Optimal. Leaf size=268 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {a+a k^2-\sqrt {a} \sqrt {-b+2 a k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{\sqrt {a} (-1+x) \left (-1+k^2 x\right )}\right )}{\sqrt {a} \sqrt {a+a k^2-\sqrt {a} \sqrt {-b+2 a k^2}}}-\frac {\sqrt {\sqrt {a} \left (\sqrt {a}+\sqrt {a} k^2+\sqrt {-b+2 a k^2}\right )} \text {ArcTan}\left (\frac {\sqrt {a+a k^2+\sqrt {a} \sqrt {-b+2 a k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{\sqrt {a} (-1+x) \left (-1+k^2 x\right )}\right )}{a \left (\sqrt {a}+\sqrt {a} k^2+\sqrt {-b+2 a k^2}\right )} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 4.25, antiderivative size = 583, normalized size of antiderivative = 2.18, number of steps
used = 28, number of rules used = 10, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6850, 6820,
6860, 728, 116, 6857, 948, 12, 174, 551} \begin {gather*} \frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {\sqrt {2} \sqrt {a}}{\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}};\text {ArcSin}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {\sqrt {2} \sqrt {a}}{\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}};\text {ArcSin}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {\sqrt {2} \sqrt {a}}{\sqrt {\sqrt {b^2-4 a^2 k^4}-b}};\text {ArcSin}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {\sqrt {2} \sqrt {a}}{\sqrt {\sqrt {b^2-4 a^2 k^4}-b}};\text {ArcSin}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\text {ArcSin}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
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Rule 12
Rule 116
Rule 174
Rule 551
Rule 728
Rule 948
Rule 6820
Rule 6850
Rule 6857
Rule 6860
Rubi steps
\begin {align*} \int \frac {-1+k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (a+b x^2+a k^4 x^4\right )} \, dx &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {-1+k^4 x^4}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (a+b x^2+a k^4 x^4\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {-1+k^4 x^4}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (a+b x^2+a k^4 x^4\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {1}{a \sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {2 a+b x^2}{a \sqrt {1-k^2 x} \sqrt {x-x^2} \left (a+b x^2+a k^4 x^4\right )}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {2 a+b x^2}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (a+b x^2+a k^4 x^4\right )} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {b-\sqrt {b^2-4 a^2 k^4}}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (b-\sqrt {b^2-4 a^2 k^4}+2 a k^4 x^2\right )}+\frac {b+\sqrt {b^2-4 a^2 k^4}}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (b+\sqrt {b^2-4 a^2 k^4}+2 a k^4 x^2\right )}\right ) \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (b-\sqrt {b^2-4 a^2 k^4}+2 a k^4 x^2\right )} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (b+\sqrt {b^2-4 a^2 k^4}+2 a k^4 x^2\right )} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}}{2 \left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-k^2 x} \left (\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}-\sqrt {2} \sqrt {a} k^2 x\right ) \sqrt {x-x^2}}+\frac {\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}}{2 \left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-k^2 x} \left (\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}+\sqrt {2} \sqrt {a} k^2 x\right ) \sqrt {x-x^2}}\right ) \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}}{2 \left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-k^2 x} \left (\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}-\sqrt {2} \sqrt {a} k^2 x\right ) \sqrt {x-x^2}}+\frac {\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}}{2 \left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-k^2 x} \left (\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}+\sqrt {2} \sqrt {a} k^2 x\right ) \sqrt {x-x^2}}\right ) \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}-\sqrt {2} \sqrt {a} k^2 x\right ) \sqrt {x-x^2}} \, dx}{2 a \sqrt {-b+\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}+\sqrt {2} \sqrt {a} k^2 x\right ) \sqrt {x-x^2}} \, dx}{2 a \sqrt {-b+\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}-\sqrt {2} \sqrt {a} k^2 x\right ) \sqrt {x-x^2}} \, dx}{2 a \sqrt {-b-\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}+\sqrt {2} \sqrt {a} k^2 x\right ) \sqrt {x-x^2}} \, dx}{2 a \sqrt {-b-\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}-\sqrt {2} \sqrt {a} k^2 x\right )} \, dx}{\sqrt {2} a \sqrt {-b+\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}+\sqrt {2} \sqrt {a} k^2 x\right )} \, dx}{\sqrt {2} a \sqrt {-b+\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}-\sqrt {2} \sqrt {a} k^2 x\right )} \, dx}{\sqrt {2} a \sqrt {-b-\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}+\sqrt {2} \sqrt {a} k^2 x\right )} \, dx}{\sqrt {2} a \sqrt {-b-\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}-\sqrt {2} \sqrt {a} k^2 x\right )} \, dx}{2 a \sqrt {-b+\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}+\sqrt {2} \sqrt {a} k^2 x\right )} \, dx}{2 a \sqrt {-b+\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}-\sqrt {2} \sqrt {a} k^2 x\right )} \, dx}{2 a \sqrt {-b-\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}+\sqrt {2} \sqrt {a} k^2 x\right )} \, dx}{2 a \sqrt {-b-\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \sqrt {1+k^2 x^2} \left (\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}-\sqrt {2} \sqrt {a} k^2 x^2\right )} \, dx,x,\sqrt {-x}\right )}{a \sqrt {-b+\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \sqrt {1+k^2 x^2} \left (\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}+\sqrt {2} \sqrt {a} k^2 x^2\right )} \, dx,x,\sqrt {-x}\right )}{a \sqrt {-b+\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \sqrt {1+k^2 x^2} \left (\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}-\sqrt {2} \sqrt {a} k^2 x^2\right )} \, dx,x,\sqrt {-x}\right )}{a \sqrt {-b-\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^4}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \sqrt {1+k^2 x^2} \left (\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}+\sqrt {2} \sqrt {a} k^2 x^2\right )} \, dx,x,\sqrt {-x}\right )}{a \sqrt {-b-\sqrt {b^2-4 a^2 k^4}} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {\sqrt {2} \sqrt {a}}{\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {\sqrt {2} \sqrt {a}}{\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {\sqrt {2} \sqrt {a}}{\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {\sqrt {2} \sqrt {a}}{\sqrt {-b+\sqrt {b^2-4 a^2 k^4}}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 35.07, size = 13957, normalized size = 52.08 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.12, size = 314, normalized size = 1.17
method | result | size |
elliptic | \(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{a \,k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,k^{4} \textit {\_Z}^{4}+b \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (b \,\underline {\hspace {1.25 ex}}\alpha ^{2}+2 a \right ) \left (a \,k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+k^{2} \underline {\hspace {1.25 ex}}\alpha a +k^{2} \underline {\hspace {1.25 ex}}\alpha b +a +b \right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {a \,k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+k^{2} \underline {\hspace {1.25 ex}}\alpha a +k^{2} \underline {\hspace {1.25 ex}}\alpha b +a +b}{a \,k^{4}+a +b}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+b \right ) \left (a \,k^{4}+a +b \right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}}{a}\) | \(312\) |
default | \(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{a \,k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,k^{4} \textit {\_Z}^{4}+b \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-b \,\underline {\hspace {1.25 ex}}\alpha ^{2}-2 a \right ) \left (a \,k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+k^{2} \underline {\hspace {1.25 ex}}\alpha a +k^{2} \underline {\hspace {1.25 ex}}\alpha b +a +b \right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {a \,k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+k^{2} \underline {\hspace {1.25 ex}}\alpha a +k^{2} \underline {\hspace {1.25 ex}}\alpha b +a +b}{a \,k^{4}+a +b}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+b \right ) \left (a \,k^{4}+a +b \right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}}{a}\) | \(314\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2407 vs.
\(2 (222) = 444\).
time = 3.52, size = 2407, normalized size = 8.98 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {k^4\,x^4-1}{\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}\,\left (a\,k^4\,x^4+b\,x^2+a\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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