Integrand size = 65, antiderivative size = 1886 \[ \int \frac {(-b+x) (-b-a c+(1+c) x)}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx =\text {Too large to display} \]
(b-x)^(4/3)*(-a+x)^(2/3)*((b-x)^(1/3)-d^(1/6)*(-a+x)^(1/3))*((b-x)^(1/3)+d ^(1/6)*(-a+x)^(1/3))*((b-x)^(2/3)-d^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+d^(1/3) *(-a+x)^(2/3))*((b-x)^(2/3)+d^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+d^(1/3)*(-a+x )^(2/3))*(1/2*3^(1/2)*b*(-1+d^(1/2))*arctan(3^(1/2)*d^(1/6)*(-a+x)^(1/3)/( -2*(b-x)^(1/3)+d^(1/6)*(-a+x)^(1/3)))/(a-b)^2/d^(2/3)+1/2*3^(1/2)*a*c*(-1+ d^(1/2))*arctan(3^(1/2)*d^(1/6)*(-a+x)^(1/3)/(-2*(b-x)^(1/3)+d^(1/6)*(-a+x )^(1/3)))/(a-b)^2/d^(2/3)-1/2*3^(1/2)*(-b+a*d^(1/2))*arctan(3^(1/2)*d^(1/6 )*(-a+x)^(1/3)/(-2*(b-x)^(1/3)+d^(1/6)*(-a+x)^(1/3)))/(a-b)^2/d^(2/3)-1/2* 3^(1/2)*c*(-b+a*d^(1/2))*arctan(3^(1/2)*d^(1/6)*(-a+x)^(1/3)/(-2*(b-x)^(1/ 3)+d^(1/6)*(-a+x)^(1/3)))/(a-b)^2/d^(2/3)-1/2*3^(1/2)*b*(1+d^(1/2))*arctan (3^(1/2)*d^(1/6)*(-a+x)^(1/3)/(2*(b-x)^(1/3)+d^(1/6)*(-a+x)^(1/3)))/(a-b)^ 2/d^(2/3)-1/2*3^(1/2)*a*c*(1+d^(1/2))*arctan(3^(1/2)*d^(1/6)*(-a+x)^(1/3)/ (2*(b-x)^(1/3)+d^(1/6)*(-a+x)^(1/3)))/(a-b)^2/d^(2/3)+1/2*3^(1/2)*(b+a*d^( 1/2))*arctan(3^(1/2)*d^(1/6)*(-a+x)^(1/3)/(2*(b-x)^(1/3)+d^(1/6)*(-a+x)^(1 /3)))/(a-b)^2/d^(2/3)+1/2*3^(1/2)*c*(b+a*d^(1/2))*arctan(3^(1/2)*d^(1/6)*( -a+x)^(1/3)/(2*(b-x)^(1/3)+d^(1/6)*(-a+x)^(1/3)))/(a-b)^2/d^(2/3)+1/2*b*(1 +d^(1/2))*ln((b-x)^(1/3)-d^(1/6)*(-a+x)^(1/3))/(a-b)^2/d^(2/3)+1/2*a*c*(1+ d^(1/2))*ln((b-x)^(1/3)-d^(1/6)*(-a+x)^(1/3))/(a-b)^2/d^(2/3)+1/2*(-b-a*d^ (1/2))*ln((b-x)^(1/3)-d^(1/6)*(-a+x)^(1/3))/(a-b)^2/d^(2/3)-1/2*c*(b+a*d^( 1/2))*ln((b-x)^(1/3)-d^(1/6)*(-a+x)^(1/3))/(a-b)^2/d^(2/3)-1/2*b*(-1+d^...
