Integrand size = 41, antiderivative size = 249 \[ \int \frac {-a x+x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2-2 a x+(1-d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{\sqrt [3]{d} x^2+2 \left (-a x^2+x^3\right )^{2/3}}\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (-\sqrt [6]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (\sqrt [6]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{2 a \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{d}} \]
-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/3)*x^2/(d^(1/3)*x^2+2*(-a*x^2+x^3)^(2/3)) )/a/d^(1/3)+1/2*ln(-d^(1/6)*x+(-a*x^2+x^3)^(1/3))/a/d^(1/3)+1/2*ln(d^(1/6) *x+(-a*x^2+x^3)^(1/3))/a/d^(1/3)-1/4*ln(d^(1/3)*x^2-d^(1/6)*x*(-a*x^2+x^3) ^(1/3)+(-a*x^2+x^3)^(2/3))/a/d^(1/3)-1/4*ln(d^(1/3)*x^2+d^(1/6)*x*(-a*x^2+ x^3)^(1/3)+(-a*x^2+x^3)^(2/3))/a/d^(1/3)
Time = 0.41 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.90 \[ \int \frac {-a x+x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2-2 a x+(1-d) x^2\right )} \, dx=-\frac {x^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x^{2/3}}{\sqrt [3]{d} x^{2/3}+2 (-a+x)^{2/3}}\right )-2 \log \left (-\sqrt [6]{d} \sqrt [3]{x}+\sqrt [3]{-a+x}\right )-2 \log \left (\sqrt [6]{d} \sqrt [3]{x}+\sqrt [3]{-a+x}\right )+\log \left (\sqrt [3]{d} x^{2/3}-\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )+\log \left (\sqrt [3]{d} x^{2/3}+\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )\right )}{4 a \sqrt [3]{d} \left (x^2 (-a+x)\right )^{2/3}} \]
-1/4*(x^(4/3)*(-a + x)^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x^(2/3))/( d^(1/3)*x^(2/3) + 2*(-a + x)^(2/3))] - 2*Log[-(d^(1/6)*x^(1/3)) + (-a + x) ^(1/3)] - 2*Log[d^(1/6)*x^(1/3) + (-a + x)^(1/3)] + Log[d^(1/3)*x^(2/3) - d^(1/6)*x^(1/3)*(-a + x)^(1/3) + (-a + x)^(2/3)] + Log[d^(1/3)*x^(2/3) + d ^(1/6)*x^(1/3)*(-a + x)^(1/3) + (-a + x)^(2/3)]))/(a*d^(1/3)*(x^2*(-a + x) )^(2/3))
Time = 0.69 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {2027, 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.
| \(\displaystyle \int \frac {x^2-a x}{\left (x^2 (x-a)\right )^{2/3} \left (a^2-2 a x+(1-d) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x (x-a)}{\left (x^2 (x-a)\right )^{2/3} \left (a^2-2 a x+(1-d) x^2\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{4/3} (x-a)^{2/3} \int \frac {\sqrt [3]{x-a}}{\sqrt [3]{x} \left (a^2-2 x a+(1-d) x^2\right )}dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 1205 |
| \(\displaystyle \frac {x^{4/3} (x-a)^{2/3} \int \left (\frac {\sqrt [3]{x-a} (1-d)}{a \sqrt {d} \sqrt [3]{x} \left (-2 \sqrt {d} a+2 a-2 (1-d) x\right )}+\frac {\sqrt [3]{x-a} (1-d)}{a \sqrt {d} \sqrt [3]{x} \left (-2 \sqrt {d} a-2 a+2 (1-d) x\right )}\right )dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^{4/3} (x-a)^{2/3} \left (\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-a}}\right )}{2 a \sqrt [3]{d}}+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 a \sqrt [3]{d}}-\frac {\log \left (2 a \left (1-\sqrt {d}\right )-2 (1-d) x\right )}{4 a \sqrt [3]{d}}-\frac {\log \left (2 (1-d) x-2 a \left (\sqrt {d}+1\right )\right )}{4 a \sqrt [3]{d}}+\frac {3 \log \left (-\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{x}\right )}{4 a \sqrt [3]{d}}+\frac {3 \log \left (\sqrt [6]{d} \sqrt [3]{x}-\sqrt [3]{x-a}\right )}{4 a \sqrt [3]{d}}\right )}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
(x^(4/3)*(-a + x)^(2/3)*((Sqrt[3]*ArcTan[1/Sqrt[3] - (2*d^(1/6)*x^(1/3))/( Sqrt[3]*(-a + x)^(1/3))])/(2*a*d^(1/3)) + (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*d ^(1/6)*x^(1/3))/(Sqrt[3]*(-a + x)^(1/3))])/(2*a*d^(1/3)) - Log[2*a*(1 - Sq rt[d]) - 2*(1 - d)*x]/(4*a*d^(1/3)) - Log[-2*a*(1 + Sqrt[d]) + 2*(1 - d)*x ]/(4*a*d^(1/3)) + (3*Log[-(d^(1/6)*x^(1/3)) - (-a + x)^(1/3)])/(4*a*d^(1/3 )) + (3*Log[d^(1/6)*x^(1/3) - (-a + x)^(1/3)])/(4*a*d^(1/3))))/(-((a - x)* x^2))^(2/3)
3.28.17.3.1 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_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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 = 0.75 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.51
