3.32.52 \(\int \frac {(-b+x) (-b-a c+(1+c) x)}{((-a+x) (-b+x)^2)^{2/3} (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2)} \, dx\) [3152]

Optimal. Leaf size=1886 \[ \frac {(b-x)^{4/3} (-a+x)^{2/3} \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right ) \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right ) \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right ) \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right ) \left (\frac {\sqrt {3} b \left (-1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\sqrt {3} a c \left (-1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} \left (-b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} c \left (-b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} b \left (1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} a c \left (1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\sqrt {3} \left (b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\sqrt {3} c \left (b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {b \left (1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {a c \left (1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-b-a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {c \left (b+a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {b \left (-1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {a c \left (-1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-b+a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {c \left (-b+a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {b \left (-1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {a c \left (-1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (b-a \sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}-\frac {c \left (-b+a \sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}-\frac {b \left (1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}-\frac {a c \left (1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (b+a \sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {c \left (b+a \sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}\right )}{\left ((b-x)^2 (-a+x)\right )^{2/3} \left (b^2-a^2 d-2 b x+2 a d x-(-1+d) x^2\right )} \]

[Out]

(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^(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)+1/4
*b*(-1+d^(1/2))*ln((b-x)^(2/3)-d^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+d^(1/3)*(-a+x)^(2/3))/(a-b)^2/d^(2/3)+1/4*a*c*
(-1+d^(1/2))*ln((b-x)^(2/3)-d^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+d^(1/3)*(-a+x)^(2/3))/(a-b)^2/d^(2/3)+1/4*(b-a*d^
(1/2))*ln((b-x)^(2/3)-d^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+d^(1/3)*(-a+x)^(2/3))/(a-b)^2/d^(2/3)-1/4*c*(-b+a*d^(1/
2))*ln((b-x)^(2/3)-d^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+d^(1/3)*(-a+x)^(2/3))/(a-b)^2/d^(2/3)-1/4*b*(1+d^(1/2))*ln
((b-x)^(2/3)+d^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+d^(1/3)*(-a+x)^(2/3))/(a-b)^2/d^(2/3)-1/4*a*c*(1+d^(1/2))*ln((b-
x)^(2/3)+d^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+d^(1/3)*(-a+x)^(2/3))/(a-b)^2/d^(2/3)+1/4*(b+a*d^(1/2))*ln((b-x)^(2/
3)+d^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+d^(1/3)*(-a+x)^(2/3))/(a-b)^2/d^(2/3)+1/4*c*(b+a*d^(1/2))*ln((b-x)^(2/3)+d
^(1/6)*(b-x)^(1/3)*(-a+x)^(1/3)+d^(1/3)*(-a+x)^(2/3))/(a-b)^2/d^(2/3))/((b-x)^2*(-a+x))^(2/3)/(b^2-a^2*d-2*b*x
+2*a*d*x-(-1+d)*x^2)

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Rubi [A]
time = 1.66, antiderivative size = 561, normalized size of antiderivative = 0.30, number of steps used = 5, number of rules used = 3, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {6851, 6860, 93} \begin {gather*} \frac {\sqrt {3} (x-a)^{2/3} (x-b)^{4/3} \left (c-\sqrt {d}\right ) \text {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) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {\sqrt {3} (x-a)^{2/3} (x-b)^{4/3} \left (c+\sqrt {d}\right ) \text {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) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {(x-a)^{2/3} (x-b)^{4/3} \left (c+\sqrt {d}\right ) \log \left (2 \left (\sqrt {d}+1\right ) \left (b-a \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{2/3} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {(x-a)^{2/3} (x-b)^{4/3} \left (c-\sqrt {d}\right ) \log \left (2 \left (1-\sqrt {d}\right ) \left (a \sqrt {d}+b\right )-2 (1-d) x\right )}{4 d^{2/3} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 (x-a)^{2/3} (x-b)^{4/3} \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) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 (x-a)^{2/3} (x-b)^{4/3} \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) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

