Integrand size = 83, antiderivative size = 51 \[ \int \frac {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} \left (-2+x^2\right ) \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx=-2 \arctan \left (\frac {\sqrt [4]{\frac {1+x}{-2+x^2}}}{-1+x^2}\right )+2 \text {arctanh}\left (\frac {-1+x^2}{\sqrt [4]{\frac {1+x}{-2+x^2}}}\right ) \]
\[ \int \frac {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} \left (-2+x^2\right ) \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx=\int \frac {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} \left (-2+x^2\right ) \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx \]
Integrate[(2 + 16*x - x^2 - 9*x^3)/(((1 + x)/(-2 + x^2))^(1/4)*(-2 + x^2)* (-3 + 2*x + 7*x^2 - 7*x^3 - 9*x^4 + 9*x^5 + 5*x^6 - 5*x^7 - x^8 + x^9)),x]
Integrate[(2 + 16*x - x^2 - 9*x^3)/(((1 + x)/(-2 + x^2))^(1/4)*(-2 + x^2)* (-3 + 2*x + 7*x^2 - 7*x^3 - 9*x^4 + 9*x^5 + 5*x^6 - 5*x^7 - x^8 + x^9)), x ]
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.
| \(\displaystyle \int \frac {-9 x^3-x^2+16 x+2}{\sqrt [4]{\frac {x+1}{x^2-2}} \left (x^2-2\right ) \left (x^9-x^8-5 x^7+5 x^6+9 x^5-9 x^4-7 x^3+7 x^2+2 x-3\right )} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt [4]{x+1} \int -\frac {-9 x^3-x^2+16 x+2}{\sqrt [4]{x+1} \left (x^2-2\right )^{3/4} \left (-x^9+x^8+5 x^7-5 x^6-9 x^5+9 x^4+7 x^3-7 x^2-2 x+3\right )}dx}{\sqrt [4]{-\frac {x+1}{2-x^2}} \sqrt [4]{x^2-2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x+1} \int \frac {-9 x^3-x^2+16 x+2}{\sqrt [4]{x+1} \left (x^2-2\right )^{3/4} \left (-x^9+x^8+5 x^7-5 x^6-9 x^5+9 x^4+7 x^3-7 x^2-2 x+3\right )}dx}{\sqrt [4]{-\frac {x+1}{2-x^2}} \sqrt [4]{x^2-2}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x+1} \int \left (\frac {9 x^3}{\sqrt [4]{x+1} \left (x^2-2\right )^{3/4} \left (x^9-x^8-5 x^7+5 x^6+9 x^5-9 x^4-7 x^3+7 x^2+2 x-3\right )}+\frac {x^2}{\sqrt [4]{x+1} \left (x^2-2\right )^{3/4} \left (x^9-x^8-5 x^7+5 x^6+9 x^5-9 x^4-7 x^3+7 x^2+2 x-3\right )}-\frac {16 x}{\sqrt [4]{x+1} \left (x^2-2\right )^{3/4} \left (x^9-x^8-5 x^7+5 x^6+9 x^5-9 x^4-7 x^3+7 x^2+2 x-3\right )}-\frac {2}{\sqrt [4]{x+1} \left (x^2-2\right )^{3/4} \left (x^9-x^8-5 x^7+5 x^6+9 x^5-9 x^4-7 x^3+7 x^2+2 x-3\right )}\right )dx}{\sqrt [4]{-\frac {x+1}{2-x^2}} \sqrt [4]{x^2-2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [4]{x+1} \left (36 \text {Subst}\left (\int \frac {x^{14}}{\left (x^8-2 x^4-1\right )^{3/4} \left (x^{36}-10 x^{32}+39 x^{28}-72 x^{24}+56 x^{20}-16 x^{12}-1\right )}dx,x,\sqrt [4]{x+1}\right )-104 \text {Subst}\left (\int \frac {x^{10}}{\left (x^8-2 x^4-1\right )^{3/4} \left (x^{36}-10 x^{32}+39 x^{28}-72 x^{24}+56 x^{20}-16 x^{12}-1\right )}dx,x,\sqrt [4]{x+1}\right )+36 \text {Subst}\left (\int \frac {x^6}{\left (x^8-2 x^4-1\right )^{3/4} \left (x^{36}-10 x^{32}+39 x^{28}-72 x^{24}+56 x^{20}-16 x^{12}-1\right )}dx,x,\sqrt [4]{x+1}\right )+24 \text {Subst}\left (\int \frac {x^2}{\left (x^8-2 x^4-1\right )^{3/4} \left (x^{36}-10 x^{32}+39 x^{28}-72 x^{24}+56 x^{20}-16 x^{12}-1\right )}dx,x,\sqrt [4]{x+1}\right )\right )}{\sqrt [4]{-\frac {x+1}{2-x^2}} \sqrt [4]{x^2-2}}\) |
Int[(2 + 16*x - x^2 - 9*x^3)/(((1 + x)/(-2 + x^2))^(1/4)*(-2 + x^2)*(-3 + 2*x + 7*x^2 - 7*x^3 - 9*x^4 + 9*x^5 + 5*x^6 - 5*x^7 - x^8 + x^9)),x]
3.7.49.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 11.06 (sec) , antiderivative size = 1014, normalized size of antiderivative = 19.88
int((-9*x^3-x^2+16*x+2)/((1+x)/(x^2-2))^(1/4)/(x^2-2)/(x^9-x^8-5*x^7+5*x^6 +9*x^5-9*x^4-7*x^3+7*x^2+2*x-3),x,method=_RETURNVERBOSE)
-ln(-(1+2*(-(-1-x)/(x^2-2))^(3/4)*x^3+2*(-(-1-x)/(x^2-2))^(1/2)*x^4-8*(-(- 1-x)/(x^2-2))^(1/4)*x^5-2*(-(-1-x)/(x^2-2))^(3/4)*x^2+6*(-(-1-x)/(x^2-2))^ (1/2)*x^3+8*(-(-1-x)/(x^2-2))^(1/4)*x^4-4*(-(-1-x)/(x^2-2))^(3/4)*x-6*(-(- 1-x)/(x^2-2))^(1/2)*x^2+10*(-(-1-x)/(x^2-2))^(1/4)*x^3-4*(-(-1-x)/(x^2-2)) ^(1/2)*x-10*(-(-1-x)/(x^2-2))^(1/4)*x^2-4*(-(-1-x)/(x^2-2))^(1/4)*x-2*x+9* x^4+5*x^7-9*x^5-7*x^2+7*x^3+x^8-5*x^6-x^9+4*(-(-1-x)/(x^2-2))^(1/4)+4*(-(- 1-x)/(x^2-2))^(3/4)+4*(-(-1-x)/(x^2-2))^(1/2)+2*(-(-1-x)/(x^2-2))^(1/4)*x^ 7-2*(-(-1-x)/(x^2-2))^(1/2)*x^5-2*(-(-1-x)/(x^2-2))^(1/4)*x^6)/(x^9-x^8-5* x^7+5*x^6+9*x^5-9*x^4-7*x^3+7*x^2+2*x-3))+RootOf(_Z^2+1)*ln((-2*(-(-1-x)/( x^2-2))^(3/4)*x^3-8*(-(-1-x)/(x^2-2))^(1/4)*x^5+2*(-(-1-x)/(x^2-2))^(3/4)* x^2+8*(-(-1-x)/(x^2-2))^(1/4)*x^4+4*(-(-1-x)/(x^2-2))^(3/4)*x+10*(-(-1-x)/ (x^2-2))^(1/4)*x^3-10*(-(-1-x)/(x^2-2))^(1/4)*x^2-4*(-(-1-x)/(x^2-2))^(1/4 )*x-4*RootOf(_Z^2+1)*(-(-1-x)/(x^2-2))^(1/2)+RootOf(_Z^2+1)*x^8-7*RootOf(_ Z^2+1)*x^2-2*RootOf(_Z^2+1)*x+7*RootOf(_Z^2+1)*x^3+9*RootOf(_Z^2+1)*x^4-Ro otOf(_Z^2+1)*x^9-9*RootOf(_Z^2+1)*x^5+RootOf(_Z^2+1)+5*RootOf(_Z^2+1)*x^7- 5*RootOf(_Z^2+1)*x^6-2*RootOf(_Z^2+1)*(-(-1-x)/(x^2-2))^(1/2)*x^4-6*RootOf (_Z^2+1)*(-(-1-x)/(x^2-2))^(1/2)*x^3+6*RootOf(_Z^2+1)*(-(-1-x)/(x^2-2))^(1 /2)*x^2+4*RootOf(_Z^2+1)*(-(-1-x)/(x^2-2))^(1/2)*x+4*(-(-1-x)/(x^2-2))^(1/ 4)-4*(-(-1-x)/(x^2-2))^(3/4)+2*(-(-1-x)/(x^2-2))^(1/4)*x^7-2*(-(-1-x)/(x^2 -2))^(1/4)*x^6+2*RootOf(_Z^2+1)*(-(-1-x)/(x^2-2))^(1/2)*x^5)/(x^9-x^8-5...
Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (47) = 94\).
Time = 62.29 (sec) , antiderivative size = 331, normalized size of antiderivative = 6.49 \[ \int \frac {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} \left (-2+x^2\right ) \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx=-\arctan \left (\frac {2 \, {\left ({\left (x^{3} - x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}} + {\left (x^{7} - x^{6} - 4 \, x^{5} + 4 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}}\right )}}{x^{9} - x^{8} - 5 \, x^{7} + 5 \, x^{6} + 9 \, x^{5} - 9 \, x^{4} - 7 \, x^{3} + 7 \, x^{2} + 2 \, x - 3}\right ) + \log \left (-\frac {x^{9} - x^{8} - 5 \, x^{7} + 5 \, x^{6} + 9 \, x^{5} - 9 \, x^{4} - 7 \, x^{3} + 7 \, x^{2} + 2 \, {\left (x^{3} - x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}} + 2 \, {\left (x^{5} - x^{4} - 3 \, x^{3} + 3 \, x^{2} + 2 \, x - 2\right )} \sqrt {\frac {x + 1}{x^{2} - 2}} + 2 \, {\left (x^{7} - x^{6} - 4 \, x^{5} + 4 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}} + 2 \, x - 1}{x^{9} - x^{8} - 5 \, x^{7} + 5 \, x^{6} + 9 \, x^{5} - 9 \, x^{4} - 7 \, x^{3} + 7 \, x^{2} + 2 \, x - 3}\right ) \]
integrate((-9*x^3-x^2+16*x+2)/((1+x)/(x^2-2))^(1/4)/(x^2-2)/(x^9-x^8-5*x^7 +5*x^6+9*x^5-9*x^4-7*x^3+7*x^2+2*x-3),x, algorithm="fricas")
-arctan(2*((x^3 - x^2 - 2*x + 2)*((x + 1)/(x^2 - 2))^(3/4) + (x^7 - x^6 - 4*x^5 + 4*x^4 + 5*x^3 - 5*x^2 - 2*x + 2)*((x + 1)/(x^2 - 2))^(1/4))/(x^9 - x^8 - 5*x^7 + 5*x^6 + 9*x^5 - 9*x^4 - 7*x^3 + 7*x^2 + 2*x - 3)) + log(-(x ^9 - x^8 - 5*x^7 + 5*x^6 + 9*x^5 - 9*x^4 - 7*x^3 + 7*x^2 + 2*(x^3 - x^2 - 2*x + 2)*((x + 1)/(x^2 - 2))^(3/4) + 2*(x^5 - x^4 - 3*x^3 + 3*x^2 + 2*x - 2)*sqrt((x + 1)/(x^2 - 2)) + 2*(x^7 - x^6 - 4*x^5 + 4*x^4 + 5*x^3 - 5*x^2 - 2*x + 2)*((x + 1)/(x^2 - 2))^(1/4) + 2*x - 1)/(x^9 - x^8 - 5*x^7 + 5*x^6 + 9*x^5 - 9*x^4 - 7*x^3 + 7*x^2 + 2*x - 3))