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x^{2}+2 \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}\right )}{3 d^{\frac {1}{3}} x^{2}}\right )+2 \ln \left (\frac {-d^{\frac {1}{3}} x^{2}+\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {d^{\frac {1}{3}} \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}+\left (-a +x \right ) \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}+d^{\frac {2}{3}} x^{2}}{x^{2}}\right )}{4 a \,d^{\frac {1}{3}}}\) | \(126\) |
1/4*(2*3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/3)*x^2+2*(-x^2*(a-x))^(2/3))/d^(1/ 3)/x^2)+2*ln((-d^(1/3)*x^2+(-x^2*(a-x))^(2/3))/x^2)-ln((d^(1/3)*(-x^2*(a-x ))^(2/3)+(-a+x)*(-x^2*(a-x))^(1/3)+d^(2/3)*x^2)/x^2))/a/d^(1/3)
Time = 0.26 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.43 \[ \int \frac {-a x+x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2-2 a x+(1-d) x^2\right )} \, dx=\left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {{\left (d + 2\right )} x^{2} + 2 \, a^{2} - 4 \, a x - \sqrt {3} {\left (d^{\frac {4}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )} d^{\frac {2}{3}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}}{{\left (d - 1\right )} x^{2} - a^{2} + 2 \, a x}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {1}{3}}}{x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{4 \, a d}, \frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}\right )}}{3 \, d^{\frac {1}{3}} x^{2}}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {1}{3}}}{x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{4 \, a d}\right ] \]
[1/4*(sqrt(3)*d*sqrt(-1/d^(2/3))*log(-((d + 2)*x^2 + 2*a^2 - 4*a*x - sqrt( 3)*(d^(4/3)*x^2 + 2*(-a*x^2 + x^3)^(1/3)*(a - x)*d^(2/3) + (-a*x^2 + x^3)^ (2/3)*d)*sqrt(-1/d^(2/3)) - 3*(-a*x^2 + x^3)^(2/3)*d^(2/3))/((d - 1)*x^2 - a^2 + 2*a*x)) - d^(2/3)*log((d^(2/3)*x^2 - (-a*x^2 + x^3)^(1/3)*(a - x) + (-a*x^2 + x^3)^(2/3)*d^(1/3))/x^2) + 2*d^(2/3)*log(-(d^(1/3)*x^2 - (-a*x^ 2 + x^3)^(2/3))/x^2))/(a*d), 1/4*(2*sqrt(3)*d^(2/3)*arctan(1/3*sqrt(3)*(d^ (1/3)*x^2 + 2*(-a*x^2 + x^3)^(2/3))/(d^(1/3)*x^2)) - d^(2/3)*log((d^(2/3)* x^2 - (-a*x^2 + x^3)^(1/3)*(a - x) + (-a*x^2 + x^3)^(2/3)*d^(1/3))/x^2) + 2*d^(2/3)*log(-(d^(1/3)*x^2 - (-a*x^2 + x^3)^(2/3))/x^2))/(a*d)]
\[ \int \frac {-a x+x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2-2 a x+(1-d) x^2\right )} \, dx=- \int \frac {x^{2}}{- a^{2} \left (- a x^{2} + x^{3}\right )^{\frac {2}{3}} + 2 a x \left (- a x^{2} + x^{3}\right )^{\frac {2}{3}} + d x^{2} \left (- a x^{2} + x^{3}\right )^{\frac {2}{3}} - x^{2} \left (- a x^{2} + x^{3}\right )^{\frac {2}{3}}}\, dx - \int \left (- \frac {a x}{- a^{2} \left (- a x^{2} + x^{3}\right )^{\frac {2}{3}} + 2 a x \left (- a x^{2} + x^{3}\right )^{\frac {2}{3}} + d x^{2} \left (- a x^{2} + x^{3}\right )^{\frac {2}{3}} - x^{2} \left (- a x^{2} + x^{3}\right )^{\frac {2}{3}}}\right )\, dx \]
-Integral(x**2/(-a**2*(-a*x**2 + x**3)**(2/3) + 2*a*x*(-a*x**2 + x**3)**(2 /3) + d*x**2*(-a*x**2 + x**3)**(2/3) - x**2*(-a*x**2 + x**3)**(2/3)), x) - Integral(-a*x/(-a**2*(-a*x**2 + x**3)**(2/3) + 2*a*x*(-a*x**2 + x**3)**(2 /3) + d*x**2*(-a*x**2 + x**3)**(2/3) - x**2*(-a*x**2 + x**3)**(2/3)), x)
\[ \int \frac {-a x+x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2-2 a x+(1-d) x^2\right )} \, dx=\int { \frac {a x - x^{2}}{\left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}} {\left ({\left (d - 1\right )} x^{2} - a^{2} + 2 \, a x\right )}} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.40 \[ \int \frac {-a x+x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2-2 a x+(1-d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + d^{\frac {1}{3}}\right )}}{3 \, d^{\frac {1}{3}}}\right )}{2 \, a d^{\frac {1}{3}}} - \frac {\log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + d^{\frac {1}{3}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + d^{\frac {2}{3}}\right )}{4 \, a d^{\frac {1}{3}}} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} - d^{\frac {1}{3}} \right |}\right )}{2 \, a d^{\frac {1}{3}}} \]
1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-a/x + 1)^(2/3) + d^(1/3))/d^(1/3))/(a* d^(1/3)) - 1/4*log((-a/x + 1)^(4/3) + d^(1/3)*(-a/x + 1)^(2/3) + d^(2/3))/ (a*d^(1/3)) + 1/2*log(abs((-a/x + 1)^(2/3) - d^(1/3)))/(a*d^(1/3))
Timed out. \[ \int \frac {-a x+x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2-2 a x+(1-d) x^2\right )} \, dx=\int \frac {a\,x-x^2}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (-a^2+2\,a\,x+\left (d-1\right )\,x^2\right )} \,d x \]