\(\int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx\) [516]

   Optimal result
   Rubi [C] (warning: unable to verify)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [B] (verification not implemented)
   Sympy [F]
   Maxima [F]
   Giac [F]
   Mupad [B] (verification not implemented)

Optimal result

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]

-2*arctan((-3-a)^(1/2)*x/(a*x^2+x^3+3*x-1)^(1/2))/(-3-a)^(1/2)

Rubi [C] (warning: unable to verify)

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]

Int[(2 + x)/((-1 + x)*Sqrt[-1 + 3*x + a*x^2 + x^3]),x]

[Out]

(2^(1/3)*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)
^2*(-15 + 4*a))])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3) - 2^(1/3)
*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(4/3)]*Sqrt[(-18*2^(1/3) + 2*2^(1/3)*a^2 - 2*a
*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(1/3) + (54 + 54*a - 4*a^3 + 6*Sqrt[3]*Sqrt[-(
(3 + a)^2*(-15 + 4*a))])^(2/3) - 6*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(1/3)*x)/(-5
4 + 6*a^2 + 3*2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3) + 2^(1/6)*Sqrt[3]*S
qrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 +
4*a))])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3) - 2^(1/3)*(27 + 27*
a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(4/3)])]*Sqrt[-((2*(9 - a^2) + (2*2^(2/3)*(9 - a^2)^2)/(
27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[
(15 - 4*a)*(3 + a)^2])^(2/3) + 18*(a/3 + x)^2 - (2^(1/3)*(18 - 2*a^2 - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*
Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3))*(a + 3*x))/(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(1/3)
)/((18 - 2*a^2 - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3))^2/(18*2^(1/3)*(27 +
 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3)) - (2*(2*(9 - a^2) + (2*2^(2/3)*(9 - a^2)^2)/(27 +
 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15
- 4*a)*(3 + a)^2])^(2/3)))/9))]*EllipticF[ArcSin[Sqrt[(-18*2^(1/3) + 2*2^(1/3)*a^2 + 4*a*(27 + 27*a - 2*a^3 +
3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(1/3) + (54 + 54*a - 4*a^3 + 6*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))
])^(2/3) + Sqrt[6]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt
[-((3 + a)^2*(-15 + 4*a))])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3)
 - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(4/3)] + 12*(27 + 27*a - 2*a^3 + 3*S
qrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(1/3)*x)/Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 +
27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((
3 + a)^2*(-15 + 4*a))])^(2/3) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(4/3)]]
/(2^(3/4)*3^(1/4))], (-2*2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a -
2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3
+ a)^2])^(2/3) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(4/3)])/(54 - 6*a^2 - 3*2^
(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) - 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 3
6*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 4*a^2*(2
7 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(
15 - 4*a)*(3 + a)^2])^(4/3)])])/(3*Sqrt[3]*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(1/3
)*Sqrt[-1 + 3*x + a*x^2 + x^3]) - (3*2^(2/3)*Sqrt[54 - 6*a^2 - 3*2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(
15 - 4*a)*(3 + a)^2])^(2/3) + 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27
*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a
)*(3 + a)^2])^(2/3) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(4/3)]]*Sqrt[2*a + (2
^(1/3)*(18 - 2*a^2 - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3)))/(27 + 27*a - 2
*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(1/3) + 6*x]*Sqrt[(2^(1/3)*(-18 + 2*a^2 + 2^(1/3)*(27 + 27*a - 2*
a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3)))/(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 +
 4*a))])^(1/3) - (Sqrt[6]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*Sqrt[
3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))]
)^(2/3) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(4/3)])/(27 + 27*a - 2*a^3 +
3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(1/3) + 4*(a + 3*x)]*Sqrt[(2^(1/3)*(-18 + 2*a^2 + 2^(1/3)*(27 + 27*a
 - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3)))/(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(
-15 + 4*a))])^(1/3) + (Sqrt[6]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*
Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4
*a))])^(2/3) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(4/3)])/(27 + 27*a - 2*a
^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(1/3) + 4*(a + 3*x)]*Sqrt[1 + (2*(18 - 2*a^2 - 2^(1/3)*(27 + 27
*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3) + 2^(2/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((
3 + a)^2*(-15 + 4*a))])^(1/3)*(a + 3*x)))/(-54 + 6*a^2 + 3*2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 +
a)^2*(-15 + 4*a))])^(2/3) - 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a
 - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*
(3 + a)^2])^(2/3) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(4/3)])]*Sqrt[1 + (2*(1
8 - 2*a^2 - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3) + 2^(2/3)*(27 + 27*a
- 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(1/3)*(a + 3*x)))/(-54 + 6*a^2 + 3*2^(1/3)*(27 + 27*a - 2*
a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3) + 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2
*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3
+ 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)
^2])^(4/3)])]*EllipticPi[(54 - 6*a^2 - 3*2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2
/3) + 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*S
qrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) - 2^
(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(4/3)])/(2^(2/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[
3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(1/3)*(2*(3 + a) + (2^(1/3)*(18 - 2*a^2 - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3
]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3)))/(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(1/3))), ArcS
in[(2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(1/6)*Sqrt[2*a + (2^(1/3)*(18 - 2*a^2 -
 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3)))/(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqr
t[(15 - 4*a)*(3 + a)^2])^(1/3) + 6*x])/Sqrt[54 - 6*a^2 - 3*2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4
*a)*(3 + a)^2])^(2/3) + 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2
*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 +
 a)^2])^(2/3) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(4/3)]]], (54 - 6*a^2 - 3*2
^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) +
36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 4*a^2*(
27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[
(15 - 4*a)*(3 + a)^2])^(4/3)])/(54 - 6*a^2 - 3*2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^
2])^(2/3) - 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*Sqr
t[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3
) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(4/3)])])/((27 + 27*a - 2*a^3 + 3*Sqrt[
3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(1/6)*(2*(3 + a) + (2^(1/3)*(18 - 2*a^2 - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3
]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3)))/(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(1/3))*Sqrt[-
1 + 3*x + a*x^2 + x^3]*Sqrt[(-18*2^(1/3) + 2*2^(1/3)*a^2 + (54 + 54*a - 4*a^3 + 6*Sqrt[3]*Sqrt[-((3 + a)^2*(-1
5 + 4*a))])^(2/3) - Sqrt[6]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*Sqr
t[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3
) - 2^(1/3)*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(4/3)] + 4*(27 + 27*a - 2*a^3 + 3*Sqrt[
3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(1/3)*(a + 3*x))/(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a
))])^(1/3)]*Sqrt[(-18*2^(1/3) + 2*2^(1/3)*a^2 + (54 + 54*a - 4*a^3 + 6*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])
^(2/3) + Sqrt[6]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(
15 - 4*a)*(3 + a)^2])^(2/3) + 4*a^2*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3) - 2^(1/3)
*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(4/3)] + 4*(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((
3 + a)^2*(-15 + 4*a))])^(1/3)*(a + 3*x))/(27 + 27*a - 2*a^3 + 3*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(1/3)]
)

