Integrand size = 63, antiderivative size = 60 \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (a+b c d-(1+b d+c d) x+d x^2\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right )}{\sqrt {d}} \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 33.10 (sec) , antiderivative size = 308, normalized size of antiderivative = 5.13 \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (a+b c d-(1+b d+c d) x+d x^2\right )} \, dx=-\frac {2 i \sqrt {\frac {b-x}{a-x}} \sqrt {\frac {c-x}{a-x}} (-a+x)^{3/2} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right ),\frac {a-c}{a-b}\right )-\operatorname {EllipticPi}\left (\frac {2 (a-c) d}{-1+2 a d-b d-c d+\sqrt {-4 a d+b^2 d^2-2 b d (-1+c d)+(1+c d)^2}},i \text {arcsinh}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right ),\frac {a-c}{a-b}\right )-\operatorname {EllipticPi}\left (-\frac {2 (a-c) d}{1-2 a d+b d+c d+\sqrt {-4 a d+b^2 d^2-2 b d (-1+c d)+(1+c d)^2}},i \text {arcsinh}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right ),\frac {a-c}{a-b}\right )\right )}{\sqrt {a-b} d \sqrt {(-a+x) (-b+x) (-c+x)}} \]
Integrate[(a*b + a*c - b*c - 2*a*x + x^2)/(Sqrt[(-a + x)*(-b + x)*(-c + x) ]*(a + b*c*d - (1 + b*d + c*d)*x + d*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)*d)/(-1 + 2*a*d - b*d - c*d + Sqrt[-4*a*d + b^2*d^2 - 2*b*d*(-1 + c *d) + (1 + c*d)^2]), I*ArcSinh[Sqrt[a - b]/Sqrt[-a + x]], (a - c)/(a - b)] - EllipticPi[(-2*(a - c)*d)/(1 - 2*a*d + b*d + c*d + Sqrt[-4*a*d + b^2*d^ 2 - 2*b*d*(-1 + c*d) + (1 + c*d)^2]), I*ArcSinh[Sqrt[a - b]/Sqrt[-a + x]], (a - c)/(a - b)]))/(Sqrt[a - b]*d*Sqrt[(-a + x)*(-b + x)*(-c + x)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 13.84 (sec) , antiderivative size = 469, normalized size of antiderivative = 7.82, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.063, 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.
\(\displaystyle \int \frac {a b+a c-2 a x-b c+x^2}{\sqrt {(x-a) (x-b) (x-c)} \left (a-x (b d+c d+1)+b c d+d x^2\right )} \, dx\) |
\(\Big \downarrow \) 7269 |
\(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int -\frac {-x^2+2 a x+b c-a (b+c)}{\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \left (d x^2-(b d+c d+1) x+a+b c d\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {-x^2+2 a x+b c-a (b+c)}{\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \left (d x^2-(b d+c d+1) x+a+b c d\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle -\frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {2 b c d+a (-b d-c d+1)-(-2 a d+b d+c d+1) x}{d \sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \left (d x^2+(-b d-c d-1) x+a+b c d\right )}-\frac {1}{d \sqrt {x-a} \sqrt {x-b} \sqrt {x-c}}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \left (\frac {2 \sqrt {b-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \operatorname {EllipticPi}\left (-\frac {2 (a-b) d}{-2 a d+b d+c d-\sqrt {b^2 d^2-4 a d+2 b (1-c d) d+(c d+1)^2}+1},\arcsin \left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right ),\frac {a-b}{a-c}\right )}{d \sqrt {x-b} \sqrt {x-c}}+\frac {2 \sqrt {b-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \operatorname {EllipticPi}\left (-\frac {2 (a-b) d}{-2 a d+b d+c d+\sqrt {b^2 d^2-4 a d+2 b (1-c d) d+(c d+1)^2}+1},\arcsin \left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right ),\frac {a-b}{a-c}\right )}{d \sqrt {x-b} \sqrt {x-c}}-\frac {2 \sqrt {b-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right ),\frac {a-b}{a-c}\right )}{d \sqrt {x-b} \sqrt {x-c}}\right )}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
