Integrand size = 45, antiderivative size = 150 \[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-2 x+2 x^3-x^8}}\right )}{2^{2/3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-2 x+2 x^3-x^8}\right )}{2^{2/3}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-2 x+2 x^3-x^8}+\sqrt [3]{2} \left (-2 x+2 x^3-x^8\right )^{2/3}\right )}{2\ 2^{2/3}} \]
-1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(-x^8+2*x^3-2*x)^(1/3)))*2^(1/3)- 1/2*ln(-2*x+2^(2/3)*(-x^8+2*x^3-2*x)^(1/3))*2^(1/3)+1/4*ln(2*x^2+2^(2/3)*x *(-x^8+2*x^3-2*x)^(1/3)+2^(1/3)*(-x^8+2*x^3-2*x)^(2/3))*2^(1/3)
Time = 3.52 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.20 \[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\frac {x^{2/3} \left (2-2 x^2+x^7\right )^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2^{2/3} \sqrt [3]{2-2 x^2+x^7}}\right )-2 \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{2-2 x^2+x^7}\right )+\log \left (-2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{2-2 x^2+x^7}-\sqrt [3]{2} \left (2-2 x^2+x^7\right )^{2/3}\right )\right )}{2\ 2^{2/3} \left (-x \left (2-2 x^2+x^7\right )\right )^{2/3}} \]
(x^(2/3)*(2 - 2*x^2 + x^7)^(2/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^( 2/3) - 2^(2/3)*(2 - 2*x^2 + x^7)^(1/3))] - 2*Log[2*x^(2/3) + 2^(2/3)*(2 - 2*x^2 + x^7)^(1/3)] + Log[-2*x^(4/3) + 2^(2/3)*x^(2/3)*(2 - 2*x^2 + x^7)^( 1/3) - 2^(1/3)*(2 - 2*x^2 + x^7)^(2/3)]))/(2*2^(2/3)*(-(x*(2 - 2*x^2 + x^7 )))^(2/3))
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.
| \(\displaystyle \int \frac {\left (5 x^7-4\right ) \sqrt [3]{-x^8+2 x^3-2 x}}{\left (x^7+2\right ) \left (x^7-2 x^2+2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{-x^8+2 x^3-2 x} \int -\frac {\sqrt [3]{x} \left (4-5 x^7\right ) \sqrt [3]{-x^7+2 x^2-2}}{\left (x^7+2\right ) \left (x^7-2 x^2+2\right )}dx}{\sqrt [3]{x} \sqrt [3]{-x^7+2 x^2-2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{-x^8+2 x^3-2 x} \int \frac {\sqrt [3]{x} \left (4-5 x^7\right ) \sqrt [3]{-x^7+2 x^2-2}}{\left (x^7+2\right ) \left (x^7-2 x^2+2\right )}dx}{\sqrt [3]{x} \sqrt [3]{-x^7+2 x^2-2}}\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle -\frac {\sqrt [3]{-x^8+2 x^3-2 x} \int -\frac {\sqrt [3]{x} \left (4-5 x^7\right )}{\left (-x^7+2 x^2-2\right )^{2/3} \left (x^7+2\right )}dx}{\sqrt [3]{x} \sqrt [3]{-x^7+2 x^2-2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{-x^8+2 x^3-2 x} \int \frac {\sqrt [3]{x} \left (4-5 x^7\right )}{\left (-x^7+2 x^2-2\right )^{2/3} \left (x^7+2\right )}dx}{\sqrt [3]{x} \sqrt [3]{-x^7+2 x^2-2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{-x^8+2 x^3-2 x} \int \frac {x \left (4-5 x^7\right )}{\left (-x^7+2 x^2-2\right )^{2/3} \left (x^7+2\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{-x^7+2 x^2-2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{-x^8+2 x^3-2 x} \int \left (\frac {14 x}{\left (-x^7+2 x^2-2\right )^{2/3} \left (x^7+2\right )}-\frac {5 x}{\left (-x^7+2 x^2-2\right )^{2/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{-x^7+2 x^2-2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{-x^8+2 x^3-2 x} \left (\frac {1}{9} (-2)^{4/21} \int \frac {1}{\left (\sqrt [21]{-2}-\sqrt [3]{x}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} 2^{4/21} \int \frac {1}{\left (-\sqrt [3]{x}-\sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{8/21} 2^{4/21} \int \frac {1}{\left (-\sqrt [3]{x}-(-1)^{2/21} \sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{4/7} 2^{4/21} \int \frac {1}{\left (\sqrt [7]{-1} \sqrt [21]{2}-\sqrt [3]{x}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{16/21} 