Optimal. Leaf size=40 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {-a-3} x}{\sqrt {a x^2+x^3+3 x-1}}\right )}{\sqrt {-a-3}} \]
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Rubi [C] time = 137.42, antiderivative size = 5379, normalized size of antiderivative = 134.48, number of steps used = 13, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, 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 {2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x} \sqrt {2 \left (9-a^2\right )+\frac {2\ 2^{2/3} \left (9-a^2\right )^2}{\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}}\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 {(15-4 a) (3+a)^2}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+x} \sqrt {\frac {1}{18} \left (2 \left (9-a^2\right )+\frac {2\ 2^{2/3} \left (9-a^2\right )^2}{\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )-\frac {\left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right ) x}{3\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+x^2}} \, dx,x,\frac {a}{3}+x\right )}{6 \sqrt {3} \sqrt {-1+3 x+a x^2+x^3}}+\frac {\left (\sqrt {2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x} \sqrt {2 \left (9-a^2\right )+\frac {2\ 2^{2/3} \left (9-a^2\right )^2}{\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}}\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 {(15-4 a) (3+a)^2}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+x} \sqrt {\frac {1}{18} \left (2 \left (9-a^2\right )+\frac {2\ 2^{2/3} \left (9-a^2\right )^2}{\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )-\frac {\left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right ) x}{3\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+x^2}} \, dx,x,\frac {a}{3}+x\right )}{2 \sqrt {3} \sqrt {-1+3 x+a x^2+x^3}}\\ &=\frac {\left (\sqrt {2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x} \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 [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}-\frac {\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}}}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+4 (a+3 x)} \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 [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+\frac {\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}}}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+4 (a+3 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 {(15-4 a) (3+a)^2}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+x} \sqrt {-\frac {18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}-\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 {(15-4 a) (3+a)^2}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+\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 {6} \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 )+\frac {2\ 2^{2/3} \left (9-a^2\right )^2}{\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}}{\frac {\left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )^2}{18 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}-\frac {2}{9} \left (2 \left (9-a^2\right )+\frac {2\ 2^{2/3} \left (9-a^2\right )^2}{\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{3 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}-\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}}-\left (54+54 a-4 a^3+6 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}+6 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}} \sqrt {\frac {\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3} \left (2 \left (9-a^2\right )+\frac {2\ 2^{2/3} \left (9-a^2\right )^2}{\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+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 {(15-4 a) (3+a)^2}\right )^{2/3}-4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}}-\left (54+54 a-4 a^3+6 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}-12 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} 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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}\right )}{3 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} \sqrt {-1+3 x+a x^2+x^3}}-\frac {\left (\sqrt {2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x} \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 [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}-\frac {\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}}}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+4 (a+3 x)} \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 [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+\frac {\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}}}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+4 (a+3 x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{6} \left (2 (3+a)+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}\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 {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}{3\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+2 x^2} \sqrt {-\frac {54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}{3\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+2 x^2}} \, dx,x,\frac {\sqrt {2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x}}{\sqrt {6}}\right )}{\sqrt {6} \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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}}-\left (54+54 a-4 a^3+6 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}+6 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}} \sqrt {\frac {\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3} \left (2 \left (9-a^2\right )+\frac {2\ 2^{2/3} \left (9-a^2\right )^2}{\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+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 {(15-4 a) (3+a)^2}\right )^{2/3}-4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}}-\left (54+54 a-4 a^3+6 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}-12 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} 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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}\right )}{3 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} \sqrt {-1+3 x+a x^2+x^3}}-\frac {\left (\sqrt {2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x} \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 [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}-\frac {\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}}}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+4 (a+3 x)} \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 [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+\frac {\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}}}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+4 (a+3 x)} \sqrt {1-\frac {2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} \left (2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{6} \left (2 (3+a)+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}\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 {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}{3\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+2 x^2} \sqrt {1-\frac {6\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} x^2}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}}} \, dx,x,\frac {\sqrt {2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x}}{\sqrt {6}}\right )}{\sqrt {-1+3 x+a x^2+x^3} \sqrt {-\frac {\sqrt [3]{2} \left (54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+2 \left (2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x\right )}}\\ &=\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}}-\left (54+54 a-4 a^3+6 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}+6 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}} \sqrt {\frac {\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3} \left (2 \left (9-a^2\right )+\frac {2\ 2^{2/3} \left (9-a^2\right )^2}{\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+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 {(15-4 a) (3+a)^2}\right )^{2/3}-4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}}-\left (54+54 a-4 a^3+6 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}-12 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} 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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}\right )}{3 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} \sqrt {-1+3 x+a x^2+x^3}}-\frac {\left (\sqrt {6} \sqrt {2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x} \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 [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}-\frac {\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}}}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+4 (a+3 x)} \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 [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+\frac {\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}}}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+4 (a+3 x)} \sqrt {1-\frac {2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} \left (2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}} \sqrt {1-\frac {2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} \left (2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{6} \left (2 (3+a)+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}\right )-x^2\right ) \sqrt {1-\frac {6\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} x^2}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}} \sqrt {1-\frac {6\ 2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} x^2}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}}} \, dx,x,\frac {\sqrt {2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x}}{\sqrt {6}}\right )}{\sqrt {-1+3 x+a x^2+x^3} \sqrt {-\frac {\sqrt [3]{2} \left (54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+2 \left (2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x\right )} \sqrt {-\frac {\sqrt [3]{2} \left (54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+2 \left (2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x\right )}}\\ &=\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}}-\left (54+54 a-4 a^3+6 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}+6 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}} \sqrt {\frac {\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3} \left (2 \left (9-a^2\right )+\frac {2\ 2^{2/3} \left (9-a^2\right )^2}{\left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+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 {(15-4 a) (3+a)^2}\right )^{2/3}-4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}+\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}}-\left (54+54 a-4 a^3+6 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}-12 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} 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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}\right )}{3 \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} \sqrt {-1+3 x+a x^2+x^3}}-\frac {3\ 2^{2/3} \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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}} \sqrt {2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x} \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 [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}-\frac {\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}}}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+4 (a+3 x)} \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 [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+\frac {\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}}}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}}}+4 (a+3 x)} \sqrt {1-\frac {2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} \left (2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}} \sqrt {1-\frac {2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} \left (2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}{2^{2/3} \sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}} \left (2 (3+a)+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}\right )};\sin ^{-1}\left (\frac {\sqrt [3]{2} \sqrt [6]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}} \sqrt {2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}}\right )|\frac {54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}}\right )}{\sqrt [6]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {-(3+a)^2 (-15+4 a)}} \left (2 (3+a)+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}\right ) \sqrt {-1+3 x+a x^2+x^3} \sqrt {-\frac {\sqrt [3]{2} \left (54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+2 \left (2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x\right )} \sqrt {-\frac {\sqrt [3]{2} \left (54-6 a^2-3 \sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\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 {(15-4 a) (3+a)^2}\right )^{2/3}+4 a^2 \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{4/3}}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+2 \left (2 a+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}\right )^{2/3}\right )}{\sqrt [3]{27+27 a-2 a^3+3 \sqrt {3} \sqrt {(15-4 a) (3+a)^2}}}+6 x\right )}}\\ \end {align*}
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Mathematica [C] time = 1.05, size = 1015, normalized size = 25.38
result too large to display
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.10, size = 40, 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}}\right )}{\sqrt {-3-a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 225, normalized size = 5.62 \begin {gather*} \left [\frac {\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 \, \sqrt {a + 3}}, \frac {\sqrt {-a - 3} \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 )}{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 = 0.73, size = 3006, normalized size = 75.15
method | result | size |
default | \(\text {Expression too large to display}\) | \(3006\) |
elliptic | \(\text {Expression too large to display}\) | \(3006\) |
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 [B] time = 0.71, size = 62, normalized size = 1.55 \begin {gather*} \frac {\ln \left (\frac {\left (\sqrt {x^3+a\,x^2+3\,x-1}+x\,\sqrt {a+3}\right )\,{\left (\sqrt {x^3+a\,x^2+3\,x-1}-x\,\sqrt {a+3}\right )}^3}{{\left (x-1\right )}^6}\right )}{\sqrt {a+3}} \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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