Integrand size = 46, antiderivative size = 207 \[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2-2 \sqrt [3]{-a x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2+2 \sqrt [3]{-a x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \left (-a x^2+x^3\right )^{2/3}}{a-x}\right )}{d^{5/6}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} x^4+\frac {\left (-a x^2+x^3\right )^{2/3}}{\sqrt [6]{d}}}{x^2 \sqrt [3]{-a x^2+x^3}}\right )}{2 d^{5/6}} \]
-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*x^2/(d^(1/6)*x^2-2*(-a*x^2+x^3)^(1/3)) )/d^(5/6)+1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*x^2/(d^(1/6)*x^2+2*(-a*x^2+x^ 3)^(1/3)))/d^(5/6)+arctanh(d^(1/6)*(-a*x^2+x^3)^(2/3)/(a-x))/d^(5/6)-1/2*a rctanh((d^(1/6)*x^4+(-a*x^2+x^3)^(2/3)/d^(1/6))/x^2/(-a*x^2+x^3)^(1/3))/d^ (5/6)
Time = 0.30 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.85 \[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=\frac {x^{4/3} (-a+x)^{2/3} \left (\sqrt {3} \left (\arctan \left (\frac {1-\frac {2 \sqrt [3]{-a+x}}{\sqrt [6]{d} x^{4/3}}}{\sqrt {3}}\right )-\arctan \left (\frac {1+\frac {2 \sqrt [3]{-a+x}}{\sqrt [6]{d} x^{4/3}}}{\sqrt {3}}\right )\right )-2 \text {arctanh}\left (\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d} x^{4/3}}\right )-\text {arctanh}\left (\frac {\sqrt [6]{d} x^{4/3} \sqrt [3]{-a+x}}{\sqrt [3]{d} x^{8/3}+(-a+x)^{2/3}}\right )\right )}{2 d^{5/6} \left (x^2 (-a+x)\right )^{2/3}} \]
(x^(4/3)*(-a + x)^(2/3)*(Sqrt[3]*(ArcTan[(1 - (2*(-a + x)^(1/3))/(d^(1/6)* x^(4/3)))/Sqrt[3]] - ArcTan[(1 + (2*(-a + x)^(1/3))/(d^(1/6)*x^(4/3)))/Sqr t[3]]) - 2*ArcTanh[(-a + x)^(1/3)/(d^(1/6)*x^(4/3))] - ArcTanh[(d^(1/6)*x^ (4/3)*(-a + x)^(1/3))/(d^(1/3)*x^(8/3) + (-a + x)^(2/3))]))/(2*d^(5/6)*(x^ 2*(-a + x))^(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.
| \(\displaystyle \int \frac {x^7 (3 x-4 a)}{\left (x^2 (x-a)\right )^{2/3} \left (-a^2+2 a x+d x^8-x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{4/3} (x-a)^{2/3} \int \frac {(4 a-3 x) x^{17/3}}{(x-a)^{2/3} \left (-d x^8+x^2-2 a x+a^2\right )}dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 x^{4/3} (x-a)^{2/3} \int \frac {(4 a-3 x) x^{19/3}}{(x-a)^{2/3} \left (-d x^8+x^2-2 a x+a^2\right )}d\sqrt [3]{x}}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{4/3} (x-a)^{2/3} \int \left (\frac {3 x^{22/3}}{(x-a)^{2/3} \left (d x^8-x^2+2 a x-a^2\right )}+\frac {4 a x^{19/3}}{(x-a)^{2/3} \left (-d x^8+x^2-2 a x+a^2\right )}\right )d\sqrt [3]{x}}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 x^{4/3} (x-a)^{2/3} \left (4 a \int \frac {x^{19/3}}{(x-a)^{2/3} \left (-d x^8+x^2-2 a x+a^2\right )}d\sqrt [3]{x}+3 \int \frac {x^{22/3}}{(x-a)^{2/3} \left (d x^8-x^2+2 a x-a^2\right )}d\sqrt [3]{x}\right )}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
3.25.97.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.42
| method | result | size |
| pseudoelliptic | \(-\frac {a^{5} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{24}-6 \textit {\_Z}^{21}+15 \textit {\_Z}^{18}-20 \textit {\_Z}^{15}+15 \textit {\_Z}^{12}-6 \textit {\_Z}^{9}-a^{6} d +\textit {\_Z}^{6}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}}{x}\right )}{\left (\textit {\_R} -1\right )^{5} \left (\textit {\_R}^{2}+\textit {\_R} +1\right )^{5} \textit {\_R}^{5}}\right )}{2}\) | \(87\) |
-1/2*a^5*sum(ln((-_R*x+(-x^2*(a-x))^(1/3))/x)/(_R-1)^5/(_R^2+_R+1)^5/_R^5, _R=RootOf(_Z^24-6*_Z^21+15*_Z^18-20*_Z^15+15*_Z^12-6*_Z^9-a^6*d+_Z^6))
