Integrand size = 40, antiderivative size = 243 \[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}}\right )}{2 a d^{2/3}}+\frac {\log \left (x-\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a d^{2/3}}+\frac {\log \left (x+\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a d^{2/3}}-\frac {\log \left (x^2-\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{2/3}}-\frac {\log \left (x^2+\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{2/3}} \]
-1/2*3^(1/2)*arctan(3^(1/2)*x^2/(x^2+2*d^(1/3)*(-a*x^2+x^3)^(2/3)))/a/d^(2 /3)+1/2*ln(x-d^(1/6)*(-a*x^2+x^3)^(1/3))/a/d^(2/3)+1/2*ln(x+d^(1/6)*(-a*x^ 2+x^3)^(1/3))/a/d^(2/3)-1/4*ln(x^2-d^(1/6)*x*(-a*x^2+x^3)^(1/3)+d^(1/3)*(- a*x^2+x^3)^(2/3))/a/d^(2/3)-1/4*ln(x^2+d^(1/6)*x*(-a*x^2+x^3)^(1/3)+d^(1/3 )*(-a*x^2+x^3)^(2/3))/a/d^(2/3)
Time = 0.42 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.89 \[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=-\frac {x^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{d} (-a+x)^{2/3}}\right )-2 \log \left (\sqrt [3]{x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )-2 \log \left (\sqrt [3]{x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )+\log \left (x^{2/3}-\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )+\log \left (x^{2/3}+\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )\right )}{4 a d^{2/3} \left (x^2 (-a+x)\right )^{2/3}} \]
-1/4*(x^(4/3)*(-a + x)^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*d^(1/3)*(-a + x)^(2/3))] - 2*Log[x^(1/3) - d^(1/6)*(-a + x)^(1/3)] - 2 *Log[x^(1/3) + d^(1/6)*(-a + x)^(1/3)] + Log[x^(2/3) - d^(1/6)*x^(1/3)*(-a + x)^(1/3) + d^(1/3)*(-a + x)^(2/3)] + Log[x^(2/3) + d^(1/6)*x^(1/3)*(-a + x)^(1/3) + d^(1/3)*(-a + x)^(2/3)]))/(a*d^(2/3)*(x^2*(-a + x))^(2/3))
Time = 0.67 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, 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.
| \(\displaystyle \int \frac {x (x-a)}{\left (x^2 (x-a)\right )^{2/3} \left (a^2 d-2 a d x+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{4/3} (x-a)^{2/3} \int \frac {\sqrt [3]{x-a}}{\sqrt [3]{x} \left (d a^2-2 d x a-(1-d) x^2\right )}dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 1205 |
| \(\displaystyle \frac {x^{4/3} (x-a)^{2/3} \int \left (\frac {\sqrt [3]{x-a} (d-1)}{a \sqrt {d} \sqrt [3]{x} \left (-2 d a-2 \sqrt {d} a-2 (1-d) x\right )}+\frac {\sqrt [3]{x-a} (d-1)}{a \sqrt {d} \sqrt [3]{x} \left (2 d a-2 \sqrt {d} a+2 (1-d) x\right )}\right )dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^{4/3} (x-a)^{2/3} \left (\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}\right )}{2 a d^{2/3}}+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 a d^{2/3}}-\frac {\log \left (-2 a \sqrt {d} \left (\sqrt {d}+1\right )-2 (1-d) x\right )}{4 a d^{2/3}}-\frac {\log \left (2 (1-d) x-2 a \left (1-\sqrt {d}\right ) \sqrt {d}\right )}{4 a d^{2/3}}+\frac {3 \log \left (-\sqrt [3]{x-a}-\frac {\sqrt [3]{x}}{\sqrt [6]{d}}\right )}{4 a d^{2/3}}+\frac {3 \log \left (\frac {\sqrt [3]{x}}{\sqrt [6]{d}}-\sqrt [3]{x-a}\right )}{4 a d^{2/3}}\right )}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
