Integrand size = 25, antiderivative size = 17 \[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=2 \text {arctanh}\left (\frac {(-2+x) x}{\sqrt {x+x^4}}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(44\) vs. \(2(17)=34\).
Time = 7.87 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.59 \[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\frac {2 \sqrt {x} \sqrt {1+x^3} \text {arctanh}\left (\frac {(-2+x) \sqrt {x}}{\sqrt {1+x^3}}\right )}{\sqrt {x+x^4}} \]
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.
\(\displaystyle \int \frac {2 x^2+x+2}{(2 x-1) \sqrt {x^4+x}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^3+1} \int -\frac {2 x^2+x+2}{(1-2 x) \sqrt {x} \sqrt {x^3+1}}dx}{\sqrt {x^4+x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^3+1} \int \frac {2 x^2+x+2}{(1-2 x) \sqrt {x} \sqrt {x^3+1}}dx}{\sqrt {x^4+x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^3+1} \int \frac {2 x^2+x+2}{(1-2 x) \sqrt {x^3+1}}d\sqrt {x}}{\sqrt {x^4+x}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^3+1} \int \left (-\frac {x}{\sqrt {x^3+1}}+\frac {3}{(1-2 x) \sqrt {x^3+1}}-\frac {1}{\sqrt {x^3+1}}\right )d\sqrt {x}}{\sqrt {x^4+x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^3+1} \left (\frac {3}{2} \int \frac {1}{\left (1-\sqrt {2} \sqrt {x}\right ) \sqrt {x^3+1}}d\sqrt {x}+\frac {3}{2} \int \frac {1}{\left (\sqrt {2} \sqrt {x}+1\right ) \sqrt {x^3+1}}d\sqrt {x}-\frac {\sqrt {x} (x+1) \sqrt {\frac {x^2-x+1}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x+1}{\left (1+\sqrt {3}\right ) x+1}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \sqrt {\frac {x (x+1)}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {x^3+1}}-\frac {1}{3} \text {arcsinh}\left (x^{3/2}\right )\right )}{\sqrt {x^4+x}}\) |
3.2.16.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(44\) vs. \(2(15)=30\).
Time = 3.66 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.65
method | result | size |
trager | \(\ln \left (-\frac {2 x^{3}+2 x \sqrt {x^{4}+x}-4 x^{2}-4 \sqrt {x^{4}+x}+4 x +1}{\left (-1+2 x \right )^{2}}\right )\) | \(45\) |
default | \(-\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}-\frac {\ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{3}+\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+2 \operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {3}{2}+\frac {3 i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(512\) |
elliptic | \(\text {Expression too large to display}\) | \(781\) |
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (15) = 30\).
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 4.12 \[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\frac {1}{3} \, \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) + \frac {2}{3} \, \log \left (-\frac {10 \, x^{3} - 6 \, x^{2} - 6 \, \sqrt {x^{4} + x} {\left (x + 1\right )} + 12 \, x + 1}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}\right ) \]
1/3*log(-2*x^3 - 2*sqrt(x^4 + x)*x - 1) + 2/3*log(-(10*x^3 - 6*x^2 - 6*sqr t(x^4 + x)*(x + 1) + 12*x + 1)/(8*x^3 - 12*x^2 + 6*x - 1))
\[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int \frac {2 x^{2} + x + 2}{\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (2 x - 1\right )}\, dx \]
\[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int { \frac {2 \, x^{2} + x + 2}{\sqrt {x^{4} + x} {\left (2 \, x - 1\right )}} \,d x } \]
\[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int { \frac {2 \, x^{2} + x + 2}{\sqrt {x^{4} + x} {\left (2 \, x - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int \frac {2\,x^2+x+2}{\left (2\,x-1\right )\,\sqrt {x^4+x}} \,d x \]