Integrand size = 42, antiderivative size = 201 \[ \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{d^{2/3}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}\right )}{d^{2/3}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}+d^{2/3} \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{2 d^{2/3}} \]
3^(1/2)*arctan(3^(1/2)*d^(1/3)*(a*b+(-a-b)*x+x^2)^(1/3)/(-2*a+2*x+d^(1/3)* (a*b+(-a-b)*x+x^2)^(1/3)))/d^(2/3)+ln(a-x+d^(1/3)*(a*b+(-a-b)*x+x^2)^(1/3) )/d^(2/3)-1/2*ln(a^2-2*a*x+x^2+(-a*d^(1/3)+d^(1/3)*x)*(a*b+(-a-b)*x+x^2)^( 1/3)+d^(2/3)*(a*b+(-a-b)*x+x^2)^(2/3))/d^(2/3)
Time = 4.83 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.90 \[ \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\frac {\sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-b+x}}{2 (-a+x)^{2/3}+\sqrt [3]{d} \sqrt [3]{-b+x}}\right )+2 \log \left ((-a+x)^{2/3}-\sqrt [3]{d} \sqrt [3]{-b+x}\right )-\log \left ((-a+x)^{4/3}+\sqrt [3]{d} (-a+x)^{2/3} \sqrt [3]{-b+x}+d^{2/3} (-b+x)^{2/3}\right )\right )}{2 d^{2/3} \sqrt [3]{(-a+x) (-b+x)}} \]
((-a + x)^(1/3)*(-b + x)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(-b + x) ^(1/3))/(2*(-a + x)^(2/3) + d^(1/3)*(-b + x)^(1/3))] + 2*Log[(-a + x)^(2/3 ) - d^(1/3)*(-b + x)^(1/3)] - Log[(-a + x)^(4/3) + d^(1/3)*(-a + x)^(2/3)* (-b + x)^(1/3) + d^(2/3)*(-b + x)^(2/3)]))/(2*d^(2/3)*((-a + x)*(-b + x))^ (1/3))
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.
\(\displaystyle \int \frac {a-2 b+x}{\sqrt [3]{(x-a) (x-b)} \left (a^2-x (2 a+d)+b d+x^2\right )} \, dx\) |
\(\Big \downarrow \) 2048 |
\(\displaystyle \int \frac {a-2 b+x}{\sqrt [3]{x (-a-b)+a b+x^2} \left (a^2-x (2 a+d)+b d+x^2\right )}dx\) |
\(\Big \downarrow \) 1375 |
\(\displaystyle \int \frac {a-2 b+x}{\sqrt [3]{x (-a-b)+a b+x^2} \left (a^2+x (-2 a-d)+b d+x^2\right )}dx\) |
3.25.67.3.1 Defintions of rubi rules used
Int[((g_.) + (h_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Unintegrable[(g + h*x)*(a + b *x + c*x^2)^p*(d + e*x + f*x^2)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q}, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) , x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F reeQ[{a, b, c, d, e, n, p}, x]
\[\int \frac {a -2 b +x}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a^{2}+b d -\left (2 a +d \right ) x +x^{2}\right )}d x\]
Timed out. \[ \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\int { \frac {a - 2 \, b + x}{{\left (a^{2} + b d - {\left (2 \, a + d\right )} x + x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\int { \frac {a - 2 \, b + x}{{\left (a^{2} + b d - {\left (2 \, a + d\right )} x + x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\int \frac {a-2\,b+x}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (b\,d-x\,\left (2\,a+d\right )+a^2+x^2\right )} \,d x \]