Optimal. Leaf size=54 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a-3} x \sqrt {-a x^2+x^3+3 x-1}}{a x^2-x^3-3 x+1}\right )}{\sqrt {a-3}} \]
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Rubi [C] time = 137.71, antiderivative size = 5514, normalized size of antiderivative = 102.11, number of steps used = 13, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6742, 2067, 2066, 718, 419, 2081, 2080, 934, 169, 538, 537}
result too large to display
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 169
Rule 419
Rule 537
Rule 538
Rule 718
Rule 934
Rule 2066
Rule 2067
Rule 2080
Rule 2081
Rule 6742
Rubi steps
\begin {align*} \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx &=\int \left (\frac {1}{\sqrt {-1+3 x-a x^2+x^3}}+\frac {3}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}}\right ) \, dx\\ &=3 \int \frac {1}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx+\int \frac {1}{\sqrt {-1+3 x-a x^2+x^3}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{3} (-3+a)+x\right ) \sqrt {\frac {1}{27} \left (-27+27 a-2 a^3\right )+\frac {1}{3} \left (9-a^2\right ) x+x^3}} \, dx,x,-\frac {a}{3}+x\right )+\operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {1}{27} \left (-27+27 a-2 a^3\right )+\frac {1}{3} \left (9-a^2\right ) x+x^3}} \, dx,x,-\frac {a}{3}+x\right )\\ &=\frac {\left (\sqrt {-\frac {a}{3}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{6 \sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+x} \sqrt {2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+18 \left (-\frac {a}{3}+x\right )^2-\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (-a+3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+x} \sqrt {\frac {1}{18} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}\right )-\frac {\left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) x}{3\ 2^{2/3} \sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+x^2}} \, dx,x,-\frac {a}{3}+x\right )}{3 \sqrt {2} \sqrt {-1+3 x-a x^2+x^3}}+\frac {\left (\sqrt {-\frac {a}{3}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{6 \sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+x} \sqrt {2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+18 \left (-\frac {a}{3}+x\right )^2-\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (-a+3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{3} (-3+a)+x\right ) \sqrt {\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+x} \sqrt {\frac {1}{18} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}\right )-\frac {\left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) x}{3\ 2^{2/3} \sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+x^2}} \, dx,x,-\frac {a}{3}+x\right )}{\sqrt {2} \sqrt {-1+3 x-a x^2+x^3}}\\ &=\frac {\left (\sqrt {\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}-4 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} (a-3 x)}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}-4 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} (a-3 x)}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {-\frac {a}{3}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{6 \sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{3} (-3+a)+x\right ) \sqrt {\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+x} \sqrt {-\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}-\frac {\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt {6} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+2 x} \sqrt {-\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+\frac {\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt {6} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+2 x}} \, dx,x,-\frac {a}{3}+x\right )}{2 \sqrt {-1+3 x-a x^2+x^3}}+\frac {\left (\sqrt [3]{2} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}} \sqrt {\frac {-18 \sqrt [3]{2}+2 \sqrt [3]{2} a^2+2 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}+\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-6 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{-54+6 a^2+3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {-\frac {2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+18 \left (-\frac {a}{3}+x\right )^2-\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (-a+3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}}{\frac {\left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right )^2}{18 \sqrt [3]{2} \left (-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}-\frac {2}{9} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}\right )}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {\sqrt {\frac {2}{3}} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}} x^2}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} \left (\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right )}{3 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}-\frac {\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt {6} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}\right )}}} \, dx,x,\frac {\sqrt {\frac {-18 \sqrt [3]{2}+2 \sqrt [3]{2} a^2-4 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}+\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}}}{2^{3/4} \sqrt [4]{3}}\right )}{3 \sqrt {3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} \sqrt {-1+3 x-a x^2+x^3}}\\ &=\frac {2^{5/6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}} \sqrt {\frac {\left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (a-3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+2 (a-3 x)^2\right )}{162\ 2^{2/3}-36\ 2^{2/3} a^2+2\ 2^{2/3} a^4+36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}} \sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-2 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+6 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} F\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2+4 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}-12 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}}{2^{3/4} \sqrt [4]{3}}\right )|-\frac {2 \sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}\right )}{3 \sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )} \sqrt {-1+3 x-a x^2+x^3}}-\frac {\left (\sqrt {\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}-4 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} (a-3 x)}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}-4 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} (a-3 x)}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {-\frac {a}{3}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{6 \sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{6} \left (6-2 a+\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}\right )-x^2\right ) \sqrt {-\frac {54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+2 x^2} \sqrt {-\frac {54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+2 x^2}} \, dx,x,\sqrt {-\frac {a}{3}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{6 \sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+x}\right )}{\sqrt {-1+3 x-a x^2+x^3}}\\ &=\frac {2^{5/6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}} \sqrt {\frac {\left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (a-3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+2 (a-3 x)^2\right )}{162\ 2^{2/3}-36\ 2^{2/3} a^2+2\ 2^{2/3} a^4+36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}} \sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-2 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+6 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} F\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2+4 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}-12 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}}{2^{3/4} \sqrt [4]{3}}\right )|-\frac {2 \sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}\right )}{3 \sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )} \sqrt {-1+3 x-a x^2+x^3}}-\frac {\left (\sqrt {6} \sqrt {\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}-4 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} (a-3 x)}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}-4 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} (a-3 x)}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {-\frac {a}{3}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{6 \sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+x} \sqrt {1-\frac {2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \left (-2 a-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{6} \left (6-2 a+\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}\right )-x^2\right ) \sqrt {-\frac {54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+2 x^2} \sqrt {1-\frac {6\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x^2}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}} \, dx,x,\sqrt {-\frac {a}{3}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{6 \sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+x}\right )}{\sqrt {\frac {-18 \sqrt [3]{2}+2 \sqrt [3]{2} a^2-4 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}+\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {-1+3 x-a x^2+x^3}}\\ &=\frac {2^{5/6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}} \sqrt {\frac {\left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (a-3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+2 (a-3 x)^2\right )}{162\ 2^{2/3}-36\ 2^{2/3} a^2+2\ 2^{2/3} a^4+36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}} \sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-2 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+6 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} F\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2+4 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}-12 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}}{2^{3/4} \sqrt [4]{3}}\right )|-\frac {2 \sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}\right )}{3 \sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )} \sqrt {-1+3 x-a x^2+x^3}}-\frac {\left (6 \sqrt {\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}-4 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} (a-3 x)}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}-4 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} (a-3 x)}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {-\frac {a}{3}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{6 \sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+x} \sqrt {1-\frac {2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \left (-2 a-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {1-\frac {2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \left (-2 a-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{6} \left (6-2 a+\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}\right )-x^2\right ) \sqrt {1-\frac {6\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x^2}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {1-\frac {6\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x^2}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}} \, dx,x,\sqrt {-\frac {a}{3}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{6 \sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+x}\right )}{\sqrt {\frac {-18 \sqrt [3]{2}+2 \sqrt [3]{2} a^2-4 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}+\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {\frac {-18 \sqrt [3]{2}+2 \sqrt [3]{2} a^2-4 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}+\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {-1+3 x-a x^2+x^3}}\\ &=\frac {2^{5/6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}} \sqrt {\frac {\left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (a-3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+2 (a-3 x)^2\right )}{162\ 2^{2/3}-36\ 2^{2/3} a^2+2\ 2^{2/3} a^4+36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}} \sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-2 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+6 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} F\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2+4 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}-12 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}}{2^{3/4} \sqrt [4]{3}}\right )|-\frac {2 \sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}\right )}{3 \sqrt [3]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )} \sqrt {-1+3 x-a x^2+x^3}}-\frac {6 \sqrt [6]{2} \sqrt {3} \sqrt {54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}} \sqrt {\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}-4 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} (a-3 x)}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}-4 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} (a-3 x)}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {-\frac {a}{3}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{6 \sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+x} \sqrt {1-\frac {2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \left (-2 a-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {1-\frac {2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \left (-2 a-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \Pi \left (\frac {54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{2 \left (18-2 a^2+3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-2^{2/3} a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right )};\sin ^{-1}\left (\frac {2^{5/6} \sqrt {3} \sqrt [6]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \sqrt {-\frac {a}{3}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{6 \sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+x}}{\sqrt {54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}\right )|\frac {54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}\right )}{\sqrt [6]{-27 a+2 a^3+3 \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )} \left (6-2 a+\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9+\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}\right ) \sqrt {\frac {-18 \sqrt [3]{2}+2 \sqrt [3]{2} a^2-4 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}+\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {\frac {-18 \sqrt [3]{2}+2 \sqrt [3]{2} a^2-4 a \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}+\left (54-54 a+4 a^3+6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{\sqrt [3]{27-27 a+2 a^3+3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {-1+3 x-a x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 1.05, size = 1148, normalized size = 21.26
result too large to display
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.11, size = 54, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {-3+a} x \sqrt {-1+3 x-a x^2+x^3}}{1-3 x+a x^2-x^3}\right )}{\sqrt {-3+a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 234, normalized size = 4.33 \begin {gather*} \left [-\frac {\sqrt {-a + 3} \log \left (-\frac {2 \, {\left (4 \, a - 9\right )} x^{5} - x^{6} - {\left (8 \, a^{2} - 24 \, a + 15\right )} x^{4} + 4 \, {\left (6 \, a - 13\right )} x^{3} - {\left (8 \, a - 9\right )} x^{2} - 4 \, {\left ({\left (2 \, a - 3\right )} x^{3} - x^{4} - 3 \, x^{2} + x\right )} \sqrt {-a x^{2} + x^{3} + 3 \, x - 1} \sqrt {-a + 3} + 6 \, x - 1}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1}\right )}{2 \, {\left (a - 3\right )}}, \frac {\arctan \left (-\frac {{\left ({\left (2 \, a - 3\right )} x^{2} - x^{3} - 3 \, x + 1\right )} \sqrt {-a x^{2} + x^{3} + 3 \, x - 1} \sqrt {a - 3}}{2 \, {\left ({\left (a - 3\right )} x^{4} - {\left (a^{2} - 3 \, a\right )} x^{3} + 3 \, {\left (a - 3\right )} x^{2} - {\left (a - 3\right )} x\right )}}\right )}{\sqrt {a - 3}}\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 {x + 2}{\sqrt {-a x^{2} + x^{3} + 3 \, x - 1} {\left (x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.74, size = 3008, normalized size = 55.70
method | result | size |
default | \(\text {Expression too large to display}\) | \(3008\) |
elliptic | \(\text {Expression too large to display}\) | \(3008\) |
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*} \int \frac {x + 2}{\sqrt {-a x^{2} + x^{3} + 3 \, x - 1} {\left (x - 1\right )}}\,{d x} \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 = -0.02 \begin {gather*} \int \frac {x+2}{\left (x-1\right )\,\sqrt {x^3-a\,x^2+3\,x-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 2}{\left (x - 1\right ) \sqrt {- a x^{2} + x^{3} + 3 x - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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