Integrand size = 23, antiderivative size = 23 \[ \int \frac {1+2 x}{\sqrt {3+x^2+2 x^3+x^4}} \, dx=\log \left (x+x^2+\sqrt {3+x^2+2 x^3+x^4}\right ) \]
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x}{\sqrt {3+x^2+2 x^3+x^4}} \, dx=\log \left (x+x^2+\sqrt {3+x^2+2 x^3+x^4}\right ) \]
Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2459, 27, 27, 1432, 1090, 222}
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.
\(\displaystyle \int \frac {2 x+1}{\sqrt {x^4+2 x^3+x^2+3}} \, dx\) |
\(\Big \downarrow \) 2459 |
\(\displaystyle \int \frac {2 \left (x+\frac {1}{2}\right )}{\sqrt {\left (x+\frac {1}{2}\right )^4-\frac {1}{2} \left (x+\frac {1}{2}\right )^2+\frac {49}{16}}}d\left (x+\frac {1}{2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {4 \left (x+\frac {1}{2}\right )}{\sqrt {16 \left (x+\frac {1}{2}\right )^4-8 \left (x+\frac {1}{2}\right )^2+49}}d\left (x+\frac {1}{2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 8 \int \frac {x+\frac {1}{2}}{\sqrt {16 \left (x+\frac {1}{2}\right )^4-8 \left (x+\frac {1}{2}\right )^2+49}}d\left (x+\frac {1}{2}\right )\) |
\(\Big \downarrow \) 1432 |
\(\displaystyle 4 \int \frac {1}{\sqrt {16 \left (x+\frac {1}{2}\right )^4-8 \left (x+\frac {1}{2}\right )^2+49}}d\left (x+\frac {1}{2}\right )^2\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {\frac {\left (x+\frac {1}{2}\right )^4}{3072}+1}}d\left (32 \left (x+\frac {1}{2}\right )^2-8\right )}{32 \sqrt {3}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \text {arcsinh}\left (\frac {32 \left (x+\frac {1}{2}\right )^2-8}{32 \sqrt {3}}\right )\) |
3.3.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Time = 2.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48
method | result | size |
default | \(\operatorname {arcsinh}\left (\frac {\sqrt {3}\, \left (1+x \right ) x}{3}\right )\) | \(11\) |
pseudoelliptic | \(\operatorname {arcsinh}\left (\frac {\sqrt {3}\, \left (1+x \right ) x}{3}\right )\) | \(11\) |
trager | \(-\ln \left (-x^{2}+\sqrt {x^{4}+2 x^{3}+x^{2}+3}-x \right )\) | \(28\) |
elliptic | \(\frac {i \left (2-i \sqrt {3}\right ) \sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {2}\, \sqrt {-\frac {i \sqrt {3}\, \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}{x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )}{6 \sqrt {\left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {i \left (2-i \sqrt {3}\right ) \sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {2}\, \sqrt {-\frac {i \sqrt {3}\, \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}{x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )-i \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, 1-\frac {i \sqrt {3}}{2}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )\right )}{3 \sqrt {\left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(528\) |
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1+2 x}{\sqrt {3+x^2+2 x^3+x^4}} \, dx=\log \left (x^{2} + x + \sqrt {x^{4} + 2 \, x^{3} + x^{2} + 3}\right ) \]
\[ \int \frac {1+2 x}{\sqrt {3+x^2+2 x^3+x^4}} \, dx=\int \frac {2 x + 1}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + 3 x + 3\right )}}\, dx \]
\[ \int \frac {1+2 x}{\sqrt {3+x^2+2 x^3+x^4}} \, dx=\int { \frac {2 \, x + 1}{\sqrt {x^{4} + 2 \, x^{3} + x^{2} + 3}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {1+2 x}{\sqrt {3+x^2+2 x^3+x^4}} \, dx=\frac {1}{2} \, \sqrt {{\left (x^{2} + x\right )}^{2} + 3} {\left (x^{2} + x\right )} - \frac {3}{2} \, \log \left (-x^{2} - x + \sqrt {{\left (x^{2} + x\right )}^{2} + 3}\right ) \]
Timed out. \[ \int \frac {1+2 x}{\sqrt {3+x^2+2 x^3+x^4}} \, dx=\int \frac {2\,x+1}{\sqrt {x^4+2\,x^3+x^2+3}} \,d x \]