Integrand size = 30, antiderivative size = 33 \[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=-\sqrt {2} \arctan \left (\frac {\sqrt {1+x^3}}{\sqrt {2} \left (1-x+x^2\right )}\right ) \]
Time = 1.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=-\sqrt {2} \arctan \left (\frac {\sqrt {1+x^3}}{\sqrt {2} \left (1-x+x^2\right )}\right ) \]
Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2570, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2+2 x-2}{\left (2 x^2-x+3\right ) \sqrt {x^3+1}} \, dx\) |
| \(\Big \downarrow \) 2570 |
| \(\displaystyle -4 \int \frac {1}{\frac {2 (x+1)^2}{x^3+1}+4}d\frac {x+1}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\sqrt {2} \arctan \left (\frac {x+1}{\sqrt {2} \sqrt {x^3+1}}\right )\) |
3.4.96.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)* Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Simp[-2*g*h Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + 2*h*(x/g))/Sqrt[a + b*x^3]], x] /; Fre eQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8 *a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.50 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82
| method | result | size |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {x^{3}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{2 x^{2}-x +3}\right )}{2}\) | \(60\) |
| default | \(\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {\left (\frac {5}{4}+\frac {i \sqrt {23}}{4}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{12}+\frac {i \sqrt {23}}{12}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}-\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \sqrt {3}\, \left (\frac {1}{4}+\frac {i \sqrt {23}}{4}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {\left (\frac {5}{4}-\frac {i \sqrt {23}}{4}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{12}-\frac {i \sqrt {23}}{12}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}+\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \sqrt {3}\, \left (\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(432\) |
| elliptic | \(\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (\frac {5}{8}+\frac {i \sqrt {23}}{8}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{12}+\frac {i \sqrt {23}}{12}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}-\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \sqrt {3}\, \left (\frac {1}{4}+\frac {i \sqrt {23}}{4}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (\frac {5}{8}-\frac {i \sqrt {23}}{8}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{12}-\frac {i \sqrt {23}}{12}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}+\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \sqrt {3}\, \left (\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(434\) |
-1/2*RootOf(_Z^2+2)*ln(-(2*RootOf(_Z^2+2)*x^2-3*RootOf(_Z^2+2)*x+4*(x^3+1) ^(1/2)+RootOf(_Z^2+2))/(2*x^2-x+3))
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{2} - 3 \, x + 1\right )}}{4 \, \sqrt {x^{3} + 1}}\right ) \]
\[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=\int \frac {x^{2} + 2 x - 2}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (2 x^{2} - x + 3\right )}\, dx \]
\[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x^{2} + 2 \, x - 2}{\sqrt {x^{3} + 1} {\left (2 \, x^{2} - x + 3\right )}} \,d x } \]
\[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x^{2} + 2 \, x - 2}{\sqrt {x^{3} + 1} {\left (2 \, x^{2} - x + 3\right )}} \,d x } \]
Time = 5.21 (sec) , antiderivative size = 274, normalized size of antiderivative = 8.30 \[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {5}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {5}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{2\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
-((3^(1/2)*1i + 3)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/ 2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^ (1/2)*1i)/2 + 3/2))^(1/2)*(ellipticPi(((3^(1/2)*1i)/2 + 3/2)/((23^(1/2)*1i )/4 + 5/4), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)) - ellipticF(asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)) + elliptic Pi(-((3^(1/2)*1i)/2 + 3/2)/((23^(1/2)*1i)/4 - 5/4), asin(((x + 1)/((3^(1/2 )*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))))/ (2*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2 )*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))