\(\int \frac {-2+x}{(1+x) \sqrt {1+3 x+a x^2+x^3}} \, dx\) [517]

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 = 124.05 (sec) , antiderivative size = 5437, normalized size of antiderivative = 135.92, 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 \left (9-a^2\right )^2}{\left (a^3-\frac {27 a}{2}+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}\right )\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 {(3-a)^2 (4 a+15)}+27\right )^{2/3}+18\right ) (a+3 x)}{\sqrt [3]{2 a^3-27 a+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}\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 (9-\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}\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 (9-\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}\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 {6 \sqrt [6]{2} \sqrt {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 {\frac {a}{3}+\frac {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}{3\ 2^{2/3} \sqrt [3]{2 a^3-27 a-3 \sqrt {3} \sqrt {(a-3)^2 (4 a+15)}+27}}+x} \sqrt {-\frac {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}{\sqrt [3]{a^3-\frac {27 a}{2}-\frac {3}{2} \left (\sqrt {3} \sqrt {(a-3)^2 (4 a+15)}-9\right )}}+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 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}{\sqrt [3]{a^3-\frac {27 a}{2}-\frac {3}{2} \left (\sqrt {3} \sqrt {(a-3)^2 (4 a+15)}-9\right )}}+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 {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 {(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 {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 {(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}+1} \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 {\frac {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}{\sqrt [3]{a^3-\frac {27 a}{2}-\frac {3}{2} \left (\sqrt {3} \sqrt {(a-3)^2 (4 a+15)}-9\right )}}+2 (a+3 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 (9-\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}\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 (9-\sqrt {3} \sqrt {(3-a)^2 (4 a+15)}\right )}}+6\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 {(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 [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 {(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 [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 + (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)/(-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[(-3 + a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(2
7 - 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^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(
15 + 4*a)])^(2/3) + (2*(9 - a^2)^2)/((-27*a)/2 + a^3 + (3*(9 - Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*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[(3 - a)^2*(15 + 4*a)])^(2
/3))*(a + 3*x))/(-27*a + 2*a^3 + 3*(9 - Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)]))^(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))^2/(18*2^(1/3)*(-27*a + 2*a^3 + 3*(9 - Sqrt[3]*S
qrt[(3 - a)^2*(15 + 4*a)]))^(2/3)) - (2*(2*(9 - a^2) + 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(
15 + 4*a)])^(2/3) + (2*(9 - a^2)^2)/((-27*a)/2 + a^3 + (3*(9 - Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*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]*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*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[(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)])/(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[(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)])])/(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[(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[(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[a/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))/(3*2^(2/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(
1/3)) + x]*Sqrt[-((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)) - (-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*a)/2 + a^3 - (3*(-9 + Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)]))/2)^(1/3) + 4*(a + 3*x)]*
Sqrt[(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) - (-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*a)/2 + a^3 - (3*(-9 + Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)]))/2)^(1/3) + 4*(a + 3*x)]*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[(-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[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[(-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)])]*EllipticPi[(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[(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*(18 - 2*a^2 + 3*2^(
2/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(1/3) - 2^(2/3)*a*(27 - 27*a + 2*a^3 - 3*Sqrt[
3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(1/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3
))), ArcSin[(2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(1/6)*Sqrt[(-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*a)/2 + a^3 - (3*(-9 + Sqrt[3]*Sqr
t[(-3 + a)^2*(15 + 4*a)]))/2)^(1/3) + 2*(a + 3*x)])/Sqrt[-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[(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)]]], (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[(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)])/(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[(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*a + 2*a^3
+ 3*(9 - Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)]))^(1/6)*(6 - 2*a + (18 - 2*a^2 - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*S
qrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3))/((-27*a)/2 + a^3 + (3*(9 - Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*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*(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)] + 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[(
-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)] + 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 \\ & = -\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 {(-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 {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}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+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 {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 {(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*}

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 = 5.45 (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)/(1+x)/(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. 89 vs. \(2 (34) = 68\).

Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 5.52 \[ \int \frac {-2+x}{(1+x) \sqrt {1+3 x+a x^2+x^3}} \, dx=\left [\frac {\log \left (\frac {2 \, {\left (4 \, a - 9\right )} x^{5} + x^{6} + {\left (8 \, a^{2} - 24 \, a + 15\right )} x^{4} + 4 \, {\left (6 \, a - 13\right )} x^{3} + {\left (8 \, a - 9\right )} x^{2} - 4 \, {\left ({\left (2 \, a - 3\right )} x^{3} + x^{4} + 3 \, x^{2} + x\right )} \sqrt {a x^{2} + x^{3} + 3 \, x + 1} \sqrt {a - 3} + 6 \, x + 1}{x^{6} + 6 \, x^{5} + 15 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} + 6 \, x + 1}\right )}{2 \, \sqrt {a - 3}}, \frac {\sqrt {-a + 3} \arctan \left (\frac {{\left ({\left (2 \, a - 3\right )} x^{2} + x^{3} + 3 \, x + 1\right )} \sqrt {a x^{2} + x^{3} + 3 \, x + 1} \sqrt {-a + 3}}{2 \, {\left ({\left (a - 3\right )} x^{4} + {\left (a^{2} - 3 \, a\right )} x^{3} + 3 \, {\left (a - 3\right )} x^{2} + {\left (a - 3\right )} x\right )}}\right )}{a - 3}\right ] \]

[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 = 5.63 (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)