3.32.51 \(\int \frac {(-b+x) (-a-b c+(1+c) x)}{((-a+x) (-b+x)^2)^{2/3} (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2)} \, dx\)
Optimal. Leaf size=1835 \[ \frac {(b-x)^{4/3} (x-a)^{2/3} \left (\sqrt [6]{d} \sqrt [3]{b-x}-\sqrt [3]{x-a}\right ) \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right ) \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right ) \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right ) \left (\frac {\sqrt {3} b c \left (\sqrt {d}-1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} a \left (\sqrt {d}-1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} c \left (a-b \sqrt {d}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} \left (a-b \sqrt {d}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} b c \left (\sqrt {d}+1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} a \left (\sqrt {d}+1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{5/6}}-\frac {\sqrt {3} c \left (a+b \sqrt {d}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{5/6}}-\frac {\sqrt {3} \left (a+b \sqrt {d}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{5/6}}-\frac {b c \left (\sqrt {d}+1\right ) \log \left (\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{b-x}\right )}{2 (a-b)^2 d^{5/6}}-\frac {a \left (\sqrt {d}+1\right ) \log \left (\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{b-x}\right )}{2 (a-b)^2 d^{5/6}}+\frac {c \left (a+b \sqrt {d}\right ) \log \left (\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{b-x}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\left (a+b \sqrt {d}\right ) \log \left (\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{b-x}\right )}{2 (a-b)^2 d^{5/6}}-\frac {b c \left (\sqrt {d}-1\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}-\frac {a \left (\sqrt {d}-1\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}-\frac {c \left (a-b \sqrt {d}\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\left (b \sqrt {d}-a\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}+\frac {b c \left (\sqrt {d}-1\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {a \left (\sqrt {d}-1\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {c \left (a-b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {\left (a-b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {b c \left (\sqrt {d}+1\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {a \left (\sqrt {d}+1\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {\left (-a-b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}-\frac {c \left (a+b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}\right )}{\left ((b-x)^2 (x-a)\right )^{2/3} \left (-a^2+2 x a+(d-1) x^2+b^2 d-2 b d x\right )} \]
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Rubi [A] time = 2.28, antiderivative size = 561, normalized size of antiderivative = 0.31,
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 =
{6719, 6728, 91} \begin {gather*} \frac {(x-a)^{2/3} (x-b)^{4/3} \left (c+\sqrt {d}\right ) \log \left (2 \left (\sqrt {d}+1\right ) \left (a-b \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{5/6} (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+b \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{5/6} (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}-\sqrt [6]{d} \sqrt [3]{x-b}\right )}{4 d^{5/6} (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 [6]{d} \sqrt [3]{x-b}-\sqrt [3]{x-a}\right )}{4 d^{5/6} (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 ) \tan ^{-1}\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) \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 ) \tan ^{-1}\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) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((-b + x)*(-a - b*c + (1 + c)*x))/(((-a + x)*(-b + x)^2)^(2/3)*(-a^2 + b^2*d + 2*(a - b*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*d^(1/6)*(-b + x)^(1/3))/(Sqrt[3]*(-
a + x)^(1/3))])/(2*(a - b)*d^(5/6)*(-((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*d^(1/6)*(-b + x)^(1/3))/(Sqrt[3]*(-a + x)^(1/3))])/(2*(a - b)*d^(5/6)*(-((a -
x)*(b - x)^2))^(2/3)) + ((c + Sqrt[d])*(-a + x)^(2/3)*(-b + x)^(4/3)*Log[2*(1 + Sqrt[d])*(a - b*Sqrt[d]) - 2*(
1 - d)*x])/(4*(a - b)*d^(5/6)*(-((a - x)*(b - x)^2))^(2/3)) - ((c - Sqrt[d])*(-a + x)^(2/3)*(-b + x)^(4/3)*Log
