Integrand size = 36, antiderivative size = 205 \[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x-2 \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x+2 \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} x}{\sqrt [3]{-a x^2+x^3}}\right )}{a d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} x^2+\frac {\left (-a x^2+x^3\right )^{2/3}}{\sqrt [6]{d}}}{x \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}} \]
1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*x/(d^(1/6)*x-2*(-a*x^2+x^3)^(1/3)))/a/d ^(5/6)-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*x/(d^(1/6)*x+2*(-a*x^2+x^3)^(1/3 )))/a/d^(5/6)+arctanh(d^(1/6)*x/(-a*x^2+x^3)^(1/3))/a/d^(5/6)+1/2*arctanh( (d^(1/6)*x^2+(-a*x^2+x^3)^(2/3)/d^(1/6))/x/(-a*x^2+x^3)^(1/3))/a/d^(5/6)
Time = 0.61 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\frac {x^{4/3} (-a+x)^{2/3} \left (\sqrt {3} \left (\arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x}-2 \sqrt [3]{-a+x}}\right )-\arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x}+2 \sqrt [3]{-a+x}}\right )\right )+2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{-a+x}}\right )+\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{-a+x}}+\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d} \sqrt [3]{x}}\right )\right )}{2 a d^{5/6} \left (x^2 (-a+x)\right )^{2/3}} \]
(x^(4/3)*(-a + x)^(2/3)*(Sqrt[3]*(ArcTan[(Sqrt[3]*d^(1/6)*x^(1/3))/(d^(1/6 )*x^(1/3) - 2*(-a + x)^(1/3))] - ArcTan[(Sqrt[3]*d^(1/6)*x^(1/3))/(d^(1/6) *x^(1/3) + 2*(-a + x)^(1/3))]) + 2*ArcTanh[(d^(1/6)*x^(1/3))/(-a + x)^(1/3 )] + ArcTanh[(d^(1/6)*x^(1/3))/(-a + x)^(1/3) + (-a + x)^(1/3)/(d^(1/6)*x^ (1/3))]))/(2*a*d^(5/6)*(x^2*(-a + x))^(2/3))
Time = 0.60 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.32, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2467, 25, 1205, 2009}
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.
| \(\displaystyle \int \frac {x^2}{\left (x^2 (x-a)\right )^{2/3} \left (-a^2+2 a x+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{4/3} (x-a)^{2/3} \int -\frac {x^{2/3}}{(x-a)^{2/3} \left (a^2-2 x a+(1-d) x^2\right )}dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{4/3} (x-a)^{2/3} \int \frac {x^{2/3}}{(x-a)^{2/3} \left (a^2-2 x a+(1-d) x^2\right )}dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 1205 |
| \(\displaystyle -\frac {x^{4/3} (x-a)^{2/3} \int \left (\frac {x^{2/3} (1-d)}{a \sqrt {d} (x-a)^{2/3} \left (-2 \sqrt {d} a+2 a-2 (1-d) x\right )}+\frac {x^{2/3} (1-d)}{a \sqrt {d} (x-a)^{2/3} \left (-2 \sqrt {d} a-2 a+2 (1-d) x\right )}\right )dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x^{4/3} (x-a)^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-a}}\right )}{2 a d^{5/6}}+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 a d^{5/6}}+\frac {\log \left (2 a \left (1-\sqrt {d}\right )-2 (1-d) x\right )}{4 a d^{5/6}}-\frac {\log \left (2 (1-d) x-2 a \left (\sqrt {d}+1\right )\right )}{4 a d^{5/6}}-\frac {3 \log \left (-\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{x}\right )}{4 a d^{5/6}}+\frac {3 \log \left (\sqrt [6]{d} \sqrt [3]{x}-\sqrt [3]{x-a}\right )}{4 a d^{5/6}}\right )}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
-((x^(4/3)*(-a + x)^(2/3)*(-1/2*(Sqrt[3]*ArcTan[1/Sqrt[3] - (2*d^(1/6)*x^( 1/3))/(Sqrt[3]*(-a + x)^(1/3))])/(a*d^(5/6)) + (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*d^(1/6)*x^(1/3))/(Sqrt[3]*(-a + x)^(1/3))])/(2*a*d^(5/6)) + Log[2*a*(1 - Sqrt[d]) - 2*(1 - d)*x]/(4*a*d^(5/6)) - Log[-2*a*(1 + Sqrt[d]) + 2*(1 - d)*x]/(4*a*d^(5/6)) - (3*Log[-(d^(1/6)*x^(1/3)) - (-a + x)^(1/3)])/(4*a*d ^(5/6)) + (3*Log[d^(1/6)*x^(1/3) - (-a + x)^(1/3)])/(4*a*d^(5/6))))/(-((a - x)*x^2))^(2/3))
3.25.88.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^ n, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && !IntegerQ[m] && !IntegerQ[n]
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]
Time = 1.24 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.09
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x +2 \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{6}} x}\right )-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x -2 \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{6}} x}\right )-2 \ln \left (\frac {d^{\frac {1}{6}} x -\left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {d^{\frac {1}{3}} x^{2}+d^{\frac {1}{6}} \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}} x +\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {d^{\frac {1}{3}} x^{2}-d^{\frac {1}{6}} \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}} x +\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+2 \ln \left (\frac {d^{\frac {1}{6}} x +\left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}}{x}\right )}{4 d^{\frac {5}{6}} a}\) | \(223\) |
1/4*(2*3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/6)*x+2*(-x^2*(a-x))^(1/3))/d^(1/6) /x)-2*3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/6)*x-2*(-x^2*(a-x))^(1/3))/d^(1/6)/ x)-2*ln((d^(1/6)*x-(-x^2*(a-x))^(1/3))/x)+ln((d^(1/3)*x^2+d^(1/6)*(-x^2*(a -x))^(1/3)*x+(-x^2*(a-x))^(2/3))/x^2)-ln((d^(1/3)*x^2-d^(1/6)*(-x^2*(a-x)) ^(1/3)*x+(-x^2*(a-x))^(2/3))/x^2)+2*ln((d^(1/6)*x+(-x^2*(a-x))^(1/3))/x))/ d^(5/6)/a
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (159) = 318\).
