3.32.49 \(\int \frac {-b-a c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2)} \, dx\) [3149]

Optimal. Leaf size=1707 \[ \frac {(b-x)^{2/3} \sqrt [3]{-a+x} \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^{5/6}}-\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^{5/6}}+\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^{5/6}}+\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^{5/6}}-\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^{5/6}}-\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^{5/6}}+\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^{5/6}}+\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^{5/6}}-\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^{5/6}}-\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^{5/6}}+\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^{5/6}}+\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^{5/6}}-\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^{5/6}}-\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^{5/6}}+\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^{5/6}}+\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^{5/6}}+\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^{5/6}}+\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^{5/6}}+\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^{5/6}}-\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^{5/6}}+\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^{5/6}}+\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^{5/6}}+\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^{5/6}}-\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^{5/6}}\right )}{\sqrt [3]{(b-x)^2 (-a+x)}} \]

[Out]

(b-x)^(2/3)*(-a+x)^(1/3)*(-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^(5/6)-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^(5/6)+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^(5/6)+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^(5/6)-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^(5/6)-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^(5/6)+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^(5/6)+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^(5/6)-1/2*b*(1+d^(1/2))*l
n((b-x)^(1/3)-d^(1/6)*(-a+x)^(1/3))/(a-b)^2/d^(5/6)-1/2*a*c*(1+d^(1/2))*ln((b-x)^(1/3)-d^(1/6)*(-a+x)^(1/3))/(
a-b)^2/d^(5/6)+1/2*(b+a*d^(1/2))*ln((b-x)^(1/3)-d^(1/6)*(-a+x)^(1/3))/(a-b)^2/d^(5/6)+1/2*c*(b+a*d^(1/2))*ln((
b-x)^(1/3)-d^(1/6)*(-a+x)^(1/3))/(a-b)^2/d^(5/6)-1/2*b*(-1+d^(1/2))*ln((b-x)^(1/3)+d^(1/6)*(-a+x)^(1/3))/(a-b)
^2/d^(5/6)-1/2*a*c*(-1+d^(1/2))*ln((b-x)^(1/3)+d^(1/6)*(-a+x)^(1/3))/(a-b)^2/d^(5/6)+1/2*(-b+a*d^(1/2))*ln((b-
x)^(1/3)+d^(1/6)*(-a+x)^(1/3))/(a-b)^2/d^(5/6)+1/2*c*(-b+a*d^(1/2))*ln((b-x)^(1/3)+d^(1/6)*(-a+x)^(1/3))/(a-b)
^2/d^(5/6)+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^
(5/6)+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^(5/
6)+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^(5/6)-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^(5/6)+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^(5/6)+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^(5/6)+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^(5/6)-1/4*c*(b+a*d^(1/2))*l
n((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^(5/6))/((b-x)^2*(-a+x))^(1/3)

________________________________________________________________________________________

Rubi [A]
time = 1.41, antiderivative size = 561, normalized size of antiderivative = 0.33, number of steps used = 5, number of rules used = 3, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6851, 6860, 93} \begin {gather*} -\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \left (c-\sqrt {d}\right ) \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}\right )}{2 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \left (c+\sqrt {d}\right ) \text {ArcTan}\left (\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{2 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt [3]{x-a} (x-b)^{2/3} \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^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{x-a} (x-b)^{2/3} \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^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \left (c-\sqrt {d}\right ) \log \left (-\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{4 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \left (c+\sqrt {d}\right ) \log \left (\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{4 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-b - a*c + (1 + c)*x)/(((-a + x)*(-b + x)^2)^(1/3)*(-b^2 + a^2*d + 2*(b - a*d)*x + (-1 + d)*x^2)),x]

[Out]

-1/2*(Sqrt[3]*(c - Sqrt[d])*(-a + x)^(1/3)*(-b + x)^(2/3)*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(-a + x)^(1/3))/(Sqrt[
3]*(-b + x)^(1/3))])/((a - b)*d^(5/6)*(-((a - x)*(b - x)^2))^(1/3)) + (Sqrt[3]*(c + Sqrt[d])*(-a + x)^(1/3)*(-
b + x)^(2/3)*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(-a + x)^(1/3))/(Sqrt[3]*(-b + x)^(1/3))])/(2*(a - b)*d^(5/6)*(-((a
- x)*(b - x)^2))^(1/3)) - ((c + Sqrt[d])*(-a + x)^(1/3)*(-b + x)^(2/3)*Log[2*(1 + Sqrt[d])*(b - a*Sqrt[d]) -
2*(1 - d)*x])/(4*(a - b)*d^(5/6)*(-((a - x)*(b - x)^2))^(1/3)) + ((c - Sqrt[d])*(-a + x)^(1/3)*(-b + x)^(2/3)*
Log[2*(1 - Sqrt[d])*(b + a*Sqrt[d]) - 2*(1 - d)*x])/(4*(a - b)*d^(5/6)*(-((a - x)*(b - x)^2))^(1/3)) - (3*(c -
Sqrt[d])*(-a + x)^(1/3)*(-b + x)^(2/3)*Log[-(d^(1/6)*(-a + x)^(1/3)) - (-b + x)^(1/3)])/(4*(a - b)*d^(5/6)*(-
((a - x)*(b - x)^2))^(1/3)) + (3*(c + Sqrt[d])*(-a + x)^(1/3)*(-b + x)^(2/3)*Log[d^(1/6)*(-a + x)^(1/3) - (-b
+ x)^(1/3)])/(4*(a - b)*d^(5/6)*(-((a - x)*(b - x)^2))^(1/3))