[2*(1 - Sqrt[d])*(a + b*Sqrt[d]) - 2*(1 - d)*x])/(4*(a - b)*d^(5/6)*(-((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) - d^(1/6)*(-b + x)^(1/3)])/(4*(a - b)*d^(5/6)*(-((a -
x)*(b - x)^2))^(2/3)) - (3*(c + Sqrt[d])*(-a + x)^(2/3)*(-b + x)^(4/3)*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))
Rule 91
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 6719
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[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 6728
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) (-a-b c+(1+c) x)}{\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 {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {-a-b c+(1+c) x}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a^2+b^2 d+2 (a-b 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 (a-b 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 (a-b 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 (a-b 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 (a-b 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 [6]{d} \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6} \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 [6]{d} \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6} \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 (a-b \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{5/6} \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 (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{5/6} \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}-\sqrt [6]{d} \sqrt [3]{-b+x}\right )}{4 (a-b) d^{5/6} \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}+\sqrt [6]{d} \sqrt [3]{-b+x}\right )}{4 (a-b) d^{5/6} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.34, size = 111, normalized size = 0.06 \begin {gather*} \frac {3 (b-x)^2 \left (\left (\sqrt {d}-c\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} (b-x)}{x-a}\right )+\left (c+\sqrt {d}\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} (x-b)}{x-a}\right )\right )}{4 \sqrt {d} (a-b) \left ((x-a) (b-x)^2\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((-b + x)*(-a - b*c + (1 + c)*x))/(((-a + x)*(-b + x)^2)^(2/3)*(-a^2 + b^2*d + 2*(a - b*d)*x + (-1 +
d)*x^2)),x]
[Out]
(3*(b - x)^2*((-c + Sqrt[d])*Hypergeometric2F1[2/3, 1, 5/3, (Sqrt[d]*(b - x))/(-a + x)] + (c + Sqrt[d])*Hyperg
eometric2F1[2/3, 1, 5/3, (Sqrt[d]*(-b + x))/(-a + x)]))/(4*(a - b)*Sqrt[d]*((b - x)^2*(-a + x))^(2/3))
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IntegrateAlgebraic [A] time = 18.60, size = 571, normalized size = 0.31 \begin {gather*} \frac {(b-x)^{4/3} (-a+x)^{2/3} \left (\sqrt [6]{d} \sqrt [3]{b-x}-\sqrt [3]{-a+x}\right ) \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right ) \left (b d^{2/3} \sqrt [3]{b-x}-d^{2/3} \sqrt [3]{b-x} x+\sqrt [3]{d} (b-x)^{2/3} (-a+x)^{2/3}+(-a+x)^{4/3}\right ) \left (\frac {\sqrt {3} \left (c+\sqrt {d}\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 (a-b) d^{5/6}}-\frac {\sqrt {3} \left (c-\sqrt {d}\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 (a-b) d^{5/6}}+\frac {\left (c-\sqrt {d}\right ) \log \left (\sqrt [6]{d}-\frac {\sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )}{2 (a-b) d^{5/6}}+\frac {\left (-c-\sqrt {d}\right ) \log \left (\sqrt [6]{d}+\frac {\sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )}{2 (a-b) d^{5/6}}+\frac {\left (c+\sqrt {d}\right ) \log \left (\sqrt [3]{d}-\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {(-a+x)^{2/3}}{(b-x)^{2/3}}\right )}{4 (a-b) d^{5/6}}+\frac {\left (-c+\sqrt {d}\right ) \log \left (\sqrt [3]{d}+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {(-a+x)^{2/3}}{(b-x)^{2/3}}\right )}{4 (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 )} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((-b + x)*(-a - b*c + (1 + c)*x))/(((-a + x)*(-b + x)^2)^(2/3)*(-a^2 + b^2*d + 2*(a - b*d)*
x + (-1 + d)*x^2)),x]
[Out]
((b - x)^(4/3)*(-a + x)^(2/3)*(d^(1/6)*(b - x)^(1/3) - (-a + x)^(1/3))*(d^(1/6)*(b - x)^(1/3) + (-a + x)^(1/3)
)*(b*d^(2/3)*(b - x)^(1/3) - d^(2/3)*(b - x)^(1/3)*x + d^(1/3)*(b - x)^(2/3)*(-a + x)^(2/3) + (-a + x)^(4/3))*
((Sqrt[3]*(c + Sqrt[d])*ArcTan[1/Sqrt[3] - (2*(-a + x)^(1/3))/(Sqrt[3]*d^(1/6)*(b - x)^(1/3))])/(2*(a - b)*d^(
