Integrand size = 54, antiderivative size = 438 \[ \int \frac {(-b+x)^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\frac {(-a+x)^{2/3} (-b+x) \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right ) \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right ) \left (-b \sqrt [3]{d}+\sqrt [3]{d} x+\sqrt [3]{b-x} (-a+x)^{2/3}\right ) \left (\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6}}+\frac {\text {arctanh}\left (\frac {(b-x)^{2/3} \sqrt [3]{-a+x}}{\sqrt [6]{d} (-b+x)}\right )}{(a-b) d^{5/6}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} (b-x)^{2/3}+\frac {(-a+x)^{2/3}}{\sqrt [6]{d}}}{\sqrt [3]{b-x} \sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6}}\right )}{\left ((b-x)^2 (-a+x)\right )^{2/3} \left (-a^2+b^2 d+2 a x-2 b d x+(-1+d) x^2\right )} \]
(-a+x)^(2/3)*(-b+x)*(d^(1/3)*(b-x)^(2/3)-d^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+ (-a+x)^(2/3))*(d^(1/3)*(b-x)^(2/3)+d^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+(-a+x) ^(2/3))*(-b*d^(1/3)+d^(1/3)*x+(b-x)^(1/3)*(-a+x)^(2/3))*(1/2*3^(1/2)*arcta n(3^(1/2)*(-a+x)^(1/3)/(-2*d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3)))/(a-b)/d^(5/6 )-1/2*3^(1/2)*arctan(3^(1/2)*(-a+x)^(1/3)/(2*d^(1/6)*(b-x)^(1/3)+(-a+x)^(1 /3)))/(a-b)/d^(5/6)+arctanh((b-x)^(2/3)*(-a+x)^(1/3)/d^(1/6)/(-b+x))/(a-b) /d^(5/6)-1/2*arctanh((d^(1/6)*(b-x)^(2/3)+(-a+x)^(2/3)/d^(1/6))/(b-x)^(1/3 )/(-a+x)^(1/3))/(a-b)/d^(5/6))/((b-x)^2*(-a+x))^(2/3)/(-a^2+b^2*d+2*a*x-2* b*d*x+(-1+d)*x^2)
Time = 0.87 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.52 \[ \int \frac {(-b+x)^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\frac {(b-x)^{4/3} (-a+x)^{2/3} \left (\sqrt {3} \left (\arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )-\arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )\right )-2 \text {arctanh}\left (\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d} \sqrt [3]{b-x}}\right )-\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}+\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d} \sqrt [3]{b-x}}\right )\right )}{2 (a-b) d^{5/6} \left ((b-x)^2 (-a+x)\right )^{2/3}} \]
Integrate[(-b + 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)*(Sqrt[3]*(ArcTan[(Sqrt[3]*(-a + x)^(1/3))/(- 2*d^(1/6)*(b - x)^(1/3) + (-a + x)^(1/3))] - ArcTan[(Sqrt[3]*(-a + x)^(1/3 ))/(2*d^(1/6)*(b - x)^(1/3) + (-a + x)^(1/3))]) - 2*ArcTanh[(-a + x)^(1/3) /(d^(1/6)*(b - x)^(1/3))] - ArcTanh[(d^(1/6)*(b - x)^(1/3))/(-a + x)^(1/3) + (-a + x)^(1/3)/(d^(1/6)*(b - x)^(1/3))]))/(2*(a - b)*d^(5/6)*((b - x)^2 *(-a + x))^(2/3))
Time = 1.14 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.