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-a c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx &=\frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {-b-a c+(1+c) x}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \left (\frac {1+c+\frac {c+d}{\sqrt {d}}}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (-2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )}+\frac {1+c-\frac {c+d}{\sqrt {d}}}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\left (1+c-\frac {c+d}{\sqrt {d}}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}+\frac {\left (\left (1+c+\frac {c+d}{\sqrt {d}}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (-2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=-\frac {\sqrt {3} \left (c-\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [3]{-b+x}}\right )}{2 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt {3} \left (c+\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [3]{-b+x}}\right )}{2 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (c+\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (2 \left (1+\sqrt {d}\right ) \left (b-a \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (c-\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (b+a \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \left (c-\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (-\sqrt [6]{d} \sqrt [3]{-a+x}-\sqrt [3]{-b+x}\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {3 \left (c+\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (\sqrt [6]{d} \sqrt [3]{-a+x}-\sqrt [3]{-b+x}\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.42, size = 495, normalized size = 0.29 \begin {gather*} -\frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \left (c+\sqrt {d}\right ) \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{b-x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\right )-2 \sqrt {3} \left (c-\sqrt {d}\right ) \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{b-x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\right )+2 c \log \left (\sqrt [6]{d}-\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 \sqrt {d} \log \left (\sqrt [6]{d}-\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 c \log \left (\sqrt [6]{d}+\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 \sqrt {d} \log \left (\sqrt [6]{d}+\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+c \log \left (\sqrt [3]{d}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+\sqrt {d} \log \left (\sqrt [3]{d}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-c \log \left (\sqrt [3]{d}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+\sqrt {d} \log \left (\sqrt [3]{d}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )\right )}{4 (a-b) d^{5/6} \sqrt [3]{(b-x)^2 (-a+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-b - a*c + (1 + c)*x)/(((-a + x)*(-b + x)^2)^(1/3)*(-b^2 + a^2*d + 2*(b - a*d)*x + (-1 + d)*x^2)),x
]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b-a*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-b^2+a^2*d+2*(-a*d+b)*x+(-1+d)*x^2),x)

[Out]

int((-b-a*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-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-a*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-b^2+a^2*d+2*(-a*d+b)*x+(-1+d)*x^2),x, algorithm="maxima")

[Out]

-integrate((a*c - (c + 1)*x + b)/((-(a - x)*(b - x)^2)^(1/3)*(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. 11788 vs. \(2 (1297) = 2594\).
time = 1.09, size = 11788, normalized size = 6.91 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b-a*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-b^2+a^2*d+2*(-a*d+b)*x+(-1+d)*x^2),x, algorithm="fricas")

[Out]