5/6)) - (Sqrt[3]*(c - Sqrt[d])*ArcTan[1/Sqrt[3] + (2*(-a + x)^(1/3))/(Sqrt[3]*d^(1/6)*(b - x)^(1/3))])/(2*(a -
b)*d^(5/6)) + ((c - Sqrt[d])*Log[d^(1/6) - (-a + x)^(1/3)/(b - x)^(1/3)])/(2*(a - b)*d^(5/6)) + ((-c - Sqrt[d
])*Log[d^(1/6) + (-a + x)^(1/3)/(b - x)^(1/3)])/(2*(a - b)*d^(5/6)) + ((c + Sqrt[d])*Log[d^(1/3) - (d^(1/6)*(-
a + x)^(1/3))/(b - x)^(1/3) + (-a + x)^(2/3)/(b - x)^(2/3)])/(4*(a - b)*d^(5/6)) + ((-c + Sqrt[d])*Log[d^(1/3)
+ (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3) + (-a + x)^(2/3)/(b - x)^(2/3)])/(4*(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))
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fricas [B] time = 1.37, size = 9684, normalized size = 5.28
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-b+x)*(-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(2/3)/(-a^2+b^2*d+2*(-b*d+a)*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((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)*arc
tan(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^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)*d^5 - ((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^4 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d^5)*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)) - 2*sqrt(3)*((a^2*b - 2*a*b^2 + b
^3)*c^4*d^2 + 3*(a^2*b - 2*a*b^2 + b^3)*c^2*d^3 - ((a^2 - 2*a*b + b^2)*c^4*d^2 + 3*(a^2 - 2*a*b + b^2)*c^2*d^3
)*x))*sqrt(((c^10 + 4*c^8*d - 2*c^6*d^2 - 12*c^4*d^3 + 9*c^2*d^4)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2
)*x)^(2/3) + ((a*b - b^2)*c^9*d + 5*(a*b - b^2)*c^7*d^2 + 3*(a*b - b^2)*c^5*d^3 - 9*(a*b - b^2)*c^3*d^4 - ((a
- b)*c^9*d + 5*(a - b)*c^7*d^2 + 3*(a - b)*c^5*d^3 - 9*(a - b)*c^3*d^4)*x - ((a^4*b - 4*a^3*b^2 + 6*a^2*b^3 -
4*a*b^4 + b^5)*c^5*d^4 + 2*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c^3*d^5 - 3*(a^4*b - 4*a^3*b^2 + 6*
a^2*b^3 - 4*a*b^4 + b^5)*c*d^6 - ((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^5*d^4 + 2*(a^4 - 4*a^3*b + 6*a
^2*b^2 - 4*a*b^3 + b^4)*c^3*d^5 - 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c*d^6)*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))^(1/3) + ((a^2*b^2 - 2*a*b^3 + b^4)*c^8*d^2 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*c^6*d^3 + 15*(a^2*
b^2 - 2*a*b^3 + b^4)*c^4*d^4 + 9*(a^2*b^2 - 2*a*b^3 + b^4)*c^2*d^5 + ((a^2 - 2*a*b + b^2)*c^8*d^2 + 7*(a^2 - 2
*a*b + b^2)*c^6*d^3 + 15*(a^2 - 2*a*b + b^2)*c^4*d^4 + 9*(a^2 - 2*a*b + b^2)*c^2*d^5)*x^2 - 2*((a^2*b - 2*a*b^
2 + b^3)*c^8*d^2 + 7*(a^2*b - 2*a*b^2 + b^3)*c^6*d^3 + 15*(a^2*b - 2*a*b^2 + b^3)*c^4*d^4 + 9*(a^2*b - 2*a*b^2
+ b^3)*c^2*d^5)*x - 2*((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^4*d^5 + 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^2*d^6 + ((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^5 + 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^6)*x^2 - 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^4*d^5 + 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^6)*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))^(2/3))/(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))^(2/3) - 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(sqrt(3)*((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^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^5*d^5 - (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^6 - 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^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)) - 2*sqrt(3)*((a^2 - 2*a*b + b^2)*c^9*d^2 + 5*(a^2 - 2*a*b + b^2)*c^7
*d^3 + 3*(a^2 - 2*a*b + b^2)*c^5*d^4 - 9*(a^2 - 2*a*b + b^2)*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))^(2/3) + 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*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)*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^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)*d^5 - ((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^4 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^
5)*d^5)*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)) + 2*sqrt(3)*((a^2*b - 2*a*b^2 + b^3)*c^4*d^2 + 3*(a^2*b - 2*a*b^2 + b^3)*c^2*d^3 - ((a^2 - 2*a*b +
b^2)*c^4*d^2 + 3*(a^2 - 2*a*b + b^2)*c^2*d^3)*x))*sqrt(((c^10 + 4*c^8*d - 2*c^6*d^2 - 12*c^4*d^3 + 9*c^2*d^4)
