b/2 + c/2 + d/2 + (2*b*d - 2*b*c - 4*a*d + 2*c*d + b^2 + c^2 + d^2)^(1/2) /2)))/((x*(a*b + a*c + b*c) - x^2*(a + b + c) + x^3 - a*b*c)^(1/2)*(b/2 - c/2 + d/2 + (2*b*d - 2*b*c - 4*a*d + 2*c*d + b^2 + c^2 + d^2)^(1/2)/2)*(2* b*d - 2*b*c - 4*a*d + 2*c*d + b^2 + c^2 + d^2)^(1/2))