Integrand size = 36, antiderivative size = 267 \[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (b-a d+(-1+d) x)} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} b-\sqrt {3} x}{b-x-2 \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{(a-b) \sqrt [3]{d}}+\frac {\log \left (b-x+\sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{(a-b) \sqrt [3]{d}}-\frac {\log \left (b^2-2 b x+x^2+\left (-b \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}+d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{2 (a-b) \sqrt [3]{d}} \]
3^(1/2)*arctan((3^(1/2)*b-x*3^(1/2))/(b-x-2*d^(1/3)*(-a*b^2+(2*a*b+b^2)*x+ (-a-2*b)*x^2+x^3)^(1/3)))/(a-b)/d^(1/3)+ln(b-x+d^(1/3)*(-a*b^2+(2*a*b+b^2) *x+(-a-2*b)*x^2+x^3)^(1/3))/(a-b)/d^(1/3)-1/2*ln(b^2-2*b*x+x^2+(-b*d^(1/3) +d^(1/3)*x)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/3)+d^(2/3)*(-a*b^2+ (2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(2/3))/(a-b)/d^(1/3)
Time = 0.21 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.65 \[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (b-a d+(-1+d) x)} \, dx=-\frac {(b-x)^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}}{\sqrt {3}}\right )-2 \log \left (1+\frac {\sqrt [3]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )+\log \left (1-\frac {\sqrt [3]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {d^{2/3} (-a+x)^{2/3}}{(b-x)^{2/3}}\right )\right )}{2 (a-b) \sqrt [3]{d} \left ((b-x)^2 (-a+x)\right )^{2/3}} \]
-1/2*((b - x)^(4/3)*(-a + x)^(2/3)*(2*Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*(-a + x)^(1/3))/(b - x)^(1/3))/Sqrt[3]] - 2*Log[1 + (d^(1/3)*(-a + x)^(1/3))/(b - x)^(1/3)] + Log[1 - (d^(1/3)*(-a + x)^(1/3))/(b - x)^(1/3) + (d^(2/3)*( -a + x)^(2/3))/(b - x)^(2/3)]))/((a - b)*d^(1/3)*((b - x)^2*(-a + x))^(2/3 ))
Time = 0.66 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.63, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {7270, 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.
| \(\displaystyle \int \frac {x-b}{\left ((x-a) (x-b)^2\right )^{2/3} (-a d+b+(d-1) x)} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \int \frac {1}{(x-a)^{2/3} \sqrt [3]{x-b} (b-a d-(1-d) x)}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \left (\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{d} (a-b)}-\frac {\log (-a d+b-(1-d) x)}{2 \sqrt [3]{d} (a-b)}+\frac {3 \log \left (\frac {\sqrt [3]{x-b}}{\sqrt [3]{d}}-\sqrt [3]{x-a}\right )}{2 \sqrt [3]{d} (a-b)}\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
((-a + x)^(2/3)*(-b + x)^(4/3)*((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(-b + x)^(1 /3))/(Sqrt[3]*d^(1/3)*(-a + x)^(1/3))])/((a - b)*d^(1/3)) - Log[b - a*d - (1 - d)*x]/(2*(a - b)*d^(1/3)) + (3*Log[-(-a + x)^(1/3) + (-b + x)^(1/3)/d ^(1/3)])/(2*(a - b)*d^(1/3))))/(-((a - x)*(b - x)^2))^(2/3)
3.28.80.3.1 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[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
\[\int \frac {-b +x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (b -a d +\left (-1+d \right ) x \right )}d x\]
Time = 0.30 (sec) , antiderivative size = 715, normalized size of antiderivative = 2.68 \[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (b-a d+(-1+d) x)} \, dx=\left [-\frac {\sqrt {3} d \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \log \left (\frac {2 \, a b d + {\left (2 \, d + 1\right )} x^{2} + b^{2} - 2 \, {\left ({\left (a + b\right )} d + b\right )} x + \sqrt {3} {\left ({\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} \left (-d\right )^{\frac {2}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d + {\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}} - 3 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} \left (-d\right )^{\frac {1}{3}}}{a b d + {\left (d - 1\right )} x^{2} - b^{2} - {\left ({\left (a + b\right )} d - 2 \, b\right )} x}\right ) + \left (-d\right )^{\frac {2}{3}} \log \left (-\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} \left (-d\right )^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d + {\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (-\frac {{\left (b - x\right )} \left (-d\right )^{\frac {2}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b - x}\right )}{2 \, {\left (a - b\right )} d}, -\frac {2 \, \sqrt {3} d \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \arctan \left (-\frac {\sqrt {3} {\left ({\left (b - x\right )} \left (-d\right )^{\frac {1}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {2}{3}}\right )} \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{3 \, {\left (b - x\right )}}\right ) + \left (-d\right )^{\frac {2}{3}} \log \left (-\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} \left (-d\right )^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d + {\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (-\frac {{\left (b - x\right )} \left (-d\right )^{\frac {2}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b - x}\right )}{2 \, {\left (a - b\right )} d}\right ] \]
[-1/2*(sqrt(3)*d*sqrt((-d)^(1/3)/d)*log((2*a*b*d + (2*d + 1)*x^2 + b^2 - 2 *((a + b)*d + b)*x + sqrt(3)*((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2 )*x)^(1/3)*(b - x)*(-d)^(2/3) + 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d + (b^2 - 2*b*x + x^2)*(-d)^(1/3))*sqrt((-d)^(1/3)/d) - 3* (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b - x)*(-d)^(1/3)) /(a*b*d + (d - 1)*x^2 - b^2 - ((a + b)*d - 2*b)*x)) + (-d)^(2/3)*log(-((-a *b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b - x)*(-d)^(2/3) - ( -a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d + (b^2 - 2*b*x + x ^2)*(-d)^(1/3))/(b^2 - 2*b*x + x^2)) - 2*(-d)^(2/3)*log(-((b - x)*(-d)^(2/ 3) + (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*d)/(b - x)))/( (a - b)*d), -1/2*(2*sqrt(3)*d*sqrt(-(-d)^(1/3)/d)*arctan(-1/3*sqrt(3)*((b - x)*(-d)^(1/3) + 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3) *(-d)^(2/3))*sqrt(-(-d)^(1/3)/d)/(b - x)) + (-d)^(2/3)*log(-((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b - x)*(-d)^(2/3) - (-a*b^2 - ( a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d + (b^2 - 2*b*x + x^2)*(-d)^( 1/3))/(b^2 - 2*b*x + x^2)) - 2*(-d)^(2/3)*log(-((b - x)*(-d)^(2/3) + (-a*b ^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*d)/(b - x)))/((a - b)*d) ]
\[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (b-a d+(-1+d) x)} \, dx=\int \frac {- b + x}{\left (\left (- a + x\right ) \left (- b + x\right )^{2}\right )^{\frac {2}{3}} \left (- a d + b + d x - x\right )}\, dx \]
\[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (b-a d+(-1+d) x)} \, dx=\int { \frac {b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (a d - {\left (d - 1\right )} x - b\right )}} \,d x } \]
\[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (b-a d+(-1+d) x)} \, dx=\int { \frac {b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (a d - {\left (d - 1\right )} x - b\right )}} \,d x } \]
Timed out. \[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (b-a d+(-1+d) x)} \, dx=\int -\frac {b-x}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (b-a\,d+x\,\left (d-1\right )\right )} \,d x \]