Integrand size = 20, antiderivative size = 31 \[ \int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx=-\frac {2}{3} \text {arctanh}\left (\frac {\frac {1}{3}+\frac {2 x}{3}+\frac {x^2}{3}}{\sqrt {x+x^4}}\right ) \]
Time = 10.93 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx=-\frac {2}{3} \text {arctanh}\left (\frac {\frac {1}{3}+\frac {2 x}{3}+\frac {x^2}{3}}{\sqrt {x+x^4}}\right ) \]
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.
| \(\displaystyle \int \frac {x+1}{(2 x-1) \sqrt {x^4+x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^3+1} \int -\frac {x+1}{(1-2 x) \sqrt {x} \sqrt {x^3+1}}dx}{\sqrt {x^4+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^3+1} \int \frac {x+1}{(1-2 x) \sqrt {x} \sqrt {x^3+1}}dx}{\sqrt {x^4+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^3+1} \int \frac {x+1}{(1-2 x) \sqrt {x^3+1}}d\sqrt {x}}{\sqrt {x^4+x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^3+1} \int \left (\frac {3}{2 (1-2 x) \sqrt {x^3+1}}-\frac {1}{2 \sqrt {x^3+1}}\right )d\sqrt {x}}{\sqrt {x^4+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^3+1} \left (\frac {3}{4} \int \frac {1}{\left (1-\sqrt {2} \sqrt {x}\right ) \sqrt {x^3+1}}d\sqrt {x}+\frac {3}{4} \int \frac {1}{\left (\sqrt {2} \sqrt {x}+1\right ) \sqrt {x^3+1}}d\sqrt {x}-\frac {\sqrt {x} (x+1) \sqrt {\frac {x^2-x+1}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x+1}{\left (1+\sqrt {3}\right ) x+1}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} \sqrt {\frac {x (x+1)}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {x^3+1}}\right )}{\sqrt {x^4+x}}\) |
3.4.79.3.1 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. \(45\) vs. \(2(21)=42\).
Time = 2.76 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48
| method | result | size |
| trager | \(-\frac {\ln \left (\frac {10 x^{3}+6 x \sqrt {x^{4}+x}-6 x^{2}+6 \sqrt {x^{4}+x}+12 x +1}{\left (-1+2 x \right )^{3}}\right )}{3}\) | \(46\) |
| default | \(-\frac {\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 {\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 )}}\) | \(491\) |
| elliptic | \(-\frac {\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 {\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 )}}\) | \(491\) |
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx=\frac {1}{3} \, \log \left (\frac {10 \, x^{3} - 6 \, x^{2} - 6 \, \sqrt {x^{4} + x} {\left (x + 1\right )} + 12 \, x + 1}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}\right ) \]
\[ \int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int \frac {x + 1}{\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (2 x - 1\right )}\, dx \]
\[ \int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int { \frac {x + 1}{\sqrt {x^{4} + x} {\left (2 \, x - 1\right )}} \,d x } \]
\[ \int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int { \frac {x + 1}{\sqrt {x^{4} + x} {\left (2 \, x - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int \frac {x+1}{\left (2\,x-1\right )\,\sqrt {x^4+x}} \,d x \]