Integrand size = 26, antiderivative size = 40 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-3-a} x}{\sqrt {-1+3 x+a x^2+x^3}}\right )}{\sqrt {-3-a}} \]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 129.25 (sec) , antiderivative size = 5375, normalized size of antiderivative = 134.38, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, 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\ 2^{2/3} \left (9-a^2\right )^2}{\left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}}+2 \left (9-a^2\right )+18 \left (\frac {a}{3}+x\right )^2-\frac {\sqrt [3]{2} \left (-2 a^2-\sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}+18\right ) (a+3 x)}{\sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27}}+\sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}}{\frac {\left (-2 a^2-\sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}+18\right )^2}{18 \sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}}-\frac {2}{9} \left (\frac {2\ 2^{2/3} \left (9-a^2\right )^2}{\left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}}+2 \left (9-a^2\right )+\sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+27\right )^{4/3}-36 \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+27\right )^{4/3}-36 \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+27\right )^{4/3}-36 \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}-162\ 2^{2/3}}+54} \sqrt {2 a+\frac {\sqrt [3]{2} \left (-2 a^2-\sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}+18\right )}{\sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27}}+6 x} \sqrt {\frac {\sqrt [3]{2} \left (2 a^2+\sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {-(a+3)^2 (4 a-15)}+27\right )^{2/3}-18\right )}{\sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {-(a+3)^2 (4 a-15)}+27}}+4 (a+3 x)-\frac {\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}}}{\sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {-(a+3)^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 {-(a+3)^2 (4 a-15)}+27\right )^{2/3}-18\right )}{\sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {-(a+3)^2 (4 a-15)}+27}}+4 (a+3 x)+\frac {\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}}}{\sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {-(a+3)^2 (4 a-15)}+27}}} \sqrt {\frac {2 \left (-2 a^2+2^{2/3} \sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {-(a+3)^2 (4 a-15)}+27} (a+3 x)-\sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {-(a+3)^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 {-(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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+27\right )^{4/3}-36 \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}-162\ 2^{2/3}}-54}+1} \sqrt {\frac {2 \left (-2 a^2+2^{2/3} \sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {-(a+3)^2 (4 a-15)}+27} (a+3 x)-\sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {-(a+3)^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 {-(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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+27\right )^{4/3}-36 \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}-162\ 2^{2/3}}-54}+1} \operatorname {EllipticPi}\left (\frac {-6 a^2-3 \sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+27\right )^{4/3}-36 \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}-162\ 2^{2/3}}+54}{2^{2/3} \sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27} \left (2 (a+3)+\frac {\sqrt [3]{2} \left (-2 a^2-\sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}+18\right )}{\sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27}}\right )},\arcsin \left (\frac {\sqrt [3]{2} \sqrt [6]{-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27} \sqrt {2 a+\frac {\sqrt [3]{2} \left (-2 a^2-\sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}+18\right )}{\sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27}}+6 x}}{\sqrt {-6 a^2-3 \sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+27\right )^{4/3}-36 \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+27\right )^{4/3}-36 \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+27\right )^{4/3}-36 \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}-162\ 2^{2/3}}+54}\right )}{\sqrt [6]{-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27} \left (2 (a+3)+\frac {\sqrt [3]{2} \left (-2 a^2-\sqrt [3]{2} \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}+18\right )}{\sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27}}\right ) \sqrt {x^3+a x^2+3 x-1} \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+3 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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+27\right )^{4/3}-36 \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}-162\ 2^{2/3}}-18 \sqrt [3]{2}}{\sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {-(a+3)^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 {-(a+3)^2 (4 a-15)}+27} (a+3 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 {(15-4 a) (a+3)^2}+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 {(15-4 a) (a+3)^2}+27\right )^{4/3}-36 \left (-2 a^3+27 a+3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27\right )^{2/3}-162\ 2^{2/3}}-18 \sqrt [3]{2}}{\sqrt [3]{-2 a^3+27 a+3 \sqrt {3} \sqrt {-(a+3)^2 (4 a-15)}+27}}}} \]
[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 {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 ) \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 {(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 ) \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 {(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}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-3-a} x}{\sqrt {-1+3 x+a x^2+x^3}}\right )}{\sqrt {-3-a}} \]
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Time = 5.43 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{2}+x^{3}+3 x -1}}{x \sqrt {3+a}}\right )}{\sqrt {3+a}}\) | \(33\) |
pseudoelliptic | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{2}+x^{3}+3 x -1}}{x \sqrt {3+a}}\right )}{\sqrt {3+a}}\) | \(33\) |
elliptic | \(\text {Expression too large to display}\) | \(3006\) |
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (34) = 68\).
Time = 0.28 (sec) , antiderivative size = 225, normalized size of antiderivative = 5.62 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=\left [\frac {\log \left (\frac {2 \, {\left (4 \, a + 9\right )} x^{5} + x^{6} + {\left (8 \, a^{2} + 24 \, a + 15\right )} x^{4} + 4 \, {\left (6 \, a + 13\right )} x^{3} - {\left (8 \, a + 9\right )} x^{2} - 4 \, {\left ({\left (2 \, a + 3\right )} x^{3} + x^{4} + 3 \, x^{2} - x\right )} \sqrt {a x^{2} + x^{3} + 3 \, x - 1} \sqrt {a + 3} - 6 \, x + 1}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1}\right )}{2 \, \sqrt {a + 3}}, \frac {\sqrt {-a - 3} \arctan \left (\frac {{\left ({\left (2 \, a + 3\right )} x^{2} + x^{3} + 3 \, x - 1\right )} \sqrt {a x^{2} + x^{3} + 3 \, x - 1} \sqrt {-a - 3}}{2 \, {\left ({\left (a + 3\right )} x^{4} + {\left (a^{2} + 3 \, a\right )} x^{3} + 3 \, {\left (a + 3\right )} x^{2} - {\left (a + 3\right )} x\right )}}\right )}{a + 3}\right ] \]
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\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=\int \frac {x + 2}{\left (x - 1\right ) \sqrt {a x^{2} + x^{3} + 3 x - 1}}\, dx \]
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\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {a x^{2} + x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \]
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\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {a x^{2} + x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \]
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Time = 5.49 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.55 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=\frac {\ln \left (\frac {\left (\sqrt {x^3+a\,x^2+3\,x-1}+x\,\sqrt {a+3}\right )\,{\left (\sqrt {x^3+a\,x^2+3\,x-1}-x\,\sqrt {a+3}\right )}^3}{{\left (x-1\right )}^6}\right )}{\sqrt {a+3}} \]
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