Rule 175

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d
*g - c*h)/d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] &&  !SimplerQ[e
 + f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d/c)*x^2]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
 d, e, f}, x] &&  !GtQ[c, 0]

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*Rt[b^2 - 4*a*c, 2]*
(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*
e - e*Rt[b^2 - 4*a*c, 2])))^m)), Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2
])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 948

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[Sqrt[b - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2]), Int[1/((d +
 e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 2091

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27
*a^2*d^2], 3]}, Dist[(a + b*x + d*x^3)^p/(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12
^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p), I
nt[Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1
/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, p}, x] && NeQ[
4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]

Rule 2092

Int[(P3_)^(p_), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3
, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x,
 x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[p, x] && PolyQ[P3, x, 3]

Rule 2105

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[(a + b*x + d*x^3)^p/(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d
*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1
/3))*x + d^2*x^2, x]^p), Int[(e + f*x)^m*Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^
(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p, x],
 x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]

Rule 2106

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-3-a} x}{\sqrt {-1+3 x+a x^2+x^3}}\right )}{\sqrt {-3-a}} \]

[In]

Integrate[(2 + x)/((-1 + x)*Sqrt[-1 + 3*x + a*x^2 + x^3]),x]

[Out]

(-2*ArcTan[(Sqrt[-3 - a]*x)/Sqrt[-1 + 3*x + a*x^2 + x^3]])/Sqrt[-3 - a]

Maple [A] (verified)

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 \(\text {Expression too large to display}\) \(3006\)