Time = 0.45 (sec) , antiderivative size = 501, normalized size of antiderivative = 0.27 \[ \int \frac {(-b+x) (-b-a c+(1+c) x)}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=-\frac {(b-x)^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \left (c+\sqrt {d}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}}{\sqrt {3}}\right )+2 \sqrt {3} \left (c-\sqrt {d}\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}}{\sqrt {3}}\right )-2 c \log \left (-1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )+2 \sqrt {d} \log \left (-1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )-2 c \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )-2 \sqrt {d} \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )+c \log \left (1-\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {\sqrt [3]{d} (-a+x)^{2/3}}{(b-x)^{2/3}}\right )+\sqrt {d} \log \left (1-\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {\sqrt [3]{d} (-a+x)^{2/3}}{(b-x)^{2/3}}\right )+c \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {\sqrt [3]{d} (-a+x)^{2/3}}{(b-x)^{2/3}}\right )-\sqrt {d} \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {\sqrt [3]{d} (-a+x)^{2/3}}{(b-x)^{2/3}}\right )\right )}{4 (a-b) d^{2/3} \left ((b-x)^2 (-a+x)\right )^{2/3}} \]
Integrate[((-b + x)*(-b - a*c + (1 + c)*x))/(((-a + x)*(-b + x)^2)^(2/3)*( -b^2 + a^2*d + 2*(b - a*d)*x + (-1 + d)*x^2)),x]
-1/4*((b - x)^(4/3)*(-a + x)^(2/3)*(2*Sqrt[3]*(c + Sqrt[d])*ArcTan[(1 - (2 *d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3))/Sqrt[3]] + 2*Sqrt[3]*(c - Sqrt[d]) *ArcTan[(1 + (2*d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3))/Sqrt[3]] - 2*c*Log[ -1 + (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3)] + 2*Sqrt[d]*Log[-1 + (d^(1/6) *(-a + x)^(1/3))/(b - x)^(1/3)] - 2*c*Log[1 + (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3)] - 2*Sqrt[d]*Log[1 + (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3)] + c*Log[1 - (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3) + (d^(1/3)*(-a + x)^(2/3) )/(b - x)^(2/3)] + Sqrt[d]*Log[1 - (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3) + (d^(1/3)*(-a + x)^(2/3))/(b - x)^(2/3)] + c*Log[1 + (d^(1/6)*(-a + x)^(1 /3))/(b - x)^(1/3) + (d^(1/3)*(-a + x)^(2/3))/(b - x)^(2/3)] - Sqrt[d]*Log [1 + (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3) + (d^(1/3)*(-a + x)^(2/3))/(b - x)^(2/3)]))/((a - b)*d^(2/3)*((b - x)^2*(-a + x))^(2/3))
Time = 1.43 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.20, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {7270, 2153, 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.
| \(\displaystyle \int \frac {(x-b) (-a c-b+(c+1) x)}{\left ((x-a) (x-b)^2\right )^{2/3} \left (a^2 d+2 x (b-a d)-b^2+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \int \frac {b+a c-(c+1) x}{(x-a)^{2/3} \sqrt [3]{x-b} \left (-d a^2+b^2+(1-d) x^2-2 (b-a d) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2153 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \int \left (\frac {-c-\frac {c+d}{\sqrt {d}}-1}{(x-a)^{2/3} \sqrt [3]{x-b} \left (2 \sqrt {d} (a-b)-2 (b-a d)+2 (1-d) x\right )}+\frac {-c+\frac {c+d}{\sqrt {d}}-1}{(x-a)^{2/3} \sqrt [3]{x-b} \left (-2 \sqrt {d} (a-b)-2 (b-a d)+2 (1-d) x\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \left (\frac {\sqrt {3} \left (c-\sqrt {d}\right ) \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-b}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}\right )}{2 d^{2/3} (a-b)}+\frac {\sqrt {3} \left (c+\sqrt {d}\right ) \arctan \left (\frac {2 \sqrt [3]{x-b}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 d^{2/3} (a-b)}-\frac {\left (c+\sqrt {d}\right ) \log \left (2 (1-d) x-2 \left (\sqrt {d}+1\right ) \left (b-a \sqrt {d}\right )\right )}{4 d^{2/3} (a-b)}-\frac {\left (c-\sqrt {d}\right ) \log \left (2 (1-d) x-2 \left (1-\sqrt {d}\right ) \left (a \sqrt {d}+b\right )\right )}{4 d^{2/3} (a-b)}+\frac {3 \left (c-\sqrt {d}\right ) \log \left (-\sqrt [3]{x-a}-\frac {\sqrt [3]{x-b}}{\sqrt [6]{d}}\right )}{4 d^{2/3} (a-b)}+\frac {3 \left (c+\sqrt {d}\right ) \log \left (\frac {\sqrt [3]{x-b}}{\sqrt [6]{d}}-\sqrt [3]{x-a}\right )}{4 d^{2/3} (a-b)}\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
Int[((-b + x)*(-b - a*c + (1 + c)*x))/(((-a + x)*(-b + x)^2)^(2/3)*(-b^2 + a^2*d + 2*(b - a*d)*x + (-1 + d)*x^2)),x]
((-a + x)^(2/3)*(-b + x)^(4/3)*((Sqrt[3]*(c - Sqrt[d])*ArcTan[1/Sqrt[3] - (2*(-b + x)^(1/3))/(Sqrt[3]*d^(1/6)*(-a + x)^(1/3))])/(2*(a - b)*d^(2/3)) + (Sqrt[3]*(c + Sqrt[d])*ArcTan[1/Sqrt[3] + (2*(-b + x)^(1/3))/(Sqrt[3]*d^ (1/6)*(-a + x)^(1/3))])/(2*(a - b)*d^(2/3)) - ((c + Sqrt[d])*Log[-2*(1 + S qrt[d])*(b - a*Sqrt[d]) + 2*(1 - d)*x])/(4*(a - b)*d^(2/3)) - ((c - Sqrt[d ])*Log[-2*(1 - Sqrt[d])*(b + a*Sqrt[d]) + 2*(1 - d)*x])/(4*(a - b)*d^(2/3) ) + (3*(c - Sqrt[d])*Log[-(-a + x)^(1/3) - (-b + x)^(1/3)/d^(1/6)])/(4*(a - b)*d^(2/3)) + (3*(c + Sqrt[d])*Log[-(-a + x)^(1/3) + (-b + x)^(1/3)/d^(1 /6)])/(4*(a - b)*d^(2/3))))/(-((a - x)*(b - x)^2))^(2/3)
3.32.52.3.1 Defintions of rubi rules used
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ [m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 0])
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]
\[\int \frac {\left (-b +x \right ) \left (-b -a c +\left (1+c \right ) x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (-b^{2}+a^{2} d +2 \left (-a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 3644 vs. \(2 (1441) = 2882\).
