Integrand size = 61, antiderivative size = 273 \[ \int \frac {a b-(a+b) x+x^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a^2-b^2 d-2 (a-b d) x+(1-d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a-\sqrt {3} x}{a-x-2 \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{2 (a-b) \sqrt [3]{d}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{2 (a-b) \sqrt [3]{d}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}+d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{4 (a-b) \sqrt [3]{d}} \]
1/2*3^(1/2)*arctan((3^(1/2)*a-x*3^(1/2))/(a-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)+1/2*ln(a-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)-1/4*ln(a^2-2*a*x+x^2+(-a *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 = 0.83 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.96 \[ \int \frac {a b-(a+b) x+x^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a^2-b^2 d-2 (a-b d) x+(1-d) x^2\right )} \, dx=\frac {(b-x)^{4/3} (-a+x)^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} (-a+x)^{2/3}}{2 \sqrt [3]{d} (b-x)^{2/3}+(-a+x)^{2/3}}\right )-2 \log \left (-\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )-2 \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )+\log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )+\log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )\right )}{4 (-a+b) \sqrt [3]{d} \left ((b-x)^2 (-a+x)\right )^{2/3}} \]
Integrate[(a*b - (a + b)*x + x^2)/(((-a + x)*(-b + x)^2)^(2/3)*(a^2 - b^2* d - 2*(a - b*d)*x + (1 - d)*x^2)),x]
((b - x)^(4/3)*(-a + x)^(2/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(-a + x)^(2/3))/ (2*d^(1/3)*(b - x)^(2/3) + (-a + x)^(2/3))] - 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) - d^(1/6)*(b - x)^(1/3)*(-a + x)^(1/3) + (-a + x)^( 2/3)] + Log[d^(1/3)*(b - x)^(2/3) + d^(1/6)*(b - x)^(1/3)*(-a + x)^(1/3) + (-a + x)^(2/3)]))/(4*(-a + b)*d^(1/3)*((b - x)^2*(-a + x))^(2/3))
Time = 1.09 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.066, Rules used = {7270, 2004, 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.
| \(\displaystyle \int \frac {-x (a+b)+a b+x^2}{\left ((x-a) (x-b)^2\right )^{2/3} \left (a^2-2 x (a-b d)-b^2 d+(1-d) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \int \frac {x^2-(a+b) x+a b}{(x-a)^{2/3} (x-b)^{4/3} \left (a^2+(1-d) x^2-b^2 d-2 (a-b d) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2004 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \int \frac {\sqrt [3]{x-a}}{\sqrt [3]{x-b} \left (a^2+(1-d) x^2-b^2 d-2 (a-b d) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 1205 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \int \left (\frac {\sqrt [3]{x-a} (1-d)}{(a-b) \sqrt {d} \sqrt [3]{x-b} \left (2 a-2 b d-2 (1-d) x-2 (a-b) \sqrt {d}\right )}+\frac {\sqrt [3]{x-a} (1-d)}{(a-b) \sqrt {d} \sqrt [3]{x-b} \left (-2 a+2 b d+2 (1-d) x-2 (a-b) \sqrt {d}\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \left (\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}\right )}{2 \sqrt [3]{d} (a-b)}+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{d} (a-b)}-\frac {\log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 \sqrt [3]{d} (a-b)}-\frac {\log \left (2 (1-d) x-2 \left (\sqrt {d}+1\right ) \left (a-b \sqrt {d}\right )\right )}{4 \sqrt [3]{d} (a-b)}+\frac {3 \log \left (-\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{x-b}\right )}{4 \sqrt [3]{d} (a-b)}+\frac {3 \log \left (\sqrt [6]{d} \sqrt [3]{x-b}-\sqrt [3]{x-a}\right )}{4 \sqrt [3]{d} (a-b)}\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
Int[(a*b - (a + b)*x + x^2)/(((-a + x)*(-b + x)^2)^(2/3)*(a^2 - b^2*d - 2* (a - b*d)*x + (1 - d)*x^2)),x]
((-a + x)^(2/3)*(-b + x)^(4/3)*((Sqrt[3]*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(-b + x)^(1/3))/(Sqrt[3]*(-a + x)^(1/3))])/(2*(a - b)*d^(1/3)) + (Sqrt[3]*Arc Tan[1/Sqrt[3] + (2*d^(1/6)*(-b + x)^(1/3))/(Sqrt[3]*(-a + x)^(1/3))])/(2*( a - b)*d^(1/3)) - Log[2*(1 - Sqrt[d])*(a + b*Sqrt[d]) - 2*(1 - d)*x]/(4*(a - b)*d^(1/3)) - Log[-2*(1 + Sqrt[d])*(a - b*Sqrt[d]) + 2*(1 - d)*x]/(4*(a - b)*d^(1/3)) + (3*Log[-(-a + x)^(1/3) - d^(1/6)*(-b + x)^(1/3)])/(4*(a - b)*d^(1/3)) + (3*Log[-(-a + x)^(1/3) + d^(1/6)*(-b + x)^(1/3)])/(4*(a - b )*d^(1/3))))/(-((a - x)*(b - x)^2))^(2/3)
3.29.3.3.1 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_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) , x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b , c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
