Integrand size = 52, antiderivative size = 569 \[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} (a-b)^{2/3} d^{4/3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a d-b d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{-2 a \sqrt [3]{a-b} d^{2/3}+2 \sqrt [3]{a-b} d^{2/3} x+\sqrt [3]{a d-b d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{2 ((a-b) d)^{5/3}}+\frac {(a-b)^{2/3} d^{4/3} \log \left (a (a d-b d)^{2/3}-(a d-b d)^{2/3} x+(a-b)^{2/3} \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{2 ((a-b) d)^{5/3}}-\frac {(a-b)^{2/3} d^{4/3} \log \left (a^3 d \sqrt [3]{a d-b d}-a^2 b d \sqrt [3]{a d-b d}-2 a^2 d \sqrt [3]{a d-b d} x+2 a b d \sqrt [3]{a d-b d} x+a d \sqrt [3]{a d-b d} x^2-b d \sqrt [3]{a d-b d} x^2+\left (-a (a-b)^{2/3} \sqrt [3]{d} (a d-b d)^{2/3}+(a-b)^{2/3} \sqrt [3]{d} (a d-b d)^{2/3} x\right ) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}+(a-b)^{4/3} d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{4 ((a-b) d)^{5/3}} \]
1/2*3^(1/2)*(a-b)^(2/3)*d^(4/3)*arctan(3^(1/2)*(a*d-b*d)^(1/3)*(-a*b^2+(2* a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/3)/(-2*a*(a-b)^(1/3)*d^(2/3)+2*(a-b)^(1/3) *d^(2/3)*x+(a*d-b*d)^(1/3)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/3))) /((a-b)*d)^(5/3)+1/2*(a-b)^(2/3)*d^(4/3)*ln(a*(a*d-b*d)^(2/3)-(a*d-b*d)^(2 /3)*x+(a-b)^(2/3)*d^(1/3)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/3))/( (a-b)*d)^(5/3)-1/4*(a-b)^(2/3)*d^(4/3)*ln(a^3*d*(a*d-b*d)^(1/3)-a^2*b*d*(a *d-b*d)^(1/3)-2*a^2*d*(a*d-b*d)^(1/3)*x+2*a*b*d*(a*d-b*d)^(1/3)*x+a*d*(a*d -b*d)^(1/3)*x^2-b*d*(a*d-b*d)^(1/3)*x^2+(-a*(a-b)^(2/3)*d^(1/3)*(a*d-b*d)^ (2/3)+(a-b)^(2/3)*d^(1/3)*(a*d-b*d)^(2/3)*x)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b )*x^2+x^3)^(1/3)+(a-b)^(4/3)*d^(2/3)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^ 3)^(2/3))/((a-b)*d)^(5/3)
Time = 0.34 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.51 \[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=-\frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b-x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b-x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [6]{d}-\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 \log \left (\sqrt [6]{d}+\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+\log \left (\sqrt [3]{d}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+\log \left (\sqrt [3]{d}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )\right )}{4 (a-b) \sqrt [3]{d} \sqrt [3]{(b-x)^2 (-a+x)}} \]
Integrate[(-b + x)/(((-a + x)*(-b + x)^2)^(1/3)*(-b^2 + a^2*d + 2*(b - a*d )*x + (-1 + d)*x^2)),x]
-1/4*((b - x)^(2/3)*(-a + x)^(1/3)*(2*Sqrt[3]*ArcTan[(1 - (2*(b - x)^(1/3) )/(d^(1/6)*(-a + x)^(1/3)))/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + (2*(b - x)^(1 /3))/(d^(1/6)*(-a + x)^(1/3)))/Sqrt[3]] - 2*Log[d^(1/6) - (b - x)^(1/3)/(- a + x)^(1/3)] - 2*Log[d^(1/6) + (b - x)^(1/3)/(-a + x)^(1/3)] + Log[d^(1/3 ) + (b - x)^(2/3)/(-a + x)^(2/3) - (d^(1/6)*(b - x)^(1/3))/(-a + x)^(1/3)] + Log[d^(1/3) + (b - x)^(2/3)/(-a + x)^(2/3) + (d^(1/6)*(b - x)^(1/3))/(- a + x)^(1/3)]))/((a - b)*d^(1/3)*((b - x)^2*(-a + x))^(1/3))
Time = 0.98 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.59, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {7270, 25, 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.