Time = 0.35 (sec) , antiderivative size = 3644, normalized size of antiderivative = 1.93 \[ \int \frac {(-b+x) (-b-a c+(1+c) x)}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=\text {Too large to display} \]
integrate((-b+x)*(-b-a*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(2/3)/(-b^2+a^2*d+2*(- a*d+b)*x+(-1+d)*x^2),x, algorithm="fricas")
-1/4*(sqrt(-3) + 1)*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6* c^2*d + d^2)/((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^3)) + c^3 + 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/ 3)*log(1/2*(2*(3*c^4 - 2*c^2*d - d^2)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a *b + b^2)*x)^(1/3) + (3*(a*b - b^2)*c^2*d + (a*b - b^2)*d^2 - (3*(a - b)*c ^2*d + (a - b)*d^2)*x + sqrt(-3)*(3*(a*b - b^2)*c^2*d + (a*b - b^2)*d^2 - (3*(a - b)*c^2*d + (a - b)*d^2)*x) + ((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c*d^2*x - (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c*d^2 + sqrt(-3)*((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c*d^2*x - (a^4*b - 4 *a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c*d^2))*sqrt((9*c^4 + 6*c^2*d + d^2) /((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 ^3)))*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/( (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^3 )) + c^3 + 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3))/(b - x)) + 1/4*(sqrt(-3) - 1)*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6* c^2*d + d^2)/((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^3)) + c^3 + 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/ 3)*log(1/2*(2*(3*c^4 - 2*c^2*d - d^2)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a *b + b^2)*x)^(1/3) + (3*(a*b - b^2)*c^2*d + (a*b - b^2)*d^2 - (3*(a - b)*c ^2*d + (a - b)*d^2)*x - sqrt(-3)*(3*(a*b - b^2)*c^2*d + (a*b - b^2)*d^2...
Timed out. \[ \int \frac {(-b+x) (-b-a c+(1+c) x)}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
integrate((-b+x)*(-b-a*c+(1+c)*x)/((-a+x)*(-b+x)**2)**(2/3)/(-b**2+a**2*d+ 2*(-a*d+b)*x+(-1+d)*x**2),x)
\[ \int \frac {(-b+x) (-b-a c+(1+c) x)}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=\int { \frac {{\left (a c - {\left (c + 1\right )} x + b\right )} {\left (b - x\right )}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b^{2} - 2 \, {\left (a d - b\right )} x\right )}} \,d x } \]
integrate((-b+x)*(-b-a*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(2/3)/(-b^2+a^2*d+2*(- a*d+b)*x+(-1+d)*x^2),x, algorithm="maxima")
integrate((a*c - (c + 1)*x + b)*(b - x)/((-(a - x)*(b - x)^2)^(2/3)*(a^2*d + (d - 1)*x^2 - b^2 - 2*(a*d - b)*x)), x)
\[ \int \frac {(-b+x) (-b-a c+(1+c) x)}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=\int { \frac {{\left (a c - {\left (c + 1\right )} x + b\right )} {\left (b - x\right )}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b^{2} - 2 \, {\left (a d - b\right )} x\right )}} \,d x } \]
integrate((-b+x)*(-b-a*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(2/3)/(-b^2+a^2*d+2*(- a*d+b)*x+(-1+d)*x^2),x, algorithm="giac")
integrate((a*c - (c + 1)*x + b)*(b - x)/((-(a - x)*(b - x)^2)^(2/3)*(a^2*d + (d - 1)*x^2 - b^2 - 2*(a*d - b)*x)), x)
Timed out. \[ \int \frac {(-b+x) (-b-a c+(1+c) x)}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx=-\int -\frac {\left (b-x\right )\,\left (b+a\,c-x\,\left (c+1\right )\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (a^2\,d+2\,x\,\left (b-a\,d\right )-b^2+x^2\,\left (d-1\right )\right )} \,d x \]
int(((b - x)*(b + a*c - x*(c + 1)))/((-(a - x)*(b - x)^2)^(2/3)*(a^2*d + 2 *x*(b - a*d) - b^2 + x^2*(d - 1))),x)