Integrand size = 39, antiderivative size = 194 \[ \int \frac {-a+x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [6]{d} \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [6]{d} \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}}\right )}{a d^{5/6}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [6]{d}}+\sqrt [6]{d} \left (-a x^2+x^3\right )^{2/3}}{x \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}} \]
1/2*3^(1/2)*arctan(3^(1/2)*x/(x-2*d^(1/6)*(-a*x^2+x^3)^(1/3)))/a/d^(5/6)-1 /2*3^(1/2)*arctan(3^(1/2)*x/(x+2*d^(1/6)*(-a*x^2+x^3)^(1/3)))/a/d^(5/6)-ar ctanh(x/d^(1/6)/(-a*x^2+x^3)^(1/3))/a/d^(5/6)-1/2*arctanh((x^2/d^(1/6)+d^( 1/6)*(-a*x^2+x^3)^(2/3))/x/(-a*x^2+x^3)^(1/3))/a/d^(5/6)
Time = 0.50 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.97 \[ \int \frac {-a+x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {x^{2/3} \sqrt [3]{-a+x} \left (\sqrt {3} \left (\arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [6]{d} \sqrt [3]{-a+x}}\right )-\arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [6]{d} \sqrt [3]{-a+x}}\right )\right )-2 \text {arctanh}\left (\frac {\sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}\right )-\text {arctanh}\left (\frac {\sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{x}}\right )\right )}{2 a d^{5/6} \sqrt [3]{x^2 (-a+x)}} \]
(x^(2/3)*(-a + x)^(1/3)*(Sqrt[3]*(ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) - 2*d^ (1/6)*(-a + x)^(1/3))] - ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*d^(1/6)*(-a + x)^(1/3))]) - 2*ArcTanh[x^(1/3)/(d^(1/6)*(-a + x)^(1/3))] - ArcTanh[x^( 1/3)/(d^(1/6)*(-a + x)^(1/3)) + (d^(1/6)*(-a + x)^(1/3))/x^(1/3)]))/(2*a*d ^(5/6)*(x^2*(-a + x))^(1/3))
Time = 0.61 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.44, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2467, 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.
| \(\displaystyle \int \frac {x-a}{\sqrt [3]{x^2 (x-a)} \left (a^2 d-2 a d x+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x-a} \int \frac {(x-a)^{2/3}}{x^{2/3} \left (d a^2-2 d x a-(1-d) x^2\right )}dx}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
| \(\Big \downarrow \) 1205 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x-a} \int \left (\frac {(x-a)^{2/3} (d-1)}{a \sqrt {d} x^{2/3} \left (-2 d a-2 \sqrt {d} a-2 (1-d) x\right )}+\frac {(x-a)^{2/3} (d-1)}{a \sqrt {d} x^{2/3} \left (2 d a-2 \sqrt {d} a+2 (1-d) x\right )}\right )dx}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x-a} \left (-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x}}\right )}{2 a d^{5/6}}+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 a d^{5/6}}-\frac {\log \left (-2 a \sqrt {d} \left (\sqrt {d}+1\right )-2 (1-d) x\right )}{4 a d^{5/6}}+\frac {\log \left (2 (1-d) x-2 a \left (1-\sqrt {d}\right ) \sqrt {d}\right )}{4 a d^{5/6}}-\frac {3 \log \left (-\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x}\right )}{4 a d^{5/6}}+\frac {3 \log \left (\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x}\right )}{4 a d^{5/6}}\right )}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
(x^(2/3)*(-a + x)^(1/3)*(-1/2*(Sqrt[3]*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(-a + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(a*d^(5/6)) + (Sqrt[3]*ArcTan[1/Sqrt[3] + ( 2*d^(1/6)*(-a + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(2*a*d^(5/6)) - Log[-2*a*(1 + Sqrt[d])*Sqrt[d] - 2*(1 - d)*x]/(4*a*d^(5/6)) + Log[-2*a*(1 - Sqrt[d])*S qrt[d] + 2*(1 - d)*x]/(4*a*d^(5/6)) - (3*Log[-x^(1/3) - d^(1/6)*(-a + x)^( 1/3)])/(4*a*d^(5/6)) + (3*Log[-x^(1/3) + d^(1/6)*(-a + x)^(1/3)])/(4*a*d^( 5/6))))/(-((a - x)*x^2))^(1/3)
3.25.14.3.1 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 = 1.90 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.29
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x \left (\frac {1}{d}\right )^{\frac {1}{6}}-2 \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}\right )}{3 x \left (\frac {1}{d}\right )^{\frac {1}{6}}}\right )+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x \left (\frac {1}{d}\right )^{\frac {1}{6}}+2 \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}\right )}{3 x \left (\frac {1}{d}\right )^{\frac {1}{6}}}\right )+\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{6}} \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}} x -\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}-\left (-\left (a -x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{6}} \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+\left (-\left (a -x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )+2 \ln \left (\frac {x \left (\frac {1}{d}\right )^{\frac {1}{6}}-\left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )-2 \ln \left (\frac {x \left (\frac {1}{d}\right )^{\frac {1}{6}}+\left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{4 a d \left (\frac {1}{d}\right )^{\frac {1}{6}}}\) | \(250\) |
1/4*(-2*3^(1/2)*arctan(1/3*3^(1/2)*(x*(1/d)^(1/6)-2*(-(a-x)*x^2)^(1/3))/x/ (1/d)^(1/6))+2*3^(1/2)*arctan(1/3*3^(1/2)*(x*(1/d)^(1/6)+2*(-(a-x)*x^2)^(1 /3))/x/(1/d)^(1/6))+ln(((1/d)^(1/6)*(-(a-x)*x^2)^(1/3)*x-(1/d)^(1/3)*x^2-( -(a-x)*x^2)^(2/3))/x^2)-ln(((1/d)^(1/6)*(-(a-x)*x^2)^(1/3)*x+(1/d)^(1/3)*x ^2+(-(a-x)*x^2)^(2/3))/x^2)+2*ln((x*(1/d)^(1/6)-(-(a-x)*x^2)^(1/3))/x)-2*l n((x*(1/d)^(1/6)+(-(a-x)*x^2)^(1/3))/x))/a/d/(1/d)^(1/6)
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (152) = 304\).