[In]

int((x+2)/(x-1)/(a*x^2+x^3+3*x-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/(3+a)^(1/2)*arctanh((a*x^2+x^3+3*x-1)^(1/2)/x/(3+a)^(1/2))

Fricas [B] (verification not implemented)

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 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=\left [\frac {\log \left (\frac {2 \, {\left (4 \, a + 9\right )} x^{5} + x^{6} + {\left (8 \, a^{2} + 24 \, a + 15\right )} x^{4} + 4 \, {\left (6 \, a + 13\right )} x^{3} - {\left (8 \, a + 9\right )} x^{2} - 4 \, {\left ({\left (2 \, a + 3\right )} x^{3} + x^{4} + 3 \, x^{2} - x\right )} \sqrt {a x^{2} + x^{3} + 3 \, x - 1} \sqrt {a + 3} - 6 \, x + 1}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1}\right )}{2 \, \sqrt {a + 3}}, \frac {\sqrt {-a - 3} \arctan \left (\frac {{\left ({\left (2 \, a + 3\right )} x^{2} + x^{3} + 3 \, x - 1\right )} \sqrt {a x^{2} + x^{3} + 3 \, x - 1} \sqrt {-a - 3}}{2 \, {\left ({\left (a + 3\right )} x^{4} + {\left (a^{2} + 3 \, a\right )} x^{3} + 3 \, {\left (a + 3\right )} x^{2} - {\left (a + 3\right )} x\right )}}\right )}{a + 3}\right ] \]

[In]

integrate((2+x)/(-1+x)/(a*x^2+x^3+3*x-1)^(1/2),x, algorithm="fricas")

[Out]

[1/2*log((2*(4*a + 9)*x^5 + x^6 + (8*a^2 + 24*a + 15)*x^4 + 4*(6*a + 13)*x^3 - (8*a + 9)*x^2 - 4*((2*a + 3)*x^
3 + x^4 + 3*x^2 - x)*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))/sqrt(a + 3), sqrt(-a - 3)*arctan(1/2*((2*a + 3)*x^2 + x^3 + 3*x - 1)*sqrt(a*x^2 + x^3 + 3*x - 1)
*sqrt(-a - 3)/((a + 3)*x^4 + (a^2 + 3*a)*x^3 + 3*(a + 3)*x^2 - (a + 3)*x))/(a + 3)]

Sympy [F]

\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=\int \frac {x + 2}{\left (x - 1\right ) \sqrt {a x^{2} + x^{3} + 3 x - 1}}\, dx \]

[In]

integrate((2+x)/(-1+x)/(a*x**2+x**3+3*x-1)**(1/2),x)

[Out]

Integral((x + 2)/((x - 1)*sqrt(a*x**2 + x**3 + 3*x - 1)), x)

Maxima [F]

\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {a x^{2} + x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \]

[In]

integrate((2+x)/(-1+x)/(a*x^2+x^3+3*x-1)^(1/2),x, algorithm="maxima")

[Out]

integrate((x + 2)/(sqrt(a*x^2 + x^3 + 3*x - 1)*(x - 1)), x)

Giac [F]

\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {a x^{2} + x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \]

[In]

integrate((2+x)/(-1+x)/(a*x^2+x^3+3*x-1)^(1/2),x, algorithm="giac")

[Out]

integrate((x + 2)/(sqrt(a*x^2 + x^3 + 3*x - 1)*(x - 1)), x)

Mupad [B] (verification not implemented)

Time = 5.49 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.55 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+a x^2+x^3}} \, dx=\frac {\ln \left (\frac {\left (\sqrt {x^3+a\,x^2+3\,x-1}+x\,\sqrt {a+3}\right )\,{\left (\sqrt {x^3+a\,x^2+3\,x-1}-x\,\sqrt {a+3}\right )}^3}{{\left (x-1\right )}^6}\right )}{\sqrt {a+3}} \]

[In]

int((x + 2)/((x - 1)*(3*x + a*x^2 + x^3 - 1)^(1/2)),x)

[Out]

log((((3*x + a*x^2 + x^3 - 1)^(1/2) + x*(a + 3)^(1/2))*((3*x + a*x^2 + x^3 - 1)^(1/2) - x*(a + 3)^(1/2))^3)/(x
 - 1)^6)/(a + 3)^(1/2)