Optimal. Leaf size=60 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}}{\sqrt {d} (a-x)}\right )}{\sqrt {d}} \]
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Rubi [C] time = 15.47, antiderivative size = 440, normalized size of antiderivative = 7.33, number of steps used = 16, number of rules used = 7, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6718, 6728, 121, 120, 169, 538, 537} \begin {gather*} -\frac {2 \sqrt {b-a} \sqrt {x-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \Pi \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}};\sin ^{-1}\left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right )|\frac {a-b}{a-c}\right )}{\sqrt {-((a-x) (b-x) (c-x))}}-\frac {2 \sqrt {b-a} \sqrt {x-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \Pi \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}};\sin ^{-1}\left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right )|\frac {a-b}{a-c}\right )}{\sqrt {-((a-x) (b-x) (c-x))}}+\frac {2 \sqrt {b-a} \sqrt {x-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} F\left (\sin ^{-1}\left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right )|\frac {a-b}{a-c}\right )}{\sqrt {-((a-x) (b-x) (c-x))}} \end {gather*}
Antiderivative was successfully verified.
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Rule 120
Rule 121
Rule 169
Rule 537
Rule 538
Rule 6718
Rule 6728
Rubi steps
\begin {align*} \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 {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {a b+a c-b c-2 a x+x^2}{\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x} \left (b c+a d-(b+c+d) x+x^2\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \left (\frac {1}{\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}}+\frac {-2 b c+a (b+c-d)-(2 a-b-c-d) x}{\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x} \left (b c+a d+(-b-c-d) x+x^2\right )}\right ) \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {1}{\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {-2 b c+a (b+c-d)-(2 a-b-c-d) x}{\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x} \left (b c+a d+(-b-c-d) x+x^2\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \left (\frac {-2 a+b+c+d+\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}}{\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x} \left (-b-c-d-\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}+2 x\right )}+\frac {-2 a+b+c+d-\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}}{\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x} \left (-b-c-d+\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}+2 x\right )}\right ) \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (\sqrt {-a+x} \sqrt {\frac {-b+x}{a-b}} \sqrt {-c+x}\right ) \int \frac {1}{\sqrt {-a+x} \sqrt {-c+x} \sqrt {-\frac {b}{a-b}+\frac {x}{a-b}}} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\left (-2 a+b+c+d-\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}\right ) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {1}{\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x} \left (-b-c-d+\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}+2 x\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (\left (-2 a+b+c+d+\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}\right ) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \int \frac {1}{\sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x} \left (-b-c-d-\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}+2 x\right )} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}+\frac {\left (\sqrt {-a+x} \sqrt {\frac {-b+x}{a-b}} \sqrt {\frac {-c+x}{a-c}}\right ) \int \frac {1}{\sqrt {-a+x} \sqrt {-\frac {b}{a-b}+\frac {x}{a-b}} \sqrt {-\frac {c}{a-c}+\frac {x}{a-c}}} \, dx}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {2 \sqrt {-a+b} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \sqrt {-a+x} F\left (\sin ^{-1}\left (\frac {\sqrt {-a+x}}{\sqrt {-a+b}}\right )|\frac {a-b}{a-c}\right )}{\sqrt {-((a-x) (b-x) (c-x))}}-\frac {\left (2 \left (-2 a+b+c+d-\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}\right ) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2 a+b+c+d-\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}-2 x^2\right ) \sqrt {a-b+x^2} \sqrt {a-c+x^2}} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 \left (-2 a+b+c+d+\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}\right ) \sqrt {-a+x} \sqrt {-b+x} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2 a+b+c+d+\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}-2 x^2\right ) \sqrt {a-b+x^2} \sqrt {a-c+x^2}} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {2 \sqrt {-a+b} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \sqrt {-a+x} F\left (\sin ^{-1}\left (\frac {\sqrt {-a+x}}{\sqrt {-a+b}}\right )|\frac {a-b}{a-c}\right )}{\sqrt {-((a-x) (b-x) (c-x))}}-\frac {\left (2 \left (-2 a+b+c+d-\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}\right ) \sqrt {-a+x} \sqrt {\frac {-b+x}{a-b}} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2 a+b+c+d-\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}-2 x^2\right ) \sqrt {a-c+x^2} \sqrt {1+\frac {x^2}{a-b}}} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 \left (-2 a+b+c+d+\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}\right ) \sqrt {-a+x} \sqrt {\frac {-b+x}{a-b}} \sqrt {-c+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2 a+b+c+d+\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}-2 x^2\right ) \sqrt {a-c+x^2} \sqrt {1+\frac {x^2}{a-b}}} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {2 \sqrt {-a+b} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \sqrt {-a+x} F\left (\sin ^{-1}\left (\frac {\sqrt {-a+x}}{\sqrt {-a+b}}\right )|\frac {a-b}{a-c}\right )}{\sqrt {-((a-x) (b-x) (c-x))}}-\frac {\left (2 \left (-2 a+b+c+d-\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}\right ) \sqrt {-a+x} \sqrt {\frac {-b+x}{a-b}} \sqrt {\frac {-c+x}{a-c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2 a+b+c+d-\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}-2 x^2\right ) \sqrt {1+\frac {x^2}{a-b}} \sqrt {1+\frac {x^2}{a-c}}} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 \left (-2 a+b+c+d+\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}\right ) \sqrt {-a+x} \sqrt {\frac {-b+x}{a-b}} \sqrt {\frac {-c+x}{a-c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2 a+b+c+d+\sqrt {b^2-2 b c+c^2-4 a d+2 b d+2 c d+d^2}-2 x^2\right ) \sqrt {1+\frac {x^2}{a-b}} \sqrt {1+\frac {x^2}{a-c}}} \, dx,x,\sqrt {-a+x}\right )}{\sqrt {(-a+x) (-b+x) (-c+x)}}\\ &=\frac {2 \sqrt {-a+b} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \sqrt {-a+x} F\left (\sin ^{-1}\left (\frac {\sqrt {-a+x}}{\sqrt {-a+b}}\right )|\frac {a-b}{a-c}\right )}{\sqrt {-((a-x) (b-x) (c-x))}}-\frac {2 \sqrt {-a+b} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \sqrt {-a+x} \Pi \left (\frac {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}};\sin ^{-1}\left (\frac {\sqrt {-a+x}}{\sqrt {-a+b}}\right )|\frac {a-b}{a-c}\right )}{\sqrt {-((a-x) (b-x) (c-x))}}-\frac {2 \sqrt {-a+b} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \sqrt {-a+x} \Pi \left (\frac {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}};\sin ^{-1}\left (\frac {\sqrt {-a+x}}{\sqrt {-a+b}}\right )|\frac {a-b}{a-c}\right )}{\sqrt {-((a-x) (b-x) (c-x))}}\\ \end {align*}
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Mathematica [C] time = 6.91, size = 289, normalized size = 4.82 \begin {gather*} -\frac {2 i (x-a)^{3/2} \sqrt {\frac {b-x}{a-x}} \sqrt {\frac {c-x}{a-x}} \left (-\Pi \left (-\frac {2 (a-c)}{-2 a+b+c+d-\sqrt {b^2-2 c b+2 d b+c^2+d^2-4 a d+2 c d}};i \sinh ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {x-a}}\right )|\frac {a-c}{a-b}\right )-\Pi \left (-\frac {2 (a-c)}{-2 a+b+c+d+\sqrt {b^2-2 c b+2 d b+c^2+d^2-4 a d+2 c d}};i \sinh ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {x-a}}\right )|\frac {a-c}{a-b}\right )+F\left (i \sinh ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {x-a}}\right )|\frac {a-c}{a-b}\right )\right )}{\sqrt {a-b} \sqrt {(x-a) (x-b) (x-c)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.81, 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)}\right )}{\sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 22.60, size = 349, normalized size = 5.82 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 8138, normalized size = 135.63
method | result | size |
default | \(\text {Expression too large to display}\) | \(8138\) |
elliptic | \(\text {Expression too large to display}\) | \(8296\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 711, normalized size = 11.85 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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