Timed out. \[ \int \frac {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} \left (-2+x^2\right ) \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx=\text {Timed out} \]
integrate((-9*x**3-x**2+16*x+2)/((1+x)/(x**2-2))**(1/4)/(x**2-2)/(x**9-x** 8-5*x**7+5*x**6+9*x**5-9*x**4-7*x**3+7*x**2+2*x-3),x)
\[ \int \frac {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} \left (-2+x^2\right ) \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx=\int { -\frac {9 \, x^{3} + x^{2} - 16 \, x - 2}{{\left (x^{9} - x^{8} - 5 \, x^{7} + 5 \, x^{6} + 9 \, x^{5} - 9 \, x^{4} - 7 \, x^{3} + 7 \, x^{2} + 2 \, x - 3\right )} {\left (x^{2} - 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}}} \,d x } \]
integrate((-9*x^3-x^2+16*x+2)/((1+x)/(x^2-2))^(1/4)/(x^2-2)/(x^9-x^8-5*x^7 +5*x^6+9*x^5-9*x^4-7*x^3+7*x^2+2*x-3),x, algorithm="maxima")
-integrate((9*x^3 + x^2 - 16*x - 2)/((x^9 - x^8 - 5*x^7 + 5*x^6 + 9*x^5 - 9*x^4 - 7*x^3 + 7*x^2 + 2*x - 3)*(x^2 - 2)*((x + 1)/(x^2 - 2))^(1/4)), x)
\[ \int \frac {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} \left (-2+x^2\right ) \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx=\int { -\frac {9 \, x^{3} + x^{2} - 16 \, x - 2}{{\left (x^{9} - x^{8} - 5 \, x^{7} + 5 \, x^{6} + 9 \, x^{5} - 9 \, x^{4} - 7 \, x^{3} + 7 \, x^{2} + 2 \, x - 3\right )} {\left (x^{2} - 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}}} \,d x } \]
integrate((-9*x^3-x^2+16*x+2)/((1+x)/(x^2-2))^(1/4)/(x^2-2)/(x^9-x^8-5*x^7 +5*x^6+9*x^5-9*x^4-7*x^3+7*x^2+2*x-3),x, algorithm="giac")
integrate(-(9*x^3 + x^2 - 16*x - 2)/((x^9 - x^8 - 5*x^7 + 5*x^6 + 9*x^5 - 9*x^4 - 7*x^3 + 7*x^2 + 2*x - 3)*(x^2 - 2)*((x + 1)/(x^2 - 2))^(1/4)), x)
Timed out. \[ \int \frac {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} \left (-2+x^2\right ) \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx=\int \frac {-9\,x^3-x^2+16\,x+2}{\left (x^2-2\right )\,{\left (\frac {x+1}{x^2-2}\right )}^{1/4}\,\left (x^9-x^8-5\,x^7+5\,x^6+9\,x^5-9\,x^4-7\,x^3+7\,x^2+2\,x-3\right )} \,d x \]
int((16*x - x^2 - 9*x^3 + 2)/((x^2 - 2)*((x + 1)/(x^2 - 2))^(1/4)*(2*x + 7 *x^2 - 7*x^3 - 9*x^4 + 9*x^5 + 5*x^6 - 5*x^7 - x^8 + x^9 - 3)),x)