*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3) + ((a*b - b^2)*c^9*d + 5*(a*b - b^2)*c^7*d^2 + 3*(a*b
- b^2)*c^5*d^3 - 9*(a*b - b^2)*c^3*d^4 - ((a - b)*c^9*d + 5*(a - b)*c^7*d^2 + 3*(a - b)*c^5*d^3 - 9*(a - b)*c^
3*d^4)*x + ((a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c^5*d^4 + 2*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b
^4 + b^5)*c^3*d^5 - 3*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c*d^6 - ((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*
a*b^3 + b^4)*c^5*d^4 + 2*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^3*d^5 - 3*(a^4 - 4*a^3*b + 6*a^2*b^2 -
4*a*b^3 + b^4)*c*d^6)*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))^(1/3) + ((a^2*b^2 - 2*a*b^3 + b^4)*c^8*d^2 + 7*
(a^2*b^2 - 2*a*b^3 + b^4)*c^6*d^3 + 15*(a^2*b^2 - 2*a*b^3 + b^4)*c^4*d^4 + 9*(a^2*b^2 - 2*a*b^3 + b^4)*c^2*d^5
+ ((a^2 - 2*a*b + b^2)*c^8*d^2 + 7*(a^2 - 2*a*b + b^2)*c^6*d^3 + 15*(a^2 - 2*a*b + b^2)*c^4*d^4 + 9*(a^2 - 2*
a*b + b^2)*c^2*d^5)*x^2 - 2*((a^2*b - 2*a*b^2 + b^3)*c^8*d^2 + 7*(a^2*b - 2*a*b^2 + b^3)*c^6*d^3 + 15*(a^2*b -
2*a*b^2 + b^3)*c^4*d^4 + 9*(a^2*b - 2*a*b^2 + b^3)*c^2*d^5)*x + 2*((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^4*d^5 + 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^2*d^6 + ((
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^5 + 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^6)*x^2 - 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^4*d^5 + 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^6)*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))^(2/3))/(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))^(2/3) - 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*
a*b + b^2)*x)^(1/3)*(sqrt(3)*((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^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^5*d^5 - (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^6 - 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^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)) + 2*sqrt(3)*((a^2 - 2*a*b
+ b^2)*c^9*d^2 + 5*(a^2 - 2*a*b + b^2)*c^7*d^3 + 3*(a^2 - 2*a*b + b^2)*c^5*d^4 - 9*(a^2 - 2*a*b + b^2)*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))^(2/3) - 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)) - 1/4*(-((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)*log(((c^10 + 4*c^8*d - 2*c^6*d
^2 - 12*c^4*d^3 + 9*c^2*d^4)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3) + ((a*b - b^2)*c^9*d + 5*(
a*b - b^2)*c^7*d^2 + 3*(a*b - b^2)*c^5*d^3 - 9*(a*b - b^2)*c^3*d^4 - ((a - b)*c^9*d + 5*(a - b)*c^7*d^2 + 3*(a
- b)*c^5*d^3 - 9*(a - b)*c^3*d^4)*x - ((a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c^5*d^4 + 2*(a^4*b - 4
*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c^3*d^5 - 3*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c*d^6 - ((a^
4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^5*d^4 + 2*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^3*d^5 - 3*(
a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c*d^6)*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))^(1/3) + ((a^2*b^2
- 2*a*b^3 + b^4)*c^8*d^2 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*c^6*d^3 + 15*(a^2*b^2 - 2*a*b^3 + b^4)*c^4*d^4 + 9*(a^2
*b^2 - 2*a*b^3 + b^4)*c^2*d^5 + ((a^2 - 2*a*b + b^2)*c^8*d^2 + 7*(a^2 - 2*a*b + b^2)*c^6*d^3 + 15*(a^2 - 2*a*b
+ b^2)*c^4*d^4 + 9*(a^2 - 2*a*b + b^2)*c^2*d^5)*x^2 - 2*((a^2*b - 2*a*b^2 + b^3)*c^8*d^2 + 7*(a^2*b - 2*a*b^2
+ b^3)*c^6*d^3 + 15*(a^2*b - 2*a*b^2 + b^3)*c^4*d^4 + 9*(a^2*b - 2*a*b^2 + b^3)*c^2*d^5)*x - 2*((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^4*d^5 + 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^2*d^6 + ((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^5 + 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^6)*x^2 - 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^4*d^5 + 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^6)*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))^(2/3))/(b^2 - 2*b*x + x^
2)) - 1/4*(((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)*log
(((c^10 + 4*c^8*d - 2*c^6*d^2 - 12*c^4*d^3 + 9*c^2*d^4)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)
