Integrand size = 23, antiderivative size = 146 \[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{-a x^2+x^3}}\right )}{a \sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{a \sqrt [3]{d}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{2 a \sqrt [3]{d}} \]
3^(1/2)*arctan(3^(1/2)*d^(1/3)*x/(d^(1/3)*x+2*(-a*x^2+x^3)^(1/3)))/a/d^(1/ 3)-ln(-d^(1/3)*x+(-a*x^2+x^3)^(1/3))/a/d^(1/3)+1/2*ln(d^(2/3)*x^2+d^(1/3)* x*(-a*x^2+x^3)^(1/3)+(-a*x^2+x^3)^(2/3))/a/d^(1/3)
Time = 0.31 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=\frac {x^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{d} \sqrt [3]{x}+2 \sqrt [3]{-a+x}}\right )-2 \log \left (-\sqrt [3]{d} \sqrt [3]{x}+\sqrt [3]{-a+x}\right )+\log \left (d^{2/3} x^{2/3}+\sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )\right )}{2 a \sqrt [3]{d} \sqrt [3]{x^2 (-a+x)}} \]
(x^(2/3)*(-a + x)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x^(1/3))/(d^(1/ 3)*x^(1/3) + 2*(-a + x)^(1/3))] - 2*Log[-(d^(1/3)*x^(1/3)) + (-a + x)^(1/3 )] + Log[d^(2/3)*x^(2/3) + d^(1/3)*x^(1/3)*(-a + x)^(1/3) + (-a + x)^(2/3) ]))/(2*a*d^(1/3)*(x^2*(-a + x))^(1/3))
Time = 0.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2467, 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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{x^2 (x-a)} (a+(d-1) x)} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x-a} \int \frac {1}{x^{2/3} \sqrt [3]{x-a} (a-(1-d) x)}dx}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x-a} \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{d}}+\frac {\log (a-(1-d) x)}{2 a \sqrt [3]{d}}-\frac {3 \log \left (\frac {\sqrt [3]{x-a}}{\sqrt [3]{d}}-\sqrt [3]{x}\right )}{2 a \sqrt [3]{d}}\right )}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
(x^(2/3)*(-a + x)^(1/3)*(-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(-a + x)^(1/3))/ (Sqrt[3]*d^(1/3)*x^(1/3))])/(a*d^(1/3))) + Log[a - (1 - d)*x]/(2*a*d^(1/3) ) - (3*Log[-x^(1/3) + (-a + x)^(1/3)/d^(1/3)])/(2*a*d^(1/3))))/(-((a - x)* x^2))^(1/3)
3.21.40.3.1 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 = 0.75 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.78
| method | result | size |
| pseudoelliptic | \(-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )+\ln \left (\frac {-d^{\frac {1}{3}} x +\left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (-\left (a -x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}}{d^{\frac {1}{3}} a}\) | \(114\) |
-1/d^(1/3)*(3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/3)*x+2*(-(a-x)*x^2)^(1/3))/d^ (1/3)/x)+ln((-d^(1/3)*x+(-(a-x)*x^2)^(1/3))/x)-1/2*ln((d^(2/3)*x^2+d^(1/3) *(-(a-x)*x^2)^(1/3)*x+(-(a-x)*x^2)^(2/3))/x^2))/a
Time = 0.30 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=\left [\frac {\sqrt {3} d \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \log \left (-\frac {{\left (d + 2\right )} x^{2} - 2 \, a x - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {2}{3}} x - \sqrt {3} {\left (\left (-d\right )^{\frac {1}{3}} d x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d x + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} \left (-d\right )^{\frac {2}{3}}\right )} \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{{\left (d - 1\right )} x^{2} + a x}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) + \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, a d}, -\frac {2 \, \sqrt {3} d \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \arctan \left (-\frac {\sqrt {3} {\left (\left (-d\right )^{\frac {1}{3}} x - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{3 \, x}\right ) + 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, a d}\right ] \]
[1/2*(sqrt(3)*d*sqrt((-d)^(1/3)/d)*log(-((d + 2)*x^2 - 2*a*x - 3*(-a*x^2 + x^3)^(1/3)*(-d)^(2/3)*x - sqrt(3)*((-d)^(1/3)*d*x^2 - (-a*x^2 + x^3)^(1/3 )*d*x + 2*(-a*x^2 + x^3)^(2/3)*(-d)^(2/3))*sqrt((-d)^(1/3)/d))/((d - 1)*x^ 2 + a*x)) - 2*(-d)^(2/3)*log(((-d)^(1/3)*x + (-a*x^2 + x^3)^(1/3))/x) + (- d)^(2/3)*log(((-d)^(2/3)*x^2 - (-a*x^2 + x^3)^(1/3)*(-d)^(1/3)*x + (-a*x^2 + x^3)^(2/3))/x^2))/(a*d), -1/2*(2*sqrt(3)*d*sqrt(-(-d)^(1/3)/d)*arctan(- 1/3*sqrt(3)*((-d)^(1/3)*x - 2*(-a*x^2 + x^3)^(1/3))*sqrt(-(-d)^(1/3)/d)/x) + 2*(-d)^(2/3)*log(((-d)^(1/3)*x + (-a*x^2 + x^3)^(1/3))/x) - (-d)^(2/3)* log(((-d)^(2/3)*x^2 - (-a*x^2 + x^3)^(1/3)*(-d)^(1/3)*x + (-a*x^2 + x^3)^( 2/3))/x^2))/(a*d)]
\[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (- a + x\right )} \left (a + d x - x\right )}\, dx \]
\[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=\int { \frac {1}{\left (-{\left (a - x\right )} x^{2}\right )^{\frac {1}{3}} {\left ({\left (d - 1\right )} x + a\right )}} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}}{3 \, d^{\frac {1}{3}}}\right )}{a d^{\frac {1}{3}}} + \frac {\log \left (d^{\frac {2}{3}} + d^{\frac {1}{3}} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}\right )}{2 \, a d^{\frac {1}{3}}} - \frac {\log \left ({\left | -d^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \right |}\right )}{a d^{\frac {1}{3}}} \]
-sqrt(3)*arctan(1/3*sqrt(3)*(d^(1/3) + 2*(-a/x + 1)^(1/3))/d^(1/3))/(a*d^( 1/3)) + 1/2*log(d^(2/3) + d^(1/3)*(-a/x + 1)^(1/3) + (-a/x + 1)^(2/3))/(a* d^(1/3)) - log(abs(-d^(1/3) + (-a/x + 1)^(1/3)))/(a*d^(1/3))
Timed out. \[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=\int \frac {1}{\left (a+x\,\left (d-1\right )\right )\,{\left (-x^2\,\left (a-x\right )\right )}^{1/3}} \,d x \]