2^{4/21} \int \frac {1}{\left (-\sqrt [3]{x}-(-1)^{4/21} \sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{20/21} 2^{4/21} \int \frac {1}{\left ((-1)^{5/21} \sqrt [21]{2}-\sqrt [3]{x}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}-\frac {1}{9} \sqrt [7]{-1} 2^{4/21} \int \frac {1}{\left (-\sqrt [3]{x}-(-1)^{2/7} \sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-2)^{4/21} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{x}+\sqrt [21]{-2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} 2^{4/21} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{x}-\sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{8/21} 2^{4/21} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{x}-(-1)^{2/21} \sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{4/7} 2^{4/21} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{x}+\sqrt [7]{-1} \sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{16/21} 2^{4/21} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{x}-(-1)^{4/21} \sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{20/21} 2^{4/21} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{x}+(-1)^{5/21} \sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}-\frac {1}{9} \sqrt [7]{-1} 2^{4/21} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{x}-(-1)^{2/7} \sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-2)^{4/21} \int \frac {1}{\left (\sqrt [21]{-2}-(-1)^{2/3} \sqrt [3]{x}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} 2^{4/21} \int \frac {1}{\left (-(-1)^{2/3} \sqrt [3]{x}-\sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{8/21} 2^{4/21} \int \frac {1}{\left (-(-1)^{2/3} \sqrt [3]{x}-(-1)^{2/21} \sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{4/7} 2^{4/21} \int \frac {1}{\left (\sqrt [7]{-1} \sqrt [21]{2}-(-1)^{2/3} \sqrt [3]{x}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{16/21} 2^{4/21} \int \frac {1}{\left (-(-1)^{2/3} \sqrt [3]{x}-(-1)^{4/21} \sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}+\frac {1}{9} (-1)^{20/21} 2^{4/21} \int \frac {1}{\left ((-1)^{5/21} \sqrt [21]{2}-(-1)^{2/3} \sqrt [3]{x}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}-\frac {1}{9} \sqrt [7]{-1} 2^{4/21} \int \frac {1}{\left (-(-1)^{2/3} \sqrt [3]{x}-(-1)^{2/7} \sqrt [21]{2}\right ) \left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}-5 \int \frac {x}{\left (-x^7+2 x^2-2\right )^{2/3}}d\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-x^7+2 x^2-2}}\) |
3.21.78.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
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]
Time = 229.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.79
| method | result | size |
| pseudoelliptic | \(\frac {2^{\frac {1}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} {\left (-x \left (x^{7}-2 x^{2}+2\right )\right )}^{\frac {1}{3}}+x \right )}{3 x}\right )+\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} {\left (-x \left (x^{7}-2 x^{2}+2\right )\right )}^{\frac {1}{3}} x +{\left (-x \left (x^{7}-2 x^{2}+2\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {1}{3}} x +{\left (-x \left (x^{7}-2 x^{2}+2\right )\right )}^{\frac {1}{3}}}{x}\right )\right )}{4}\) | \(119\) |
| trager | \(\text {Expression too large to display}\) | \(1000\) |
1/4*2^(1/3)*(2*3^(1/2)*arctan(1/3*3^(1/2)*(2^(2/3)*(-x*(x^7-2*x^2+2))^(1/3 )+x)/x)+ln((2^(2/3)*x^2+2^(1/3)*(-x*(x^7-2*x^2+2))^(1/3)*x+(-x*(x^7-2*x^2+ 2))^(2/3))/x^2)-2*ln((-2^(1/3)*x+(-x*(x^7-2*x^2+2))^(1/3))/x))
Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (122) = 244\).