| \(\displaystyle \int \frac {x-b}{\sqrt [3]{(x-a) (x-b)^2} \left (a^2 d+2 x (b-a d)-b^2+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt [3]{x-a} (x-b)^{2/3} \int -\frac {\sqrt [3]{x-b}}{\sqrt [3]{x-a} \left (-d a^2+b^2+(1-d) x^2-2 (b-a d) x\right )}dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \int \frac {\sqrt [3]{x-b}}{\sqrt [3]{x-a} \left (-d a^2+b^2+(1-d) x^2-2 (b-a d) x\right )}dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 1205 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \int \left (\frac {\sqrt [3]{x-b} (1-d)}{(a-b) \sqrt {d} \sqrt [3]{x-a} \left (-2 \sqrt {d} (a-b)+2 b-2 a d-2 (1-d) x\right )}+\frac {\sqrt [3]{x-b} (1-d)}{(a-b) \sqrt {d} \sqrt [3]{x-a} \left (-2 \sqrt {d} (a-b)-2 b+2 a d+2 (1-d) x\right )}\right )dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}\right )}{2 \sqrt [3]{d} (a-b)}-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{d} (a-b)}+\frac {\log \left (2 \left (\sqrt {d}+1\right ) \left (b-a \sqrt {d}\right )-2 (1-d) x\right )}{4 \sqrt [3]{d} (a-b)}+\frac {\log \left (2 (1-d) x-2 \left (1-\sqrt {d}\right ) \left (a \sqrt {d}+b\right )\right )}{4 \sqrt [3]{d} (a-b)}-\frac {3 \log \left (-\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{4 \sqrt [3]{d} (a-b)}-\frac {3 \log \left (\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{4 \sqrt [3]{d} (a-b)}\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
-(((-a + x)^(1/3)*(-b + x)^(2/3)*(-1/2*(Sqrt[3]*ArcTan[1/Sqrt[3] - (2*d^(1 /6)*(-a + x)^(1/3))/(Sqrt[3]*(-b + x)^(1/3))])/((a - b)*d^(1/3)) - (Sqrt[3 ]*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(-a + x)^(1/3))/(Sqrt[3]*(-b + x)^(1/3))]) /(2*(a - b)*d^(1/3)) + Log[2*(1 + Sqrt[d])*(b - a*Sqrt[d]) - 2*(1 - d)*x]/ (4*(a - b)*d^(1/3)) + Log[-2*(1 - Sqrt[d])*(b + a*Sqrt[d]) + 2*(1 - d)*x]/ (4*(a - b)*d^(1/3)) - (3*Log[-(d^(1/6)*(-a + x)^(1/3)) - (-b + x)^(1/3)])/ (4*(a - b)*d^(1/3)) - (3*Log[d^(1/6)*(-a + x)^(1/3) - (-b + x)^(1/3)])/(4* (a - b)*d^(1/3))))/(-((a - x)*(b - x)^2))^(1/3))
3.32.1.3.1 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[(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]
\[\int \frac {-b +x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (-b^{2}+a^{2} d +2 \left (-a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
Time = 0.27 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.33 \[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=\left [-\frac {\sqrt {3} d \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \log \left (\frac {2 \, a^{2} d + {\left (2 \, d + 1\right )} x^{2} + b^{2} - 2 \, {\left (2 \, a d + b\right )} x - \sqrt {3} {\left (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 (a d - d x\right )} - {\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{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}} \left (-d\right )^{\frac {2}{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 {2}{3}} \left (-d\right )^{\frac {1}{3}}}{a^{2} d + {\left (d - 1\right )} x^{2} - b^{2} - 2 \, {\left (a d - b\right )} x}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\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}{b^{2} - 2 \, b x + x^{2}}\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 (a d - d x\right )} + {\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{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}} \left (-d\right )^{\frac {2}{3}}}{b^{2} - 2 \, b x + x^{2}}\right )}{4 \, {\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^{2} - 2 \, b x + x^{2}\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 {2}{3}} \left (-d\right )^{\frac {2}{3}}\right )} \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{3 \, {\left (b^{2} - 2 \, b x + x^{2}\right )}}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\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}{b^{2} - 2 \, b x + x^{2}}\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 (a d - d x\right )} + {\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{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}} \left (-d\right )^{\frac {2}{3}}}{b^{2} - 2 \, b x + x^{2}}\right )}{4 \, {\left (a - b\right )} d}\right ] \]
integrate((-b+x)/((-a+x)*(-b+x)^2)^(1/3)/(-b^2+a^2*d+2*(-a*d+b)*x+(-1+d)*x ^2),x, algorithm="fricas")
[-1/4*(sqrt(3)*d*sqrt((-d)^(1/3)/d)*log((2*a^2*d + (2*d + 1)*x^2 + b^2 - 2 *(2*a*d + b)*x - sqrt(3)*(2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)* x)^(1/3)*(a*d - d*x) - (b^2 - 2*b*x + x^2)*(-d)^(1/3) + (-a*b^2 - (a + 2*b )*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*(-d)^(2/3))*sqrt((-d)^(1/3)/d) + 3*(- a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*(-d)^(1/3))/(a^2*d + (d - 1)*x^2 - b^2 - 2*(a*d - b)*x)) - 2*(-d)^(2/3)*log(-((b^2 - 2*b*x + x^ 2)*(-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)^(2/3)*log(-((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2 *a*b + b^2)*x)^(1/3)*(a*d - d*x) + (b^2 - 2*b*x + x^2)*(-d)^(1/3) - (-a*b^ 2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*(-d)^(2/3))/(b^2 - 2*b*x + x^2)))/((a - b)*d), -1/4*(2*sqrt(3)*d*sqrt(-(-d)^(1/3)/d)*arctan(-1/3*sq rt(3)*((b^2 - 2*b*x + x^2)*(-d)^(1/3) - 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*(-d)^(2/3))*sqrt(-(-d)^(1/3)/d)/(b^2 - 2*b*x + x^2) ) - 2*(-d)^(2/3)*log(-((b^2 - 2*b*x + x^2)*(-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)^(2/3)* log(-((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(a*d - d*x) + (b^2 - 2*b*x + x^2)*(-d)^(1/3) - (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*(-d)^(2/3))/(b^2 - 2*b*x + x^2)))/((a - b)*d)]
Timed out. \[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b^{2} - 2 \, {\left (a d - b\right )} x\right )}} \,d x } \]
integrate((-b+x)/((-a+x)*(-b+x)^2)^(1/3)/(-b^2+a^2*d+2*(-a*d+b)*x+(-1+d)*x ^2),x, algorithm="maxima")
-integrate((b - x)/((-(a - x)*(b - x)^2)^(1/3)*(a^2*d + (d - 1)*x^2 - b^2 - 2*(a*d - b)*x)), x)
\[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b^{2} - 2 \, {\left (a d - b\right )} x\right )}} \,d x } \]
integrate((-b+x)/((-a+x)*(-b+x)^2)^(1/3)/(-b^2+a^2*d+2*(-a*d+b)*x+(-1+d)*x ^2),x, algorithm="giac")
integrate(-(b - x)/((-(a - x)*(b - x)^2)^(1/3)*(a^2*d + (d - 1)*x^2 - b^2 - 2*(a*d - b)*x)), x)
Timed out. \[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=\int -\frac {b-x}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (a^2\,d+2\,x\,\left (b-a\,d\right )-b^2+x^2\,\left (d-1\right )\right )} \,d x \]