\(\int \frac {-2+x}{(1+x) \sqrt {1+3 x-a x^2+x^3}} \, dx\) [690]

Optimal result

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]

2*arctan((3+a)^(1/2)*x*(-a*x^2+x^3+3*x+1)^(1/2)/(a*x^2-x^3-3*x-1))/(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 = 122.54 (sec) , antiderivative size = 5395, normalized size of antiderivative = 99.91, 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\ 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 (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 {(15-4 a) (a+3)^2}+27\right )^{2/3}+18\right ) (3 x-a)}{\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 {6 \sqrt [6]{2} \sqrt {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 {1-\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}} \sqrt {1-\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}} \sqrt {\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}}-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 {(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}}-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 {a}{3}+\frac {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}{3\ 2^{2/3} \sqrt [3]{-2 a^3+27 a-3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27}}+x} \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+\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\right )},\arcsin \left (\frac {2^{5/6} \sqrt {3} \sqrt [6]{-2 a^3+27 a-3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27} \sqrt {-\frac {a}{3}+\frac {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}{3\ 2^{2/3} \sqrt [3]{-2 a^3+27 a-3 \sqrt {3} \sqrt {(15-4 a) (a+3)^2}+27}}+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+\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\right ) \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}}} \sqrt {x^3-a x^2+3 x+1}} \]

[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 + (54
+ 54*a - 4*a^3 - 6*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3) - 2*a*(27 + 27*a - 2*a^3 - 3*Sqrt[3]*Sqrt[-(
(3 + a)^2*(-15 + 4*a))])^(1/3) + 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*(-1/3*a + 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 - (54 + 54*a - 4*a^3 - 6
*Sqrt[3]*Sqrt[-((3 + a)^2*(-15 + 4*a))])^(2/3) - 4*a*(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]*S
qrt[-((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*Sqrt[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) +
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)])])/(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]) - (6*2^(1/6)*Sqrt[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 - 3
6*(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)]]*Sqr
t[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 + 2
7*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*(-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[(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[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]*Sq
rt[-((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[(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[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[-1/3*a + (-18 + 2*a^2 + 2^(1/3)*(27 + 27*a - 2*a^3 - 3*Sqr
t[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(2/3))/(3*2^(2/3)*(27 + 27*a - 2*a^3 - 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^
(1/3)) + x]*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]*Sqr
t[(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)*(6 + 2*a + (2^(1/3)*(18 - 2*a^2 - 2^(1/3)*(27 + 27*a - 2*a^3 - 3*Sqrt[3]*Sq
rt[(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))), ArcSin[(
2^(5/6)*Sqrt[3]*(27 + 27*a - 2*a^3 - 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(1/6)*Sqrt[-1/3*a + (-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))/(3*2^(2/3)*(27 + 27*a - 2*a^3 - 3*Sq
rt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(1/3)) + x])/Sqrt[-54 + 6*a^2 + 3*2^(1/3)*(27 + 27*a - 2*a^3 - 3*Sqrt[3]*Sqr
t[(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*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)])])/((27 + 27*a - 2*a^3
- 3*Sqrt[3]*Sqrt[(15 - 4*a)*(3 + a)^2])^(1/6)*(6 + 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))
*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) - Sqr
t[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*(-1
5 + 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[-1
62*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)]*Sqrt[1 + 3*x - a*x^2 +
x^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 \\ & = -\left (3 \int \frac {1}{(1+x) \sqrt {1+3 x-a x^2+x^3}} \, dx\right )+\int \frac {1}{\sqrt {1+3 x-a x^2+x^3}} \, dx \\ & = -\left (3 \text {Subst}\left (\int \frac {1}{\left (\frac {3+a}{3}+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 )\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 {(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 {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 )}{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 {(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 {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 {3+a}{3}+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 )}{\sqrt {2} \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 = 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]

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 = 9.16 (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]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (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]

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

[Out]

[-1/2*sqrt(-a - 3)*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))/(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))/sqrt(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 [F(-1)]

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]

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

[Out]

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