(Sqrt[3]*(c - Sqrt[d])*(-a + x)^(2/3)*(-b + x)^(4/3)*ArcTan[1/Sqrt[3] - (2*(-b + x)^(1/3))/(Sqrt[3]*d^(1/6)*(-
a + x)^(1/3))])/(2*(a - b)*d^(2/3)*(-((a - x)*(b - x)^2))^(2/3)) + (Sqrt[3]*(c + Sqrt[d])*(-a + x)^(2/3)*(-b +
x)^(4/3)*ArcTan[1/Sqrt[3] + (2*(-b + x)^(1/3))/(Sqrt[3]*d^(1/6)*(-a + x)^(1/3))])/(2*(a - b)*d^(2/3)*(-((a -
x)*(b - x)^2))^(2/3)) - ((c + Sqrt[d])*(-a + x)^(2/3)*(-b + x)^(4/3)*Log[2*(1 + Sqrt[d])*(b - a*Sqrt[d]) - 2*(
1 - d)*x])/(4*(a - b)*d^(2/3)*(-((a - x)*(b - x)^2))^(2/3)) - ((c - Sqrt[d])*(-a + x)^(2/3)*(-b + x)^(4/3)*Log
[2*(1 - Sqrt[d])*(b + a*Sqrt[d]) - 2*(1 - d)*x])/(4*(a - b)*d^(2/3)*(-((a - x)*(b - x)^2))^(2/3)) + (3*(c - Sq
rt[d])*(-a + x)^(2/3)*(-b + x)^(4/3)*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*(c + Sqrt[d])*(-a + x)^(2/3)*(-b + x)^(4/3)*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))

Rule 93

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])*q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1
/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q*(a + b*x)^(1/3) - (c +
d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \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 {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {-b-a c+(1+c) x}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \left (\frac {1+c+\frac {c+d}{\sqrt {d}}}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (-2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )}+\frac {1+c-\frac {c+d}{\sqrt {d}}}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )}\right ) \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left (\left (1+c-\frac {c+d}{\sqrt {d}}\right ) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (\left (1+c+\frac {c+d}{\sqrt {d}}\right ) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (-2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\sqrt {3} \left (c-\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {\sqrt {3} \left (c+\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {\left (c+\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \log \left (2 \left (1+\sqrt {d}\right ) \left (b-a \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {\left (c-\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (b+a \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 \left (c-\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \log \left (-\sqrt [3]{-a+x}-\frac {\sqrt [3]{-b+x}}{\sqrt [6]{d}}\right )}{4 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 \left (c+\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \log \left (-\sqrt [3]{-a+x}+\frac {\sqrt [3]{-b+x}}{\sqrt [6]{d}}\right )}{4 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.44, size = 501, normalized size = 0.27 \begin {gather*} -\frac {(b-x)^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \left (c+\sqrt {d}\right ) \text {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 ) \text {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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

-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*Lo
g[-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))

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (-b +x \right ) \left (-b -a c +\left (1+c \right ) x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (-b^{2}+a^{2} d +2 \left (-a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-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)

[Out]

int((-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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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="m
axima")

[Out]

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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 9468 vs. \(2 (1441) = 2882\).
time = 0.79, size = 9468, normalized size = 5.02 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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="f
ricas")

[Out]