Int[(a*b + a*c - b*c - 2*a*x + x^2)/(Sqrt[(-a + x)*(-b + x)*(-c + x)]*(a + b*c*d - (1 + b*d + c*d)*x + d*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)])/(d*Sqrt[-b + x]*Sqrt[-c + x]) + (2*Sqrt[-a + b]*S qrt[-((b - x)/(a - b))]*Sqrt[-((c - x)/(a - c))]*EllipticPi[(-2*(a - b)*d) /(1 - 2*a*d + b*d + c*d - Sqrt[-4*a*d + b^2*d^2 + 2*b*d*(1 - c*d) + (1 + c *d)^2]), ArcSin[Sqrt[-a + x]/Sqrt[-a + b]], (a - b)/(a - c)])/(d*Sqrt[-b + x]*Sqrt[-c + x]) + (2*Sqrt[-a + b]*Sqrt[-((b - x)/(a - b))]*Sqrt[-((c - x )/(a - c))]*EllipticPi[(-2*(a - b)*d)/(1 - 2*a*d + b*d + c*d + Sqrt[-4*a*d + b^2*d^2 + 2*b*d*(1 - c*d) + (1 + c*d)^2]), ArcSin[Sqrt[-a + x]/Sqrt[-a + b]], (a - b)/(a - c)])/(d*Sqrt[-b + x]*Sqrt[-c + x])))/Sqrt[-((a - x)*(b - x)*(c - x))])
3.8.76.3.1 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 = 2.59 (sec) , antiderivative size = 9198, normalized size of antiderivative = 153.30
method | result | size |
default | \(\text {Expression too large to display}\) | \(9198\) |
elliptic | \(\text {Expression too large to display}\) | \(9394\) |
int((a*b+a*c-2*a*x-b*c+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/2)/(a+b*c*d-(b*d+c*d +1)*x+d*x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (49) = 98\).
Time = 16.76 (sec) , antiderivative size = 379, normalized size of antiderivative = 6.32 \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (a+b c d-(1+b d+c d) x+d x^2\right )} \, dx=\left [\frac {\log \left (\frac {b^{2} c^{2} d^{2} + d^{2} x^{4} - 6 \, a b c d - 2 \, {\left ({\left (b + c\right )} d^{2} - 3 \, d\right )} x^{3} + {\left ({\left (b^{2} + 4 \, b c + c^{2}\right )} d^{2} - 6 \, {\left (a + b + c\right )} d + 1\right )} x^{2} + a^{2} - 4 \, \sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} {\left (b c d + d x^{2} - {\left ({\left (b + c\right )} d - 1\right )} x - a\right )} \sqrt {d} - 2 \, {\left ({\left (b^{2} c + b c^{2}\right )} d^{2} - 3 \, {\left (a b + {\left (a + b\right )} c\right )} d + a\right )} x}{b^{2} c^{2} d^{2} + d^{2} x^{4} + 2 \, a b c d - 2 \, {\left ({\left (b + c\right )} d^{2} + d\right )} x^{3} + {\left ({\left (b^{2} + 4 \, b c + c^{2}\right )} d^{2} + 2 \, {\left (a + b + c\right )} d + 1\right )} x^{2} + a^{2} - 2 \, {\left ({\left (b^{2} c + b c^{2}\right )} d^{2} + {\left (a b + {\left (a + b\right )} c\right )} d + a\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (-\frac {\sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} {\left (b c d + d x^{2} - {\left ({\left (b + c\right )} d - 1\right )} x - a\right )} \sqrt {-d}}{2 \, {\left (a b c d + {\left (a + b + c\right )} d x^{2} - d x^{3} - {\left (a b + {\left (a + b\right )} c\right )} d x\right )}}\right )}{d}\right ] \]
integrate((a*b+a*c-2*a*x-b*c+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/2)/(a+b*c*d-(b *d+c*d+1)*x+d*x^2),x, algorithm="fricas")
[1/2*log((b^2*c^2*d^2 + d^2*x^4 - 6*a*b*c*d - 2*((b + c)*d^2 - 3*d)*x^3 + ((b^2 + 4*b*c + c^2)*d^2 - 6*(a + b + c)*d + 1)*x^2 + a^2 - 4*sqrt(-a*b*c - (a + b + c)*x^2 + x^3 + (a*b + (a + b)*c)*x)*(b*c*d + d*x^2 - ((b + c)*d - 1)*x - a)*sqrt(d) - 2*((b^2*c + b*c^2)*d^2 - 3*(a*b + (a + b)*c)*d + a) *x)/(b^2*c^2*d^2 + d^2*x^4 + 2*a*b*c*d - 2*((b + c)*d^2 + d)*x^3 + ((b^2 + 4*b*c + c^2)*d^2 + 2*(a + b + c)*d + 1)*x^2 + a^2 - 2*((b^2*c + b*c^2)*d^ 2 + (a*b + (a + b)*c)*d + a)*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*d + d*x^2 - ((b + c)* d - 1)*x - a)*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. \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (a+b c d-(1+b d+c d) x+d x^2\right )} \, dx=\text {Timed out} \]