+ ((a*b - b^2)*c^9*d + 5*(a*b - b^2)*c^7*d^2 + 3*(a*b - b^2)*c^5*d^3 - 9*(a*b - b^2)*c^3*d^4 - ((a - b)*c^9*d
+ 5*(a - b)*c^7*d^2 + 3*(a - b)*c^5*d^3 - 9*(a - b)*c^3*d^4)*x + ((a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 +
b^5)*c^5*d^4 + 2*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c^3*d^5 - 3*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 -
4*a*b^4 + b^5)*c*d^6 - ((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^5*d^4 + 2*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4
*a*b^3 + b^4)*c^3*d^5 - 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c*d^6)*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))^(1/3) + ((a^2*b^2 - 2*a*b^3 + b^4)*c^8*d^2 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*c^6*d^3 + 15*(a^2*b^2 - 2*a*b
^3 + b^4)*c^4*d^4 + 9*(a^2*b^2 - 2*a*b^3 + b^4)*c^2*d^5 + ((a^2 - 2*a*b + b^2)*c^8*d^2 + 7*(a^2 - 2*a*b + b^2)
*c^6*d^3 + 15*(a^2 - 2*a*b + b^2)*c^4*d^4 + 9*(a^2 - 2*a*b + b^2)*c^2*d^5)*x^2 - 2*((a^2*b - 2*a*b^2 + b^3)*c^
8*d^2 + 7*(a^2*b - 2*a*b^2 + b^3)*c^6*d^3 + 15*(a^2*b - 2*a*b^2 + b^3)*c^4*d^4 + 9*(a^2*b - 2*a*b^2 + b^3)*c^2
*d^5)*x + 2*((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^4*d^5 + 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^2*d^6 + ((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^5 + 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^6)*x^2 - 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^4*d^5 + 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^6)*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 + 1
5*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))/(b^2 - 2*b*x + x^2)) + 1/2*(-((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)*log(((c^5 + 2*c^3*d - 3*c*d^2)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3) -
((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 + ((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)))*(-((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 - x)) + 1/2*(((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)*log(((c^5 + 2*c^3*d - 3*c*d^2)*(-a*
b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3) - ((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 - ((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)))*(((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 - x))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b c - {\left (c + 1\right )} x + a\right )} {\left (b - x\right )}}{\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-b+x)*(-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(2/3)/(-a^2+b^2*d+2*(-b*d+a)*x+(-1+d)*x^2),x, algorithm="g
iac")
[Out]
integrate((b*c - (c + 1)*x + a)*(b - x)/((-(a - x)*(b - x)^2)^(2/3)*(b^2*d + (d - 1)*x^2 - a^2 - 2*(b*d - a)*x
)), x)
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (-b +x \right ) \left (-a -b c +\left (1+c \right ) x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (-a^{2}+b^{2} d +2 \left (-b d +a \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-b+x)*(-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(2/3)/(-a^2+b^2*d+2*(-b*d+a)*x+(-1+d)*x^2),x)
[Out]
int((-b+x)*(-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(2/3)/(-a^2+b^2*d+2*(-b*d+a)*x+(-1+d)*x^2),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b c - {\left (c + 1\right )} x + a\right )} {\left (b - x\right )}}{\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-b+x)*(-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(2/3)/(-a^2+b^2*d+2*(-b*d+a)*x+(-1+d)*x^2),x, algorithm="m
axima")
[Out]
integrate((b*c - (c + 1)*x + a)*(b - x)/((-(a - x)*(b - x)^2)^(2/3)*(b^2*d + (d - 1)*x^2 - a^2 - 2*(b*d - a)*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 (a+b\,c-x\,\left (c+1\right )\right )}{{\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((b - x)*(a + b*c - x*(c + 1)))/((-(a - x)*(b - x)^2)^(2/3)*(b^2*d + 2*x*(a - b*d) - a^2 + x^2*(d - 1))),x
)
[Out]
-int(-((b - x)*(a + b*c - x*(c + 1)))/((-(a - x)*(b - x)^2)^(2/3)*(b^2*d + 2*x*(a - b*d) - a^2 + x^2*(d - 1)))
, x)
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sympy [F(-1)] 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)*(-a-b*c+(1+c)*x)/((-a+x)*(-b+x)**2)**(2/3)/(-a**2+b**2*d+2*(-b*d+a)*x+(-1+d)*x**2),x)
[Out]
Timed out
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