Integrand size = 27, antiderivative size = 54 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {-3+a} x \sqrt {-1+3 x-a x^2+x^3}}{1-3 x+a x^2-x^3}\right )}{\sqrt {-3+a}} \]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 125.80 (sec) , antiderivative size = 5514, normalized size of antiderivative = 102.11, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6874, 2092, 2091, 732, 430, 2106, 2105, 948, 175, 552, 551} \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\frac {\sqrt [3]{2} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}} \sqrt {\frac {2 \sqrt [3]{2} a^2+2 \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27} a-6 \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27} x+\left (4 a^3-54 a+6 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+54\right )^{2/3}-18 \sqrt [3]{2}}{6 a^2+3 \sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}-54}} \sqrt {-\frac {\frac {2 \left (9-a^2\right )^2}{\left (a^3-\frac {27 a}{2}+\frac {3}{2} \left (\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+9\right )\right )^{2/3}}+2 \left (9-a^2\right )+18 \left (x-\frac {a}{3}\right )^2-\frac {\sqrt [3]{2} \left (-2 a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+18\right ) (3 x-a)}{\sqrt [3]{2 a^3-27 a+3 \left (\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+9\right )}}+\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}}{\frac {\left (-2 a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+18\right )^2}{18 \sqrt [3]{2} \left (2 a^3-27 a+3 \left (\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+9\right )\right )^{2/3}}-\frac {2}{9} \left (\frac {2 \left (9-a^2\right )^2}{\left (a^3-\frac {27 a}{2}+\frac {3}{2} \left (\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+9\right )\right )^{2/3}}+2 \left (9-a^2\right )+\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2 \sqrt [3]{2} a^2-4 \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27} a+12 \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27} x+\left (4 a^3-54 a+6 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+54\right )^{2/3}+\sqrt {6} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}-18 \sqrt [3]{2}}{\sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}}}}{2^{3/4} \sqrt [4]{3}}\right ),-\frac {2 \sqrt [6]{2} \sqrt {3} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}}{-6 a^2-3 \sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}+54}\right )}{3 \sqrt {3} \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27} \sqrt {x^3-a x^2+3 x-1}}-\frac {3\ 2^{2/3} \sqrt {-6 a^2-3 \sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}+54} \sqrt {1-\frac {2 \left (-2 a^2-2^{2/3} \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27} (a-3 x)-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+18\right )}{-6 a^2-3 \sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}+54}} \sqrt {1-\frac {2 \left (-2 a^2-2^{2/3} \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27} (a-3 x)-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+18\right )}{-6 a^2-3 \sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}+54}} \sqrt {-\frac {\sqrt [3]{2} \left (-2 a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+18\right )+4 \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27} (a-3 x)-\sqrt {6} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}}{\sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27}}} \sqrt {-\frac {\sqrt [3]{2} \left (-2 a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+18\right )+4 \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27} (a-3 x)+\sqrt {6} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}}{\sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27}}} \sqrt {-2 a+\frac {-2 a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+18}{\sqrt [3]{a^3-\frac {27 a}{2}+\frac {3}{2} \left (\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+9\right )}}+6 x} \operatorname {EllipticPi}\left (\frac {-6 a^2-3 \sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}+54}{2 \left (-2 a^2-2^{2/3} \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27} a-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+3\ 2^{2/3} \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27}+18\right )},\arcsin \left (\frac {\sqrt [3]{2} \sqrt [6]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27} \sqrt {-2 a+\frac {-2 a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+18}{\sqrt [3]{a^3-\frac {27 a}{2}+\frac {3}{2} \left (\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+9\right )}}+6 x}}{\sqrt {-6 a^2-3 \sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}+54}}\right ),\frac {-6 a^2-3 \sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}+54}{-6 a^2-3 \sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}+54}\right )}{\sqrt [6]{2 a^3-27 a+3 \left (\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+9\right )} \left (-2 a+\frac {-2 a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}+18}{\sqrt [3]{a^3-\frac {27 a}{2}+\frac {3}{2} \left (\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+9\right )}}+6\right ) \sqrt {-\frac {-2 \sqrt [3]{2} a^2+4 \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27} (a-3 x)-\left (4 a^3-54 a+6 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+54\right )^{2/3}-\sqrt {6} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}+18 \sqrt [3]{2}}{\sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27}}} \sqrt {-\frac {-2 \sqrt [3]{2} a^2+4 \sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27} (a-3 x)-\left (4 a^3-54 a+6 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+54\right )^{2/3}+\sqrt {6} \sqrt {-2 2^{2/3} a^4+4 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3} a^2+36\ 2^{2/3} a^2-\sqrt [3]{2} \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{4/3}-36 \left (2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27\right )^{2/3}-162\ 2^{2/3}}+18 \sqrt [3]{2}}{\sqrt [3]{2 a^3-27 a+3 \sqrt {3} \sqrt {(3-a)^2 (4 a+15)}+27}}} \sqrt {x^3-a x^2+3 x-1}} \]
[In]
[Out]
Rule 175
Rule 430
Rule 551
Rule 552
Rule 732
Rule 948
Rule 2091
Rule 2092
Rule 2105
Rule 2106
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \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 \text {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 )+\text {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 ) \text {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 ) \text {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}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.69 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-3+a} x}{\sqrt {-1+3 x-a x^2+x^3}}\right )}{\sqrt {-3+a}} \]
[In]
[Out]
Time = 6.54 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {2 \arctan \left (\frac {\sqrt {-a \,x^{2}+x^{3}+3 x -1}}{x \sqrt {-3+a}}\right )}{\sqrt {-3+a}}\) | \(34\) |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\sqrt {-a \,x^{2}+x^{3}+3 x -1}}{x \sqrt {-3+a}}\right )}{\sqrt {-3+a}}\) | \(34\) |
elliptic | \(\text {Expression too large to display}\) | \(3008\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 234, normalized size of antiderivative = 4.33 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\left [-\frac {\sqrt {-a + 3} \log \left (-\frac {2 \, {\left (4 \, a - 9\right )} x^{5} - x^{6} - {\left (8 \, a^{2} - 24 \, a + 15\right )} x^{4} + 4 \, {\left (6 \, a - 13\right )} x^{3} - {\left (8 \, a - 9\right )} x^{2} - 4 \, {\left ({\left (2 \, a - 3\right )} x^{3} - x^{4} - 3 \, x^{2} + x\right )} \sqrt {-a x^{2} + x^{3} + 3 \, x - 1} \sqrt {-a + 3} + 6 \, x - 1}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1}\right )}{2 \, {\left (a - 3\right )}}, \frac {\arctan \left (-\frac {{\left ({\left (2 \, a - 3\right )} x^{2} - x^{3} - 3 \, x + 1\right )} \sqrt {-a x^{2} + x^{3} + 3 \, x - 1} \sqrt {a - 3}}{2 \, {\left ({\left (a - 3\right )} x^{4} - {\left (a^{2} - 3 \, a\right )} x^{3} + 3 \, {\left (a - 3\right )} x^{2} - {\left (a - 3\right )} x\right )}}\right )}{\sqrt {a - 3}}\right ] \]
[In]
[Out]
\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\int \frac {x + 2}{\left (x - 1\right ) \sqrt {- a x^{2} + x^{3} + 3 x - 1}}\, dx \]
[In]
[Out]
\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {-a x^{2} + x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {-a x^{2} + x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\int \frac {x+2}{\left (x-1\right )\,\sqrt {x^3-a\,x^2+3\,x-1}} \,d x \]
[In]
[Out]