Time = 0.25 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.59 \[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} a d x + a d x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} a d x + a d x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} a d x - a d x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} a d x - a d x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) + \frac {1}{2} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a d x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{2} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {a d x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) \]
1/4*(sqrt(-3) + 1)*(1/(a^6*d^5))^(1/6)*log(1/2*((sqrt(-3)*a*d*x + a*d*x)*( 1/(a^6*d^5))^(1/6) + 2*(-a*x^2 + x^3)^(1/3))/x) - 1/4*(sqrt(-3) + 1)*(1/(a ^6*d^5))^(1/6)*log(-1/2*((sqrt(-3)*a*d*x + a*d*x)*(1/(a^6*d^5))^(1/6) - 2* (-a*x^2 + x^3)^(1/3))/x) + 1/4*(sqrt(-3) - 1)*(1/(a^6*d^5))^(1/6)*log(1/2* ((sqrt(-3)*a*d*x - a*d*x)*(1/(a^6*d^5))^(1/6) + 2*(-a*x^2 + x^3)^(1/3))/x) - 1/4*(sqrt(-3) - 1)*(1/(a^6*d^5))^(1/6)*log(-1/2*((sqrt(-3)*a*d*x - a*d* x)*(1/(a^6*d^5))^(1/6) - 2*(-a*x^2 + x^3)^(1/3))/x) + 1/2*(1/(a^6*d^5))^(1 /6)*log((a*d*x*(1/(a^6*d^5))^(1/6) + (-a*x^2 + x^3)^(1/3))/x) - 1/2*(1/(a^ 6*d^5))^(1/6)*log(-(a*d*x*(1/(a^6*d^5))^(1/6) - (-a*x^2 + x^3)^(1/3))/x)
\[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\int \frac {x^{2}}{\left (x^{2} \left (- a + x\right )\right )^{\frac {2}{3}} \left (- a^{2} + 2 a x + d x^{2} - x^{2}\right )}\, dx \]
\[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\int { \frac {x^{2}}{\left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}} {\left ({\left (d - 1\right )} x^{2} - a^{2} + 2 \, a x\right )}} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.02 \[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \log \left (\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{4 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} \log \left (-\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{4 \, a d} - \frac {\arctan \left (\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{2 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (-\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} - 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{2 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (\frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{a \left (-d\right )^{\frac {5}{6}}} \]
-1/4*sqrt(3)*log(sqrt(3)*(-d)^(1/6)*(-a/x + 1)^(1/3) + (-a/x + 1)^(2/3) + (-d)^(1/3))/(a*(-d)^(5/6)) - 1/4*sqrt(3)*(-d)^(1/6)*log(-sqrt(3)*(-d)^(1/6 )*(-a/x + 1)^(1/3) + (-a/x + 1)^(2/3) + (-d)^(1/3))/(a*d) - 1/2*arctan((sq rt(3)*(-d)^(1/6) + 2*(-a/x + 1)^(1/3))/(-d)^(1/6))/(a*(-d)^(5/6)) - 1/2*ar ctan(-(sqrt(3)*(-d)^(1/6) - 2*(-a/x + 1)^(1/3))/(-d)^(1/6))/(a*(-d)^(5/6)) - arctan((-a/x + 1)^(1/3)/(-d)^(1/6))/(a*(-d)^(5/6))
Timed out. \[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\int \frac {x^2}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (-a^2+2\,a\,x+\left (d-1\right )\,x^2\right )} \,d x \]