Integrand size = 27, antiderivative size = 139 \[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (-a d+(-1+d) x)} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}}\right )}{a \sqrt [3]{d}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{-a x^2+x^3}\right )}{a \sqrt [3]{d}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{-a x^2+x^3}+d^{2/3} \left (-a x^2+x^3\right )^{2/3}\right )}{2 a \sqrt [3]{d}} \]
3^(1/2)*arctan(3^(1/2)*x/(x+2*d^(1/3)*(-a*x^2+x^3)^(1/3)))/a/d^(1/3)+ln(x- d^(1/3)*(-a*x^2+x^3)^(1/3))/a/d^(1/3)-1/2*ln(x^2+d^(1/3)*x*(-a*x^2+x^3)^(1 /3)+d^(2/3)*(-a*x^2+x^3)^(2/3))/a/d^(1/3)
Time = 0.33 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.09 \[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (-a d+(-1+d) x)} \, dx=\frac {x^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{d} \sqrt [3]{-a+x}}\right )+2 \log \left (\sqrt [3]{x}-\sqrt [3]{d} \sqrt [3]{-a+x}\right )-\log \left (x^{2/3}+\sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+d^{2/3} (-a+x)^{2/3}\right )\right )}{2 a \sqrt [3]{d} \left (x^2 (-a+x)\right )^{2/3}} \]
(x^(4/3)*(-a + x)^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*d ^(1/3)*(-a + x)^(1/3))] + 2*Log[x^(1/3) - d^(1/3)*(-a + x)^(1/3)] - Log[x^ (2/3) + d^(1/3)*x^(1/3)*(-a + x)^(1/3) + d^(2/3)*(-a + x)^(2/3)]))/(2*a*d^ (1/3)*(x^2*(-a + x))^(2/3))
Time = 0.32 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2467, 25, 102}
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.
| \(\displaystyle \int \frac {x}{\left (x^2 (x-a)\right )^{2/3} ((d-1) x-a d)} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{4/3} (x-a)^{2/3} \int -\frac {1}{\sqrt [3]{x} (x-a)^{2/3} (a d+(1-d) x)}dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{4/3} (x-a)^{2/3} \int \frac {1}{\sqrt [3]{x} (x-a)^{2/3} (a d+(1-d) x)}dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle -\frac {x^{4/3} (x-a)^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{d}}+\frac {\log (a d+(1-d) x)}{2 a \sqrt [3]{d}}-\frac {3 \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{d}}-\sqrt [3]{x-a}\right )}{2 a \sqrt [3]{d}}\right )}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
-((x^(4/3)*(-a + x)^(2/3)*(-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt [3]*d^(1/3)*(-a + x)^(1/3))])/(a*d^(1/3))) + Log[a*d + (1 - d)*x]/(2*a*d^( 1/3)) - (3*Log[x^(1/3)/d^(1/3) - (-a + x)^(1/3)])/(2*a*d^(1/3))))/(-((a - x)*x^2))^(2/3))
3.20.63.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.) *(x_))), x_] :> With[{q = Rt[(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])* q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q *(a + b*x)^(1/3) - (c + d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, x]
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 = 0.79 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.95
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x +2 \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x +\left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{2}+\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (-\left (a -x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 a d \left (\frac {1}{d}\right )^{\frac {2}{3}}}\) | \(132\) |
1/2*(-2*3^(1/2)*arctan(1/3*3^(1/2)*((1/d)^(1/3)*x+2*(-(a-x)*x^2)^(1/3))/(1 /d)^(1/3)/x)+2*ln((-(1/d)^(1/3)*x+(-(a-x)*x^2)^(1/3))/x)-ln(((1/d)^(2/3)*x ^2+(1/d)^(1/3)*(-(a-x)*x^2)^(1/3)*x+(-(a-x)*x^2)^(2/3))/x^2))/a/d/(1/d)^(2 /3)
Time = 0.24 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.43 \[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (-a d+(-1+d) x)} \, dx=\left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (\frac {2 \, a d x - {\left (2 \, d + 1\right )} x^{2} - \sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} + 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {1}{3}} x}{a d x - {\left (d - 1\right )} x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {2}{3}} x - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right )}{2 \, a d}, -\frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} x + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}}\right )}}{3 \, d^{\frac {1}{3}} x}\right ) - 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {2}{3}} x - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x}\right ) + d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right )}{2 \, a d}\right ] \]
[1/2*(sqrt(3)*d*sqrt(-1/d^(2/3))*log((2*a*d*x - (2*d + 1)*x^2 - sqrt(3)*(d ^(1/3)*x^2 + (-a*x^2 + x^3)^(1/3)*d^(2/3)*x - 2*(-a*x^2 + x^3)^(2/3)*d)*sq rt(-1/d^(2/3)) + 3*(-a*x^2 + x^3)^(1/3)*d^(1/3)*x)/(a*d*x - (d - 1)*x^2)) + 2*d^(2/3)*log(-(d^(2/3)*x - (-a*x^2 + x^3)^(1/3)*d)/x) - d^(2/3)*log((d^ (1/3)*x^2 + (-a*x^2 + x^3)^(1/3)*d^(2/3)*x + (-a*x^2 + x^3)^(2/3)*d)/x^2)) /(a*d), -1/2*(2*sqrt(3)*d^(2/3)*arctan(1/3*sqrt(3)*(d^(1/3)*x + 2*(-a*x^2 + x^3)^(1/3)*d^(2/3))/(d^(1/3)*x)) - 2*d^(2/3)*log(-(d^(2/3)*x - (-a*x^2 + x^3)^(1/3)*d)/x) + d^(2/3)*log((d^(1/3)*x^2 + (-a*x^2 + x^3)^(1/3)*d^(2/3 )*x + (-a*x^2 + x^3)^(2/3)*d)/x^2))/(a*d)]
\[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (-a d+(-1+d) x)} \, dx=\int \frac {x}{\left (x^{2} \left (- a + x\right )\right )^{\frac {2}{3}} \left (- a d + d x - x\right )}\, dx \]
\[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (-a d+(-1+d) x)} \, dx=\int { -\frac {x}{\left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}} {\left (a d - {\left (d - 1\right )} x\right )}} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.77 \[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (-a d+(-1+d) x)} \, dx=-\frac {\sqrt {3} {\left | d \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} d^{\frac {1}{3}} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + \frac {1}{d^{\frac {1}{3}}}\right )}\right )}{a d} - \frac {{\left | d \right |}^{\frac {2}{3}} \log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{d^{\frac {1}{3}}} + \frac {1}{d^{\frac {2}{3}}}\right )}{2 \, a d} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} - \frac {1}{d^{\frac {1}{3}}} \right |}\right )}{a d^{\frac {1}{3}}} \]
-sqrt(3)*abs(d)^(2/3)*arctan(1/3*sqrt(3)*d^(1/3)*(2*(-a/x + 1)^(1/3) + 1/d ^(1/3)))/(a*d) - 1/2*abs(d)^(2/3)*log((-a/x + 1)^(2/3) + (-a/x + 1)^(1/3)/ d^(1/3) + 1/d^(2/3))/(a*d) + log(abs((-a/x + 1)^(1/3) - 1/d^(1/3)))/(a*d^( 1/3))
Timed out. \[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (-a d+(-1+d) x)} \, dx=\int -\frac {x}{\left (a\,d-x\,\left (d-1\right )\right )\,{\left (-x^2\,\left (a-x\right )\right )}^{2/3}} \,d x \]