Time = 5.09 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.60 \[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (-\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{15} - 18 \, x^{10} + 4 \, x^{8} + 36 \, x^{5} - 36 \, x^{3} + 4 \, x\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} - 12 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{14} - 6 \, x^{9} + 4 \, x^{7} - 12 \, x^{2} + 4\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (x^{21} - 36 \, x^{16} + 6 \, x^{14} + 180 \, x^{11} - 144 \, x^{9} + 12 \, x^{7} - 216 \, x^{6} + 360 \, x^{4} - 144 \, x^{2} + 8\right )}\right )}}{6 \, {\left (x^{21} + 6 \, x^{14} - 108 \, x^{11} + 12 \, x^{7} + 216 \, x^{6} - 216 \, x^{4} + 8\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}} {\left (x^{7} - 6 \, x^{2} + 2\right )} + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{14} - 18 \, x^{9} + 4 \, x^{7} + 36 \, x^{4} - 36 \, x^{2} + 4\right )} + 24 \, {\left (x^{8} - 3 \, x^{3} + 2 \, x\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}}}{x^{14} + 4 \, x^{7} + 4}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} x + 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{7} + 2\right )} + 6 \, {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}}}{x^{7} + 2}\right ) \]
1/6*4^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(-1/6*4^(1/6)*sqrt(3)*(6*4^(2/3)*(-1) ^(2/3)*(x^15 - 18*x^10 + 4*x^8 + 36*x^5 - 36*x^3 + 4*x)*(-x^8 + 2*x^3 - 2* x)^(1/3) - 12*(-1)^(1/3)*(x^14 - 6*x^9 + 4*x^7 - 12*x^2 + 4)*(-x^8 + 2*x^3 - 2*x)^(2/3) - 4^(1/3)*(x^21 - 36*x^16 + 6*x^14 + 180*x^11 - 144*x^9 + 12 *x^7 - 216*x^6 + 360*x^4 - 144*x^2 + 8))/(x^21 + 6*x^14 - 108*x^11 + 12*x^ 7 + 216*x^6 - 216*x^4 + 8)) - 1/24*4^(2/3)*(-1)^(1/3)*log(-(6*4^(1/3)*(-1) ^(2/3)*(-x^8 + 2*x^3 - 2*x)^(2/3)*(x^7 - 6*x^2 + 2) + 4^(2/3)*(-1)^(1/3)*( x^14 - 18*x^9 + 4*x^7 + 36*x^4 - 36*x^2 + 4) + 24*(x^8 - 3*x^3 + 2*x)*(-x^ 8 + 2*x^3 - 2*x)^(1/3))/(x^14 + 4*x^7 + 4)) + 1/12*4^(2/3)*(-1)^(1/3)*log( -(3*4^(2/3)*(-1)^(1/3)*(-x^8 + 2*x^3 - 2*x)^(1/3)*x + 4^(1/3)*(-1)^(2/3)*( x^7 + 2) + 6*(-x^8 + 2*x^3 - 2*x)^(2/3))/(x^7 + 2))
Timed out. \[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\int { \frac {{\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} - 4\right )}}{{\left (x^{7} - 2 \, x^{2} + 2\right )} {\left (x^{7} + 2\right )}} \,d x } \]
\[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\int { \frac {{\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} - 4\right )}}{{\left (x^{7} - 2 \, x^{2} + 2\right )} {\left (x^{7} + 2\right )}} \,d x } \]
Timed out. \[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\int \frac {\left (5\,x^7-4\right )\,{\left (-x^8+2\,x^3-2\,x\right )}^{1/3}}{\left (x^7+2\right )\,\left (x^7-2\,x^2+2\right )} \,d x \]