Integrand size = 35, antiderivative size = 243 \[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}}\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (x-\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (x+\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a \sqrt [3]{d}}-\frac {\log \left (x^2-\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{d}}-\frac {\log \left (x^2+\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{d}} \]
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^(1/ 3)+1/2*ln(x-d^(1/6)*(-a*x^2+x^3)^(1/3))/a/d^(1/3)+1/2*ln(x+d^(1/6)*(-a*x^2 +x^3)^(1/3))/a/d^(1/3)-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^(1/3)-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^(1/3)
Time = 0.41 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {x^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{d} (-a+x)^{2/3}}\right )+2 \log \left (\sqrt [3]{x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )+2 \log \left (\sqrt [3]{x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )-\log \left (x^{2/3}-\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )-\log \left (x^{2/3}+\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )\right )}{4 a \sqrt [3]{d} \sqrt [3]{x^2 (-a+x)}} \]
(x^(2/3)*(-a + x)^(1/3)*(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)]))/(4*a*d^(1/3)*(x^2*(-a + x))^(1/3))
Time = 0.55 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, 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.
| \(\displaystyle \int \frac {x}{\sqrt [3]{x^2 (x-a)} \left (a^2 d-2 a d x+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x-a} \int \frac {\sqrt [3]{x}}{\sqrt [3]{x-a} \left (d a^2-2 d x a-(1-d) x^2\right )}dx}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
| \(\Big \downarrow \) 1205 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x-a} \int \left (\frac {\sqrt [3]{x} (d-1)}{a \sqrt {d} \sqrt [3]{x-a} \left (-2 d a-2 \sqrt {d} a-2 (1-d) x\right )}+\frac {\sqrt [3]{x} (d-1)}{a \sqrt {d} \sqrt [3]{x-a} \left (2 d a-2 \sqrt {d} a+2 (1-d) x\right )}\right )dx}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x-a} \left (\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x}}\right )}{2 a \sqrt [3]{d}}+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 a \sqrt [3]{d}}-\frac {\log \left (-2 a \sqrt {d} \left (\sqrt {d}+1\right )-2 (1-d) x\right )}{4 a \sqrt [3]{d}}-\frac {\log \left (2 (1-d) x-2 a \left (1-\sqrt {d}\right ) \sqrt {d}\right )}{4 a \sqrt [3]{d}}+\frac {3 \log \left (-\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x}\right )}{4 a \sqrt [3]{d}}+\frac {3 \log \left (\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x}\right )}{4 a \sqrt [3]{d}}\right )}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
(x^(2/3)*(-a + x)^(1/3)*((Sqrt[3]*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(-a + x)^( 1/3))/(Sqrt[3]*x^(1/3))])/(2*a*d^(1/3)) + (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*d ^(1/6)*(-a + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(2*a*d^(1/3)) - Log[-2*a*(1 + S qrt[d])*Sqrt[d] - 2*(1 - d)*x]/(4*a*d^(1/3)) - Log[-2*a*(1 - Sqrt[d])*Sqrt [d] + 2*(1 - d)*x]/(4*a*d^(1/3)) + (3*Log[-x^(1/3) - d^(1/6)*(-a + x)^(1/3 )])/(4*a*d^(1/3)) + (3*Log[-x^(1/3) + d^(1/6)*(-a + x)^(1/3)])/(4*a*d^(1/3 ))))/(-((a - x)*x^2))^(1/3)
3.27.83.3.1 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 = 0.33 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.58
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+2 \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}+\left (-a +x \right ) \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}+\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{2}}{x^{2}}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{4 \left (\frac {1}{d}\right )^{\frac {2}{3}} a d}\) | \(141\) |
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)-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)+2*ln((-(1/d)^(1/3)*x^2+(-x^2*(a-x))^(2/3))/x^2)) /(1/d)^(2/3)/a/d
Time = 0.29 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.53 \[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (\frac {2 \, a^{2} d - 4 \, a d x + {\left (2 \, d + 1\right )} x^{2} + \sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a d - d x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {1}{3}}}{a^{2} d - 2 \, a d x + {\left (d - 1\right )} x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a d - d x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}}{x^{2}}\right )}{4 \, a d}, -\frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}\right )}}{3 \, d^{\frac {1}{3}} x^{2}}\right ) - 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right ) + d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a d - d x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}}{x^{2}}\right )}{4 \, a d}\right ] \]
[1/4*(sqrt(3)*d*sqrt(-1/d^(2/3))*log((2*a^2*d - 4*a*d*x + (2*d + 1)*x^2 + sqrt(3)*(d^(1/3)*x^2 + 2*(-a*x^2 + x^3)^(1/3)*(a*d - d*x) + (-a*x^2 + x^3) ^(2/3)*d^(2/3))*sqrt(-1/d^(2/3)) - 3*(-a*x^2 + x^3)^(2/3)*d^(1/3))/(a^2*d - 2*a*d*x + (d - 1)*x^2)) + 2*d^(2/3)*log(-(d^(2/3)*x^2 - (-a*x^2 + x^3)^( 2/3)*d)/x^2) - d^(2/3)*log((d^(1/3)*x^2 - (-a*x^2 + x^3)^(1/3)*(a*d - d*x) + (-a*x^2 + x^3)^(2/3)*d^(2/3))/x^2))/(a*d), -1/4*(2*sqrt(3)*d^(2/3)*arct an(1/3*sqrt(3)*(d^(1/3)*x^2 + 2*(-a*x^2 + x^3)^(2/3)*d^(2/3))/(d^(1/3)*x^2 )) - 2*d^(2/3)*log(-(d^(2/3)*x^2 - (-a*x^2 + x^3)^(2/3)*d)/x^2) + d^(2/3)* log((d^(1/3)*x^2 - (-a*x^2 + x^3)^(1/3)*(a*d - d*x) + (-a*x^2 + x^3)^(2/3) *d^(2/3))/x^2))/(a*d)]
\[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int \frac {x}{\sqrt [3]{x^{2} \left (- a + x\right )} \left (a^{2} d - 2 a d x + d x^{2} - x^{2}\right )}\, dx \]
\[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int { \frac {x}{{\left (a^{2} d - 2 \, a d x + {\left (d - 1\right )} x^{2}\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {1}{3}}} \,d x } \]
Time = 0.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.44 \[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} {\left | d \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} d^{\frac {1}{3}} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \frac {1}{d^{\frac {1}{3}}}\right )}\right )}{2 \, a d} - \frac {{\left | d \right |}^{\frac {2}{3}} \log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{d^{\frac {1}{3}}} + \frac {1}{d^{\frac {2}{3}}}\right )}{4 \, a d} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} - \frac {1}{d^{\frac {1}{3}}} \right |}\right )}{2 \, a d^{\frac {1}{3}}} \]
-1/2*sqrt(3)*abs(d)^(2/3)*arctan(1/3*sqrt(3)*d^(1/3)*(2*(-a/x + 1)^(2/3) + 1/d^(1/3)))/(a*d) - 1/4*abs(d)^(2/3)*log((-a/x + 1)^(4/3) + (-a/x + 1)^(2 /3)/d^(1/3) + 1/d^(2/3))/(a*d) + 1/2*log(abs((-a/x + 1)^(2/3) - 1/d^(1/3)) )/(a*d^(1/3))
Timed out. \[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int \frac {x}{{\left (-x^2\,\left (a-x\right )\right )}^{1/3}\,\left (d\,a^2-2\,d\,a\,x+\left (d-1\right )\,x^2\right )} \,d x \]