| \(\displaystyle \int \frac {(x-b)^2}{\left ((x-a) (x-b)^2\right )^{2/3} \left (-a^2+2 x (a-b d)+b^2 d+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \int -\frac {(x-b)^{2/3}}{(x-a)^{2/3} \left (a^2+(1-d) x^2-b^2 d-2 (a-b d) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \int \frac {(x-b)^{2/3}}{(x-a)^{2/3} \left (a^2+(1-d) x^2-b^2 d-2 (a-b d) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 1205 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \int \left (\frac {(x-b)^{2/3} (1-d)}{(a-b) \sqrt {d} (x-a)^{2/3} \left (2 a-2 b d-2 (1-d) x-2 (a-b) \sqrt {d}\right )}+\frac {(x-b)^{2/3} (1-d)}{(a-b) \sqrt {d} (x-a)^{2/3} \left (-2 a+2 b d+2 (1-d) x-2 (a-b) \sqrt {d}\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}\right )}{2 d^{5/6} (a-b)}+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 d^{5/6} (a-b)}+\frac {\log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{5/6} (a-b)}-\frac {\log \left (2 (1-d) x-2 \left (\sqrt {d}+1\right ) \left (a-b \sqrt {d}\right )\right )}{4 d^{5/6} (a-b)}-\frac {3 \log \left (-\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{x-b}\right )}{4 d^{5/6} (a-b)}+\frac {3 \log \left (\sqrt [6]{d} \sqrt [3]{x-b}-\sqrt [3]{x-a}\right )}{4 d^{5/6} (a-b)}\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
-(((-a + x)^(2/3)*(-b + x)^(4/3)*(-1/2*(Sqrt[3]*ArcTan[1/Sqrt[3] - (2*d^(1 /6)*(-b + x)^(1/3))/(Sqrt[3]*(-a + x)^(1/3))])/((a - b)*d^(5/6)) + (Sqrt[3 ]*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(-b + x)^(1/3))/(Sqrt[3]*(-a + x)^(1/3))]) /(2*(a - b)*d^(5/6)) + Log[2*(1 - Sqrt[d])*(a + b*Sqrt[d]) - 2*(1 - d)*x]/ (4*(a - b)*d^(5/6)) - Log[-2*(1 + Sqrt[d])*(a - b*Sqrt[d]) + 2*(1 - d)*x]/ (4*(a - b)*d^(5/6)) - (3*Log[-(-a + x)^(1/3) - d^(1/6)*(-b + x)^(1/3)])/(4 *(a - b)*d^(5/6)) + (3*Log[-(-a + x)^(1/3) + d^(1/6)*(-b + x)^(1/3)])/(4*( a - b)*d^(5/6))))/(-((a - x)*(b - x)^2))^(2/3))
3.31.32.3.1 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]
\[\int \frac {\left (-b +x \right )^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (-a^{2}+b^{2} d +2 \left (-b d +a \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1146 vs. \(2 (345) = 690\).
Time = 0.26 (sec) , antiderivative size = 1146, normalized size of antiderivative = 2.62 \[ \int \frac {(-b+x)^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\text {Too large to display} \]
integrate((-b+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) + 1)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b ^4 - 6*a*b^5 + b^6)*d^5))^(1/6)*log(-1/2*(((a - b)*d*x - (a*b - b^2)*d + s qrt(-3)*((a - b)*d*x - (a*b - b^2)*d))*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 2 0*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6) + 2*(-a*b^2 - (a + 2*b )*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3))/(b - x)) - 1/4*(sqrt(-3) + 1)*(1/((a ^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5)) ^(1/6)*log(1/2*(((a - b)*d*x - (a*b - b^2)*d + sqrt(-3)*((a - b)*d*x - (a* b - b^2)*d))*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6 *a*b^5 + b^6)*d^5))^(1/6) - 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2 )*x)^(1/3))/(b - x)) - 1/4*(sqrt(-3) - 1)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6)*log(-1/2*(((a - b)* d*x - (a*b - b^2)*d - sqrt(-3)*((a - b)*d*x - (a*b - b^2)*d))*(1/((a^6 - 6 *a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6) + 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3))/(b - x)) + 1/ 4*(sqrt(-3) - 1)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6)*log(1/2*(((a - b)*d*x - (a*b - b^2)*d - sqrt (-3)*((a - b)*d*x - (a*b - b^2)*d))*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a ^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6) - 2*(-a*b^2 - (a + 2*b)*x ^2 + x^3 + (2*a*b + b^2)*x)^(1/3))/(b - x)) + 1/2*(1/((a^6 - 6*a^5*b + 15* a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6)*log(-(((...
Timed out. \[ \int \frac {(-b+x)^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {(-b+x)^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\int { \frac {{\left (b - x\right )}^{2}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}} \,d x } \]
integrate((-b+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((b - x)^2/((-(a - x)*(b - x)^2)^(2/3)*(b^2*d + (d - 1)*x^2 - a^2 - 2*(b*d - a)*x)), x)
\[ \int \frac {(-b+x)^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\int { \frac {{\left (b - x\right )}^{2}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}} \,d x } \]
integrate((-b+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((b - 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. \[ \int \frac {(-b+x)^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\int \frac {{\left (b-x\right )}^2}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (b^2\,d+2\,x\,\left (a-b\,d\right )-a^2+x^2\,\left (d-1\right )\right )} \,d x \]