Time = 0.26 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.89 \[ \int \frac {-a+x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} a^{5} d^{4} x + a^{5} d^{4} x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} a^{5} d^{4} x + a^{5} d^{4} x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} a^{5} d^{4} x - a^{5} d^{4} x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} a^{5} d^{4} x - a^{5} d^{4} x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) - \frac {1}{2} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a^{5} d^{4} x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {a^{5} d^{4} x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) \]
1/4*(sqrt(-3) - 1)*(1/(a^6*d^5))^(1/6)*log(1/2*((sqrt(-3)*a^5*d^4*x + a^5* d^4*x)*(1/(a^6*d^5))^(5/6) + 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^5*d^4*x + a^5*d^4*x)*(1/(a^6* d^5))^(5/6) - 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^5*d^4*x - a^5*d^4*x)*(1/(a^6*d^5))^(5/6) + 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^5*d^4*x - a^5*d^4*x)*(1/(a^6*d^5))^(5/6) - 2*(-a*x^2 + x^3)^ (1/3))/x) - 1/2*(1/(a^6*d^5))^(1/6)*log((a^5*d^4*x*(1/(a^6*d^5))^(5/6) + ( -a*x^2 + x^3)^(1/3))/x) + 1/2*(1/(a^6*d^5))^(1/6)*log(-(a^5*d^4*x*(1/(a^6* d^5))^(5/6) - (-a*x^2 + x^3)^(1/3))/x)
\[ \int \frac {-a+x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int \frac {- a + x}{\sqrt [3]{x^{2} \left (- a + x\right )} \left (a^{2} d - 2 a d x + d x^{2} - x^{2}\right )}\, dx \]
\[ \int \frac {-a+x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int { -\frac {a - x}{{\left (a^{2} d - 2 \, a d x + {\left (d - 1\right )} x^{2}\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {1}{3}}} \,d x } \]
Time = 0.37 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.20 \[ \int \frac {-a+x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \log \left (\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{4 \, \left (-d^{5}\right )^{\frac {1}{6}} a} + \frac {\sqrt {3} \log \left (-\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{4 \, \left (-d^{5}\right )^{\frac {1}{6}} a} + \frac {\arctan \left (\frac {\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{2 \, \left (-d^{5}\right )^{\frac {1}{6}} a} + \frac {\arctan \left (-\frac {\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{6}} - 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{2 \, \left (-d^{5}\right )^{\frac {1}{6}} a} + \frac {\arctan \left (\frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{\left (-d^{5}\right )^{\frac {1}{6}} a} \]
-1/4*sqrt(3)*log(sqrt(3)*(-a/x + 1)^(1/3)*(-1/d)^(1/6) + (-a/x + 1)^(2/3) + (-1/d)^(1/3))/((-d^5)^(1/6)*a) + 1/4*sqrt(3)*log(-sqrt(3)*(-a/x + 1)^(1/ 3)*(-1/d)^(1/6) + (-a/x + 1)^(2/3) + (-1/d)^(1/3))/((-d^5)^(1/6)*a) + 1/2* arctan((sqrt(3)*(-1/d)^(1/6) + 2*(-a/x + 1)^(1/3))/(-1/d)^(1/6))/((-d^5)^( 1/6)*a) + 1/2*arctan(-(sqrt(3)*(-1/d)^(1/6) - 2*(-a/x + 1)^(1/3))/(-1/d)^( 1/6))/((-d^5)^(1/6)*a) + arctan((-a/x + 1)^(1/3)/(-1/d)^(1/6))/((-d^5)^(1/ 6)*a)
Timed out. \[ \int \frac {-a+x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int -\frac {a-x}{{\left (-x^2\,\left (a-x\right )\right )}^{1/3}\,\left (d\,a^2-2\,d\,a\,x+\left (d-1\right )\,x^2\right )} \,d x \]