-sqrt(3)*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((c^6 + 6*c^4*d + 9*c^2*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^5)) + 3*c^2 + d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3)*arct
an(1/3*(2*(sqrt(3)*((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d^4*x - (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b
^4 + b^5)*d^4)*sqrt((c^6 + 6*c^4*d + 9*c^2*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^5)) + sqrt(3)*((a*b - b^2)*c^4*d + 3*(a*b - b^2)*c^2*d^2 - ((a - b)*c^4*d + 3*(a - b)*c^2*d^2)*x))
*sqrt(((2*(a^2*b - 2*a*b^2 + b^3)*c^11*d^2 + 8*(a^2*b - 2*a*b^2 + b^3)*c^9*d^3 - 4*(a^2*b - 2*a*b^2 + b^3)*c^7
*d^4 - 24*(a^2*b - 2*a*b^2 + b^3)*c^5*d^5 + 18*(a^2*b - 2*a*b^2 + b^3)*c^3*d^6 - 2*((a^2 - 2*a*b + b^2)*c^11*d
^2 + 4*(a^2 - 2*a*b + b^2)*c^9*d^3 - 2*(a^2 - 2*a*b + b^2)*c^7*d^4 - 12*(a^2 - 2*a*b + b^2)*c^5*d^5 + 9*(a^2 -
2*a*b + b^2)*c^3*d^6)*x - ((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^9*d^4 + 2*(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^7*d^5 - 4*(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^5*d^6 - 2*(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^7 + 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^8 - ((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*
a*b^4 - b^5)*c^9*d^4 + 2*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^7*d^5 - 4*(a^5 - 5*a^4*b
+ 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^5*d^6 - 2*(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^7 + 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^8)*x)*sqrt((c^6 + 6*c^4*d + 9*c^2
*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^5)))*(-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((c^6 + 6*c^4*d + 9*c^2*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^5)) + 3*c^2 + d)/((a^3 - 3*a^2*b + 3*a*b^2 -
b^3)*d^2))^(2/3) + (c^14 + 2*c^12*d - 9*c^10*d^2 - 4*c^8*d^3 + 31*c^6*d^4 - 30*c^4*d^5 + 9*c^2*d^6)*(-a*b^2 -
(a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3) - ((a*b^2 - b^3)*c^12*d + 3*(a*b^2 - b^3)*c^10*d^2 - 6*(a*b^2 - b
^3)*c^8*d^3 - 10*(a*b^2 - b^3)*c^6*d^4 + 21*(a*b^2 - b^3)*c^4*d^5 - 9*(a*b^2 - b^3)*c^2*d^6 + ((a - b)*c^12*d
+ 3*(a - b)*c^10*d^2 - 6*(a - b)*c^8*d^3 - 10*(a - b)*c^6*d^4 + 21*(a - b)*c^4*d^5 - 9*(a - b)*c^2*d^6)*x^2 -
2*((a*b - b^2)*c^12*d + 3*(a*b - b^2)*c^10*d^2 - 6*(a*b - b^2)*c^8*d^3 - 10*(a*b - b^2)*c^6*d^4 + 21*(a*b - b^
2)*c^4*d^5 - 9*(a*b - b^2)*c^2*d^6)*x - ((a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*c^10*d^3 - 6*(a^4*b
^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*c^6*d^5 + 8*(a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*c^4*
d^6 - 3*(a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*c^2*d^7 + ((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^
4)*c^10*d^3 - 6*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^6*d^5 + 8*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 +
b^4)*c^4*d^6 - 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^2*d^7)*x^2 - 2*((a^4*b - 4*a^3*b^2 + 6*a^2*b^3
- 4*a*b^4 + b^5)*c^10*d^3 - 6*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c^6*d^5 + 8*(a^4*b - 4*a^3*b^2
+ 6*a^2*b^3 - 4*a*b^4 + b^5)*c^4*d^6 - 3*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c^2*d^7)*x)*sqrt((c^6
+ 6*c^4*d + 9*c^2*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^5)))*(((a^3
- 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((c^6 + 6*c^4*d + 9*c^2*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^5)) + 3*c^2 + d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3))/(b^2 - 2*b*x + x^2)
)*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((c^6 + 6*c^4*d + 9*c^2*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^5)) + 3*c^2 + d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3) - 2*(-a*b^2
- (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(sqrt(3)*((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^7*d^4
+ (a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^5*d^5 - 5*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^3*d^6
+ 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c*d^7)*sqrt((c^6 + 6*c^4*d + 9*c^2*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^5)) - sqrt(3)*((a - b)*c^11*d + 4*(a - b)*c^9*d^2 - 2*(a -
b)*c^7*d^3 - 12*(a - b)*c^5*d^4 + 9*(a - b)*c^3*d^5))*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((c^6 + 6*c^4
*d + 9*c^2*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^5)) + 3*c^2 + d)/((a
^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3) - sqrt(3)*(b*c^12 + 3*b*c^10*d - 6*b*c^8*d^2 - 10*b*c^6*d^3 + 21*b*c
^4*d^4 - 9*b*c^2*d^5 - (c^12 + 3*c^10*d - 6*c^8*d^2 - 10*c^6*d^3 + 21*c^4*d^4 - 9*c^2*d^5)*x))/(b*c^12 + 3*b*c
^10*d - 6*b*c^8*d^2 - 10*b*c^6*d^3 + 21*b*c^4*d^4 - 9*b*c^2*d^5 - (c^12 + 3*c^10*d - 6*c^8*d^2 - 10*c^6*d^3 +
21*c^4*d^4 - 9*c^2*d^5)*x)) + sqrt(3)*(-((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((c^6 + 6*c^4*d + 9*c^2*d^2)/
((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*...

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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-a*c+(1+c)*x)/((-a+x)*(-b+x)**2)**(1/3)/(-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-a*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-b^2+a^2*d+2*(-a*d+b)*x+(-1+d)*x^2),x, algorithm="giac")

[Out]

integrate(-(a*c - (c + 1)*x + b)/((-(a - x)*(b - x)^2)^(1/3)*(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 {b+a\,c-x\,\left (c+1\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\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 + a*c - x*(c + 1))/((-(a - x)*(b - x)^2)^(1/3)*(a^2*d + 2*x*(b - a*d) - b^2 + x^2*(d - 1))),x)

[Out]

-int((b + a*c - x*(c + 1))/((-(a - x)*(b - x)^2)^(1/3)*(a^2*d + 2*x*(b - a*d) - b^2 + x^2*(d - 1))), x)

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