(x^(4/3)*(-a + x)^(2/3)*((Sqrt[3]*ArcTan[1/Sqrt[3] - (2*x^(1/3))/(Sqrt[3]* d^(1/6)*(-a + x)^(1/3))])/(2*a*d^(2/3)) + (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*x ^(1/3))/(Sqrt[3]*d^(1/6)*(-a + x)^(1/3))])/(2*a*d^(2/3)) - Log[-2*a*(1 + S qrt[d])*Sqrt[d] - 2*(1 - d)*x]/(4*a*d^(2/3)) - Log[-2*a*(1 - Sqrt[d])*Sqrt [d] + 2*(1 - d)*x]/(4*a*d^(2/3)) + (3*Log[-(x^(1/3)/d^(1/6)) - (-a + x)^(1 /3)])/(4*a*d^(2/3)) + (3*Log[x^(1/3)/d^(1/6) - (-a + x)^(1/3)])/(4*a*d^(2/ 3))))/(-((a - x)*x^2))^(2/3)
3.27.82.3.1 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 = 0.78 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.58
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+2 \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}+\left (-a +x \right ) \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}+\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{2}}{x^{2}}\right )}{4 a d \left (\frac {1}{d}\right )^{\frac {1}{3}}}\) | \(141\) |
1/4*(2*3^(1/2)*arctan(1/3*3^(1/2)*((1/d)^(1/3)*x^2+2*(-x^2*(a-x))^(2/3))/( 1/d)^(1/3)/x^2)+2*ln((-(1/d)^(1/3)*x^2+(-x^2*(a-x))^(2/3))/x^2)-ln(((1/d)^ (1/3)*(-x^2*(a-x))^(2/3)+(-a+x)*(-x^2*(a-x))^(1/3)+(1/d)^(2/3)*x^2)/x^2))/ a/d/(1/d)^(1/3)
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.69 \[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left ({\left (d^{2}\right )}^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} {\left (d^{2}\right )}^{\frac {1}{6}}}{3 \, d x^{2}}\right ) - {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (d^{2}\right )}^{\frac {2}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {1}{3}} d - {\left (a d^{2} - d^{2} x\right )} {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (d^{2}\right )}^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right )}{4 \, a d^{2}} \]
1/4*(2*sqrt(3)*(d^2)^(1/6)*d*arctan(1/3*sqrt(3)*((d^2)^(1/3)*x^2 + 2*(-a*x ^2 + x^3)^(2/3)*d)*(d^2)^(1/6)/(d*x^2)) - (d^2)^(2/3)*log(((d^2)^(2/3)*x^2 + (-a*x^2 + x^3)^(2/3)*(d^2)^(1/3)*d - (a*d^2 - d^2*x)*(-a*x^2 + x^3)^(1/ 3))/x^2) + 2*(d^2)^(2/3)*log(-((d^2)^(1/3)*x^2 - (-a*x^2 + x^3)^(2/3)*d)/x ^2))/(a*d^2)
\[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int \frac {x \left (- a + x\right )}{\left (x^{2} \left (- a + x\right )\right )^{\frac {2}{3}} \left (a^{2} d - 2 a d x + d x^{2} - x^{2}\right )}\, dx \]
\[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int { -\frac {{\left (a - x\right )} x}{{\left (a^{2} d - 2 \, a d x + {\left (d - 1\right )} x^{2}\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}}} \,d x } \]
Time = 0.36 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.42 \[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} d^{\frac {1}{3}} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \frac {1}{d^{\frac {1}{3}}}\right )}\right )}{2 \, a {\left | d \right |}^{\frac {2}{3}}} - \frac {\log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{d^{\frac {1}{3}}} + \frac {1}{d^{\frac {2}{3}}}\right )}{4 \, a {\left | d \right |}^{\frac {2}{3}}} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} - \frac {1}{d^{\frac {1}{3}}} \right |}\right )}{2 \, a d^{\frac {2}{3}}} \]
1/2*sqrt(3)*arctan(1/3*sqrt(3)*d^(1/3)*(2*(-a/x + 1)^(2/3) + 1/d^(1/3)))/( a*abs(d)^(2/3)) - 1/4*log((-a/x + 1)^(4/3) + (-a/x + 1)^(2/3)/d^(1/3) + 1/ d^(2/3))/(a*abs(d)^(2/3)) + 1/2*log(abs((-a/x + 1)^(2/3) - 1/d^(1/3)))/(a* d^(2/3))
Timed out. \[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int -\frac {x\,\left (a-x\right )}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (d\,a^2-2\,d\,a\,x+\left (d-1\right )\,x^2\right )} \,d x \]