Time = 0.29 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.39 \[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=-\frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} d x^{2} + d x^{2}\right )} \frac {1}{d^{5}}^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x^{2}}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} d x^{2} + d x^{2}\right )} \frac {1}{d^{5}}^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x^{2}}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} d x^{2} - d x^{2}\right )} \frac {1}{d^{5}}^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x^{2}}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} d x^{2} - d x^{2}\right )} \frac {1}{d^{5}}^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x^{2}}\right ) - \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (-\frac {d \frac {1}{d^{5}}^{\frac {1}{6}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (\frac {d \frac {1}{d^{5}}^{\frac {1}{6}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) \]
-1/4*(sqrt(-3) + 1)*(d^(-5))^(1/6)*log(-1/2*((sqrt(-3)*d*x^2 + d*x^2)*(d^( -5))^(1/6) + 2*(-a*x^2 + x^3)^(1/3))/x^2) + 1/4*(sqrt(-3) + 1)*(d^(-5))^(1 /6)*log(1/2*((sqrt(-3)*d*x^2 + d*x^2)*(d^(-5))^(1/6) - 2*(-a*x^2 + x^3)^(1 /3))/x^2) - 1/4*(sqrt(-3) - 1)*(d^(-5))^(1/6)*log(-1/2*((sqrt(-3)*d*x^2 - d*x^2)*(d^(-5))^(1/6) + 2*(-a*x^2 + x^3)^(1/3))/x^2) + 1/4*(sqrt(-3) - 1)* (d^(-5))^(1/6)*log(1/2*((sqrt(-3)*d*x^2 - d*x^2)*(d^(-5))^(1/6) - 2*(-a*x^ 2 + x^3)^(1/3))/x^2) - 1/2*(d^(-5))^(1/6)*log(-(d*(d^(-5))^(1/6)*x^2 + (-a *x^2 + x^3)^(1/3))/x^2) + 1/2*(d^(-5))^(1/6)*log((d*(d^(-5))^(1/6)*x^2 - ( -a*x^2 + x^3)^(1/3))/x^2)
\[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=\int \frac {x^{7} \left (- 4 a + 3 x\right )}{\left (x^{2} \left (- a + x\right )\right )^{\frac {2}{3}} \left (- a^{2} + 2 a x + d x^{8} - x^{2}\right )}\, dx \]
\[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=\int { -\frac {{\left (4 \, a - 3 \, x\right )} x^{7}}{{\left (d x^{8} - a^{2} + 2 \, a x - x^{2}\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (161) = 322\).
Time = 0.50 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.93 \[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=\frac {1}{2} \, \sqrt {3} \frac {1}{d^{5}}^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{{\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}\right ) + \frac {1}{2} \, \sqrt {3} \frac {1}{d^{5}}^{\frac {1}{6}} \arctan \left (-\frac {\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{{\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}\right ) + \frac {1}{4} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left (\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2} + {\left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) - \frac {1}{4} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left (\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2} + {\left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) + \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) \]
1/2*sqrt(3)*(d^(-5))^(1/6)*arctan((sqrt(3)*(-a/x + 1)^(4/3) - sqrt(3)*a*d^ (1/6) - sqrt(3)*(-a/x + 1)^(1/3))/((-a/x + 1)^(4/3) + a*d^(1/6) - (-a/x + 1)^(1/3))) + 1/2*sqrt(3)*(d^(-5))^(1/6)*arctan(-(sqrt(3)*(-a/x + 1)^(4/3) + sqrt(3)*a*d^(1/6) - sqrt(3)*(-a/x + 1)^(1/3))/((-a/x + 1)^(4/3) - a*d^(1 /6) - (-a/x + 1)^(1/3))) + 1/4*(d^(-5))^(1/6)*log((sqrt(3)*(-a/x + 1)^(4/3 ) + sqrt(3)*a*d^(1/6) - sqrt(3)*(-a/x + 1)^(1/3))^2 + ((-a/x + 1)^(4/3) - a*d^(1/6) - (-a/x + 1)^(1/3))^2) - 1/4*(d^(-5))^(1/6)*log((sqrt(3)*(-a/x + 1)^(4/3) - sqrt(3)*a*d^(1/6) - sqrt(3)*(-a/x + 1)^(1/3))^2 + ((-a/x + 1)^ (4/3) + a*d^(1/6) - (-a/x + 1)^(1/3))^2) + 1/2*(d^(-5))^(1/6)*log(abs((-a/ x + 1)^(4/3) + a*d^(1/6) - (-a/x + 1)^(1/3))) - 1/2*(d^(-5))^(1/6)*log(abs ((-a/x + 1)^(4/3) - a*d^(1/6) - (-a/x + 1)^(1/3)))
Timed out. \[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=\int -\frac {x^7\,\left (4\,a-3\,x\right )}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (-a^2+2\,a\,x+d\,x^8-x^2\right )} \,d x \]