Optimal. Leaf size=60 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}}{\sqrt {d} (a-x)^2}\right )}{\sqrt {d}} \]
________________________________________________________________________________________
Rubi [F] time = 53.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {\sqrt {-a+x} \left (-a b-a c+3 b c+(2 a-2 b-2 c) x+x^2\right )}{\sqrt {-b+x} \sqrt {-c+x} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {\sqrt {-a+x} \left (-3 b c+a (b+c)-2 (a-b-c) x-x^2\right )}{\sqrt {-b+x} \sqrt {-c+x} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \left (\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )}+\frac {2 (-a+b+c) x \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )}+\frac {x^2 \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )}\right ) \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {x^2 \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {x \sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left ((-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {\sqrt {-a+x}}{\sqrt {-b+x} \sqrt {-c+x} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^2\right )^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2-b c+b x^2+c x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^2\right )}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2-b c+b x^2+c x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2-b c+b x^2+c x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^2\right )^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2-b c+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^2\right )}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2-b c+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2-b c+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^2\right )^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-a-x^2\right )}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c}{a^2}\right )-(b+c) x^2+x^4-d x^6-a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{d \sqrt {a-b+x^2} \sqrt {a-c+x^2}}+\frac {(a-b) (a-c)+\left (2 a-b-c+a^2 d\right ) x^2+(1+2 a d) x^4}{d \sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )}\right ) \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d x^6\right )}+\frac {x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d x^6\right )}\right ) \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 a (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b+x^2} \sqrt {a-c+x^2}} \, dx,x,\sqrt {-a+x}\right )}{d \sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {(a-b) (a-c)+\left (2 a-b-c+a^2 d\right ) x^2+(1+2 a d) x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{d \sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=-\frac {2 \sqrt {a-c} (b-x) \sqrt {-a+x} F\left (\tan ^{-1}\left (\frac {\sqrt {-a+x}}{\sqrt {a-c}}\right )|-\frac {b-c}{a-b}\right )}{(a-b) d \sqrt {\frac {(a-c) (b-x)}{(a-b) (c-x)}} \sqrt {-((a-x) (b-x) (c-x))}}+\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 a (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (2 \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \left (\frac {(a-b) (-a+c)}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )}+\frac {\left (-2 a+b+c-a^2 d\right ) x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )}+\frac {(-1-2 a d) x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )}\right ) \, dx,x,\sqrt {-a+x}\right )}{d \sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=-\frac {2 \sqrt {a-c} (b-x) \sqrt {-a+x} F\left (\tan ^{-1}\left (\frac {\sqrt {-a+x}}{\sqrt {a-c}}\right )|-\frac {b-c}{a-b}\right )}{(a-b) d \sqrt {\frac {(a-c) (b-x)}{(a-b) (c-x)}} \sqrt {-((a-x) (b-x) (c-x))}}+\frac {\left (4 (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 a (a-b-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )-2 a \left (1-\frac {b+c}{2 a}\right ) x^2-x^4+d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (-3 b c+a (b+c)) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (-a^2 \left (1+\frac {b c}{a^2}\right )+(b+c) x^2-x^4+d x^6+a \left (b+c-2 x^2\right )\right )} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (a-b) (a-c) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{d \sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (2 (-1-2 a d) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{d \sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (2 \left (-2 a+b+c-a^2 d\right ) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^2} \sqrt {a-c+x^2} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^2+x^4-d x^6\right )} \, dx,x,\sqrt {-a+x}\right )}{d \sqrt {(-a+x) (-b+x) (-c+x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 4.95, size = 4752, normalized size = 79.20 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 3.13, size = 60, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{\sqrt {d} (a-x)^2}\right )}{\sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 89.10, size = 651, normalized size = 10.85 \begin {gather*} \left [\frac {\log \left (\frac {a^{6} d^{2} + d^{2} x^{6} - 6 \, a^{3} b c d - 6 \, {\left (a d^{2} - d\right )} x^{5} + {\left (15 \, a^{2} d^{2} - 6 \, {\left (3 \, a + b + c\right )} d + 1\right )} x^{4} + b^{2} c^{2} - 2 \, {\left (10 \, a^{3} d^{2} - 3 \, {\left (3 \, a^{2} + 3 \, a b + {\left (3 \, a + b\right )} c\right )} d + b + c\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} + 4 \, b c + c^{2} - 6 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, {\left (a^{2} + a b\right )} c\right )} d\right )} x^{2} - 4 \, {\left (a^{4} d + d x^{4} - {\left (4 \, a d - 1\right )} x^{3} - a b c + {\left (6 \, a^{2} d - a - b - c\right )} x^{2} - {\left (4 \, a^{3} d - a b - {\left (a + b\right )} c\right )} x\right )} \sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} \sqrt {d} - 2 \, {\left (3 \, a^{5} d^{2} + b^{2} c + b c^{2} - 3 \, {\left (a^{3} b + {\left (a^{3} + 3 \, a^{2} b\right )} c\right )} d\right )} x}{a^{6} d^{2} + d^{2} x^{6} + 2 \, a^{3} b c d - 2 \, {\left (3 \, a d^{2} + d\right )} x^{5} + {\left (15 \, a^{2} d^{2} + 2 \, {\left (3 \, a + b + c\right )} d + 1\right )} x^{4} + b^{2} c^{2} - 2 \, {\left (10 \, a^{3} d^{2} + {\left (3 \, a^{2} + 3 \, a b + {\left (3 \, a + b\right )} c\right )} d + b + c\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} + 4 \, b c + c^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, {\left (a^{2} + a b\right )} c\right )} d\right )} x^{2} - 2 \, {\left (3 \, a^{5} d^{2} + b^{2} c + b c^{2} + {\left (a^{3} b + {\left (a^{3} + 3 \, a^{2} b\right )} c\right )} d\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (-\frac {{\left (a^{3} d - d x^{3} + {\left (3 \, a d - 1\right )} x^{2} - b c - {\left (3 \, a^{2} d - b - c\right )} x\right )} \sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} \sqrt {-d}}{2 \, {\left (a^{2} b c d - {\left (2 \, a + b + c\right )} d x^{3} + d x^{4} + {\left (a^{2} + 2 \, a b + {\left (2 \, a + b\right )} c\right )} d x^{2} - {\left (a^{2} b + {\left (a^{2} + 2 \, a b\right )} c\right )} d x\right )}}\right )}{d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.14, size = 534, normalized size = 8.90
method | result | size |
default | \(\frac {2 \left (b -c \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \EllipticF \left (\sqrt {\frac {-c +x}{b -c}}, \sqrt {\frac {-b +c}{c -a}}\right )}{d \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b +c \right ) \textit {\_Z} -a^{3} d -b c \right )}{\sum }\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} c d -5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d +\underline {\hspace {1.25 ex}}\alpha a b d +\underline {\hspace {1.25 ex}}\alpha a c d +3 \underline {\hspace {1.25 ex}}\alpha b c d +a^{3} d +a^{2} b d +a^{2} c d -3 a b c d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +b c \right ) \left (b -c \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 a c d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \EllipticPi \left (\sqrt {\frac {-c +x}{b -c}}, \frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 a c d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \left (b -c \right )}{d \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )}, \sqrt {\frac {-b +c}{c -a}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +6 \underline {\hspace {1.25 ex}}\alpha a d -3 a^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -b -c \right ) \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right ) \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}\right )}{d^{2}}\) | \(534\) |
elliptic | \(\frac {2 \left (b -c \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \EllipticF \left (\sqrt {\frac {-c +x}{b -c}}, \sqrt {\frac {-b +c}{c -a}}\right )}{d \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b +c \right ) \textit {\_Z} -a^{3} d -b c \right )}{\sum }\frac {\left (-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} c d +5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d -\underline {\hspace {1.25 ex}}\alpha a b d -\underline {\hspace {1.25 ex}}\alpha a c d -3 \underline {\hspace {1.25 ex}}\alpha b c d -a^{3} d -a^{2} b d -a^{2} c d +3 a b c d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b +\underline {\hspace {1.25 ex}}\alpha c -b c \right ) \left (b -c \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 a c d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \EllipticPi \left (\sqrt {\frac {-c +x}{b -c}}, \frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 a c d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \left (b -c \right )}{d \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )}, \sqrt {\frac {-b +c}{c -a}}\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -6 \underline {\hspace {1.25 ex}}\alpha a d +3 a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha +b +c \right ) \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right ) \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}\right )}{d^{2}}\) | \(536\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (a - 2 \, b - 2 \, c\right )} x^{2} + x^{3} + {\left (a b + a c - 3 \, b c\right )} a - {\left (2 \, a^{2} - a b - a c - 3 \, b c\right )} x}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} + b c - {\left (3 \, a^{2} d + b + c\right )} x\right )} \sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 19.26, size = 569, normalized size = 9.48 \begin {gather*} \frac {\ln \left (\frac {\left (a-b-c+x+a^2\,d+d\,x^2-2\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-2\,a\,d\,x\right )\,\left (b\,c^2+b^2\,c+a^4\,d-a\,x^2+2\,b\,x^2-b^2\,x+2\,c\,x^2-c^2\,x+2\,d\,x^4-x^3+a^5\,d^2-d^2\,x^5+3\,a^2\,d\,x^2+5\,a\,d^2\,x^4-5\,a^4\,d^2\,x-2\,a^2\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-a\,b\,c+a\,b\,x+a\,c\,x-3\,b\,c\,x-10\,a^2\,d^2\,x^3+10\,a^3\,d^2\,x^2-a^3\,b\,d-a^3\,c\,d-4\,a\,d\,x^3-2\,a^3\,d\,x-2\,b\,d\,x^3-2\,c\,d\,x^3+3\,a^2\,b\,c\,d+3\,a\,b\,d\,x^2+3\,a\,c\,d\,x^2+3\,b\,c\,d\,x^2+2\,a\,b\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}+2\,a\,c\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-2\,b\,c\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-6\,a\,b\,c\,d\,x\right )}{\left (b\,c-b\,x-c\,x+a^3\,d-d\,x^3+x^2+3\,a\,d\,x^2-3\,a^2\,d\,x\right )\,\left (a^4\,d^2-4\,a^3\,d^2\,x+2\,a^3\,d-2\,a^2\,b\,d-2\,a^2\,c\,d+6\,a^2\,d^2\,x^2-2\,a^2\,d\,x+a^2+4\,a\,b\,c\,d-2\,a\,b-2\,a\,c-4\,a\,d^2\,x^3+2\,a\,d\,x^2+2\,a\,x+b^2-4\,b\,c\,d\,x+2\,b\,c+2\,b\,d\,x^2-2\,b\,x+c^2+2\,c\,d\,x^2-2\,c\,x+d^2\,x^4-2\,d\,x^3+x^2\right )}\right )}{\sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________