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]
\[\int \frac {a b -\left (a +b \right ) x +x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (a^{2}-b^{2} d -2 \left (-b d +a \right ) x +\left (1-d \right ) x^{2}\right )}d x\]
Time = 0.27 (sec) , antiderivative size = 740, normalized size of antiderivative = 2.71 \[ \int \frac {a b-(a+b) x+x^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a^2-b^2 d-2 (a-b 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 {b^{2} d + {\left (d + 2\right )} x^{2} + 2 \, a^{2} - 2 \, {\left (b d + 2 \, a\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 - 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} d - 2 \, b d x + d 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 {2}{3}} \left (-d\right )^{\frac {2}{3}}}{b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x}\right ) + \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 {1}{3}} {\left (a - x\right )} - {\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}}}{b^{2} - 2 \, b x + x^{2}}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {{\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}}}{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}}\right )} \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{3 \, {\left (b^{2} - 2 \, b x + x^{2}\right )}}\right ) - \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 {1}{3}} {\left (a - x\right )} - {\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}}}{b^{2} - 2 \, b x + x^{2}}\right ) + 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {{\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}}}{b^{2} - 2 \, b x + x^{2}}\right )}{4 \, {\left (a - b\right )} d}\right ] \]
integrate((a*b-(a+b)*x+x^2)/((-a+x)*(-b+x)^2)^(2/3)/(a^2-b^2*d-2*(-b*d+a)* x+(1-d)*x^2),x, algorithm="fricas")
[-1/4*(sqrt(3)*d*sqrt((-d)^(1/3)/d)*log(-(b^2*d + (d + 2)*x^2 + 2*a^2 - 2* (b*d + 2*a)*x + sqrt(3)*(2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x )^(1/3)*(a - 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*d - 2*b*d*x + d*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)^(2/3)*(-d)^(2/3))/(b^2*d + (d - 1)*x^2 - a^2 - 2*(b*d - a)*x)) + (-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)^(1/3)*(a - x) - (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*(-d)^(1/3))/ (b^2 - 2*b*x + x^2)) - 2*(-d)^(2/3)*log(((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))/(b^2 - 2*b*x + x^2 )))/((a - b)*d), 1/4*(2*sqrt(3)*d*sqrt(-(-d)^(1/3)/d)*arctan(-1/3*sqrt(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))*sqrt(-(-d)^(1/3)/d)/(b^2 - 2*b*x + x^2)) - (-d)^(2/3)*lo g(((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)^(1/3)*(a - x) - (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x )^(2/3)*(-d)^(1/3))/(b^2 - 2*b*x + x^2)) + 2*(-d)^(2/3)*log(((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) )/(b^2 - 2*b*x + x^2)))/((a - b)*d)]
Timed out. \[ \int \frac {a b-(a+b) x+x^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a^2-b^2 d-2 (a-b d) x+(1-d) x^2\right )} \, dx=\text {Timed out} \]
integrate((a*b-(a+b)*x+x**2)/((-a+x)*(-b+x)**2)**(2/3)/(a**2-b**2*d-2*(-b* d+a)*x+(1-d)*x**2),x)
\[ \int \frac {a b-(a+b) x+x^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a^2-b^2 d-2 (a-b d) x+(1-d) x^2\right )} \, dx=\int { -\frac {a b - {\left (a + b\right )} x + x^{2}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}} \,d x } \]
integrate((a*b-(a+b)*x+x^2)/((-a+x)*(-b+x)^2)^(2/3)/(a^2-b^2*d-2*(-b*d+a)* x+(1-d)*x^2),x, algorithm="maxima")
-integrate((a*b - (a + b)*x + x^2)/((-(a - x)*(b - x)^2)^(2/3)*(b^2*d + (d - 1)*x^2 - a^2 - 2*(b*d - a)*x)), x)
\[ \int \frac {a b-(a+b) x+x^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a^2-b^2 d-2 (a-b d) x+(1-d) x^2\right )} \, dx=\int { -\frac {a b - {\left (a + b\right )} x + x^{2}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}} \,d x } \]
integrate((a*b-(a+b)*x+x^2)/((-a+x)*(-b+x)^2)^(2/3)/(a^2-b^2*d-2*(-b*d+a)* x+(1-d)*x^2),x, algorithm="giac")
integrate(-(a*b - (a + b)*x + x^2)/((-(a - x)*(b - x)^2)^(2/3)*(b^2*d + (d - 1)*x^2 - a^2 - 2*(b*d - a)*x)), x)
Timed out. \[ \int \frac {a b-(a+b) x+x^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a^2-b^2 d-2 (a-b d) x+(1-d) x^2\right )} \, dx=\int -\frac {x^2+\left (-a-b\right )\,x+a\,b}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (b^2\,d+2\,x\,\left (a-b\,d\right )-a^2+x^2\,\left (d-1\right )\right )} \,d x \]
int(-(a*b + x^2 - x*(a + b))/((-(a - x)*(b - x)^2)^(2/3)*(b^2*d + 2*x*(a - b*d) - a^2 + x^2*(d - 1))),x)