-sqrt(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)*arctan
(1/3*(2*(sqrt(3)*((a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*c^2*d^3 + (a^5*b - 5*a^4*b^2 +
10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*d^4 - ((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^2
*d^3 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d^4)*x)*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)) - 2*sqrt(3)*(3*(a^2*b - 2*a*b^2 + b^3)*c
^3*d^2 + (a^2*b - 2*a*b^2 + b^3)*c*d^3 - (3*(a^2 - 2*a*b + b^2)*c^3*d^2 + (a^2 - 2*a*b + b^2)*c*d^3)*x))*sqrt(
((9*c^8 - 12*c^6*d - 2*c^4*d^2 + 4*c^2*d^3 + d^4)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3) + (9*
(a*b - b^2)*c^6*d - 3*(a*b - b^2)*c^4*d^2 - 5*(a*b - b^2)*c^2*d^3 - (a*b - b^2)*d^4 - (9*(a - b)*c^6*d - 3*(a
- b)*c^4*d^2 - 5*(a - b)*c^2*d^3 - (a - b)*d^4)*x - (3*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c^5*d^2
- 2*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c^3*d^3 - (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)
*c*d^4 - (3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^5*d^2 - 2*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4
)*c^3*d^3 - (a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c*d^4)*x)*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*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*
x)^(1/3)*(((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) + (9*(
a^2*b^2 - 2*a*b^3 + b^4)*c^6*d + 15*(a^2*b^2 - 2*a*b^3 + b^4)*c^4*d^2 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*c^2*d^3 +
(a^2*b^2 - 2*a*b^3 + b^4)*d^4 + (9*(a^2 - 2*a*b + b^2)*c^6*d + 15*(a^2 - 2*a*b + b^2)*c^4*d^2 + 7*(a^2 - 2*a*b
+ b^2)*c^2*d^3 + (a^2 - 2*a*b + b^2)*d^4)*x^2 - 2*(9*(a^2*b - 2*a*b^2 + b^3)*c^6*d + 15*(a^2*b - 2*a*b^2 + b^
3)*c^4*d^2 + 7*(a^2*b - 2*a*b^2 + b^3)*c^2*d^3 + (a^2*b - 2*a*b^2 + b^3)*d^4)*x - 2*(3*(a^5*b^2 - 5*a^4*b^3 +
10*a^3*b^4 - 10*a^2*b^5 + 5*a*b^6 - b^7)*c^3*d^3 + (a^5*b^2 - 5*a^4*b^3 + 10*a^3*b^4 - 10*a^2*b^5 + 5*a*b^6 -
b^7)*c*d^4 + (3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^3*d^3 + (a^5 - 5*a^4*b + 10*a^3*b^
2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c*d^4)*x^2 - 2*(3*(a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6
)*c^3*d^3 + (a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*c*d^4)*x)*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))^(2/3))/(b^2 - 2*b*x + x^2))*(((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))^(2/3) - 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 +
(2*a*b + b^2)*x)^(1/3)*(sqrt(3)*(3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^6*d^3 + (a^5 -
5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^4*d^4 - 3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*
b^4 - b^5)*c^2*d^5 - (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d^6)*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)) - 2*sqrt(3)*(9*(a^2 - 2*a*b +
b^2)*c^7*d^2 - 3*(a^2 - 2*a*b + b^2)*c^5*d^3 - 5*(a^2 - 2*a*b + b^2)*c^3*d^4 - (a^2 - 2*a*b + b^2)*c*d^5))*((
(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 + 1
5*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))^(2/3) + sqrt(3)*(9*b*c^
10 - 21*b*c^8*d + 10*b*c^6*d^2 + 6*b*c^4*d^3 - 3*b*c^2*d^4 - b*d^5 - (9*c^10 - 21*c^8*d + 10*c^6*d^2 + 6*c^4*d
^3 - 3*c^2*d^4 - d^5)*x))/(9*b*c^10 - 21*b*c^8*d + 10*b*c^6*d^2 + 6*b*c^4*d^3 - 3*b*c^2*d^4 - b*d^5 - (9*c^10
- 21*c^8*d + 10*c^6*d^2 + 6*c^4*d^3 - 3*c^2*d^4 - d^5)*x)) + sqrt(3)*(-((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sq
rt((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)*arctan(1/3*(2*(sqrt(3)*((a^5*b - 5*a^4*b^2 + 10*a^3*b^3
- 10*a^2*b^4 + 5*a*b^5 - b^6)*c^2*d^3 + (a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*d^4 - (
(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^2*d^3 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 +
5*a*b^4 - b^5)*d^4)*x)*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)) + 2*sqrt(3)*(3*(a^2*b - 2*a*b^2 + b^3)*c^3*d^2 + (a^2*b - 2*a*b^2 + b^3)*c*d^3 - (3*(a^2
- 2*a*b + b^2)*c^3*d^2 + (a^2 - 2*a*b + b^2)*c*...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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="g
iac")

[Out]

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
)

[Out]

-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)

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