integrate((a*b+a*c-2*a*x-b*c+x**2)/((-a+x)*(-b+x)*(-c+x))**(1/2)/(a+b*c*d- (b*d+c*d+1)*x+d*x**2),x)
Exception generated. \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (a+b c d-(1+b d+c d) x+d x^2\right )} \, dx=\text {Exception raised: ValueError} \]
integrate((a*b+a*c-2*a*x-b*c+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/2)/(a+b*c*d-(b *d+c*d+1)*x+d*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((c*d+b*d+1)^2>0)', see `assume?` for more
\[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (a+b c d-(1+b d+c d) x+d x^2\right )} \, dx=\int { \frac {a b + a c - b c - 2 \, a x + x^{2}}{\sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}} {\left (b c d + d x^{2} - {\left (b d + c d + 1\right )} x + a\right )}} \,d x } \]
integrate((a*b+a*c-2*a*x-b*c+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/2)/(a+b*c*d-(b *d+c*d+1)*x+d*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*d + d*x^2 - (b*d + c*d + 1)*x + a)), x)
Time = 5.76 (sec) , antiderivative size = 690, normalized size of antiderivative = 11.50 \[ \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (a+b c d-(1+b d+c d) x+d x^2\right )} \, dx=\frac {2\,\left (a-c\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}}{d\,\sqrt {x^3+\left (-a-b-c\right )\,x^2+\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c}}+\frac {\left (a-c\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}\,\Pi \left (-\frac {a-c}{c-\frac {b\,d+c\,d+\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\left (b\,d-2\,a\,d+c\,d+\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1\right )}{d^2\,\left (c-\frac {b\,d+c\,d+\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b-c\right )\,x^2+\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c}}+\frac {\left (a-c\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}\,\Pi \left (-\frac {a-c}{c-\frac {b\,d+c\,d-\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\left (b\,d-2\,a\,d+c\,d-\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1\right )}{d^2\,\left (c-\frac {b\,d+c\,d-\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b-c\right )\,x^2+\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c}} \]
int((a*b + a*c - b*c - 2*a*x + x^2)/((-(a - x)*(b - x)*(c - x))^(1/2)*(a - x*(b*d + c*d + 1) + d*x^2 + b*c*d)),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))/(d*( x*(a*b + a*c + b*c) - x^2*(a + b + c) + x^3 - a*b*c)^(1/2)) + ((a - c)*((a - x)/(a - c))^(1/2)*(-(c - x)/(a - c))^(1/2)*((b - x)/(b - c))^(1/2)*elli pticPi(-(a - c)/(c - (b*d + c*d + (2*b*d - 4*a*d + 2*c*d + b^2*d^2 + c^2*d ^2 - 2*b*c*d^2 + 1)^(1/2) + 1)/(2*d)), asin((-(c - x)/(a - c))^(1/2)), (a - c)/(b - c))*(b*d - 2*a*d + c*d + (2*b*d - 4*a*d + 2*c*d + b^2*d^2 + c^2* d^2 - 2*b*c*d^2 + 1)^(1/2) + 1))/(d^2*(c - (b*d + c*d + (2*b*d - 4*a*d + 2 *c*d + b^2*d^2 + c^2*d^2 - 2*b*c*d^2 + 1)^(1/2) + 1)/(2*d))*(x*(a*b + a*c + b*c) - x^2*(a + b + c) + x^3 - a*b*c)^(1/2)) + ((a - c)*((a - x)/(a - c) )^(1/2)*(-(c - x)/(a - c))^(1/2)*((b - x)/(b - c))^(1/2)*ellipticPi(-(a - c)/(c - (b*d + c*d - (2*b*d - 4*a*d + 2*c*d + b^2*d^2 + c^2*d^2 - 2*b*c*d^ 2 + 1)^(1/2) + 1)/(2*d)), asin((-(c - x)/(a - c))^(1/2)), (a - c)/(b - c)) *(b*d - 2*a*d + c*d - (2*b*d - 4*a*d + 2*c*d + b^2*d^2 + c^2*d^2 - 2*b*c*d ^2 + 1)^(1/2) + 1))/(d^2*(c - (b*d + c*d - (2*b*d - 4*a*d + 2*c*d + b^2*d^ 2 + c^2*d^2 - 2*b*c*d^2 + 1)^(1/2) + 1)/(2*d))*(x*(a*b + a*c + b*c) - x^2* (a + b + c) + x^3 - a*b*c)^(1/2))