\(\int \frac {x}{\sqrt [3]{x^2 (-a+x)} (a^2 d-2 a d x+(-1+d) x^2)} \, dx\) [2683]

Optimal result

Integrand size = 35, antiderivative size = 243 \[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}}\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (x-\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (x+\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a \sqrt [3]{d}}-\frac {\log \left (x^2-\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{d}}-\frac {\log \left (x^2+\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{d}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.67, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {6851, 925, 129, 494, 245, 384} \[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x-a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x-a}}}{\sqrt {3}}\right )}{2 a \sqrt [3]{d} \sqrt [3]{-\left (x^2 (a-x)\right )}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x-a} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x-a}}+1}{\sqrt {3}}\right )}{2 a \sqrt [3]{d} \sqrt [3]{-\left (x^2 (a-x)\right )}}-\frac {x^{2/3} \sqrt [3]{x-a} \log \left (a \left (1-\sqrt {d}\right ) \sqrt {d}-(1-d) x\right )}{4 a \sqrt [3]{d} \sqrt [3]{-\left (x^2 (a-x)\right )}}-\frac {x^{2/3} \sqrt [3]{x-a} \log \left (a \sqrt {d} \left (\sqrt {d}+1\right )+(1-d) x\right )}{4 a \sqrt [3]{d} \sqrt [3]{-\left (x^2 (a-x)\right )}}+\frac {3 x^{2/3} \sqrt [3]{x-a} \log \left (\sqrt [3]{x}-\sqrt [6]{d} \sqrt [3]{x-a}\right )}{4 a \sqrt [3]{d} \sqrt [3]{-\left (x^2 (a-x)\right )}}+\frac {3 x^{2/3} \sqrt [3]{x-a} \log \left (\sqrt [6]{d} \sqrt [3]{x-a}+\sqrt [3]{x}\right )}{4 a \sqrt [3]{d} \sqrt [3]{-\left (x^2 (a-x)\right )}} \]

[In]

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

[Out]

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

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 494

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Dist[e^n/b, Int[
(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[a*(e^n/b), Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /;
FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b
, c, d, e, m, n, -1, q, x]

Rule 925

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x
] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-a+x} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx}{\sqrt [3]{x^2 (-a+x)}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{-a+x}\right ) \int \left (\frac {(-1+d) \sqrt [3]{x}}{a \sqrt {d} \sqrt [3]{-a+x} \left (-2 a \sqrt {d}-2 a d-2 (1-d) x\right )}+\frac {(-1+d) \sqrt [3]{x}}{a \sqrt {d} \sqrt [3]{-a+x} \left (-2 a \sqrt {d}+2 a d+2 (1-d) x\right )}\right ) \, dx}{\sqrt [3]{x^2 (-a+x)}} \\ & = -\frac {\left ((1-d) x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-a+x} \left (-2 a \sqrt {d}-2 a d-2 (1-d) x\right )} \, dx}{a \sqrt {d} \sqrt [3]{x^2 (-a+x)}}-\frac {\left ((1-d) x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-a+x} \left (-2 a \sqrt {d}+2 a d+2 (1-d) x\right )} \, dx}{a \sqrt {d} \sqrt [3]{x^2 (-a+x)}} \\ & = -\frac {\left (3 (1-d) x^{2/3} \sqrt [3]{-a+x}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-a+x^3} \left (-2 a \sqrt {d}-2 a d-2 (1-d) x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{a \sqrt {d} \sqrt [3]{x^2 (-a+x)}}-\frac {\left (3 (1-d) x^{2/3} \sqrt [3]{-a+x}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-a+x^3} \left (-2 a \sqrt {d}+2 a d+2 (1-d) x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{a \sqrt {d} \sqrt [3]{x^2 (-a+x)}} \\ & = \frac {\left (3 (1-d) x^{2/3} \sqrt [3]{-a+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-a+x^3} \left (-2 a \sqrt {d}-2 a d-2 (1-d) x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (1-\sqrt {d}\right ) \sqrt [3]{x^2 (-a+x)}}-\frac {\left (3 (1-d) x^{2/3} \sqrt [3]{-a+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-a+x^3} \left (-2 a \sqrt {d}+2 a d+2 (1-d) x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (1+\sqrt {d}\right ) \sqrt [3]{x^2 (-a+x)}} \\ & = -\frac {\sqrt {3} x^{2/3} \sqrt [3]{-a+x} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\right )}{2 a \sqrt [3]{d} \sqrt [3]{-\left ((a-x) x^2\right )}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{-a+x} \arctan \left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\right )}{2 a \sqrt [3]{d} \sqrt [3]{-\left ((a-x) x^2\right )}}-\frac {x^{2/3} \sqrt [3]{-a+x} \log \left (a \left (1-\sqrt {d}\right ) \sqrt {d}-(1-d) x\right )}{4 a \sqrt [3]{d} \sqrt [3]{-\left ((a-x) x^2\right )}}-\frac {x^{2/3} \sqrt [3]{-a+x} \log \left (a \left (1+\sqrt {d}\right ) \sqrt {d}+(1-d) x\right )}{4 a \sqrt [3]{d} \sqrt [3]{-\left ((a-x) x^2\right )}}+\frac {3 x^{2/3} \sqrt [3]{-a+x} \log \left (\sqrt [3]{x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{4 a \sqrt [3]{d} \sqrt [3]{-\left ((a-x) x^2\right )}}+\frac {3 x^{2/3} \sqrt [3]{-a+x} \log \left (\sqrt [3]{x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{4 a \sqrt [3]{d} \sqrt [3]{-\left ((a-x) x^2\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {x^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{d} (-a+x)^{2/3}}\right )+2 \log \left (\sqrt [3]{x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )+2 \log \left (\sqrt [3]{x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )-\log \left (x^{2/3}-\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )-\log \left (x^{2/3}+\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )\right )}{4 a \sqrt [3]{d} \sqrt [3]{x^2 (-a+x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.58

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+2 \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}+\left (-a +x \right ) \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}+\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{2}}{x^{2}}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{4 \left (\frac {1}{d}\right )^{\frac {2}{3}} a d}\)	\(141\)

[In]

int(x/(x^2*(-a+x))^(1/3)/(a^2*d-2*a*d*x+(-1+d)*x^2),x,method=_RETURNVERBOSE)

[Out]

1/4*(-2*3^(1/2)*arctan(1/3*3^(1/2)*((1/d)^(1/3)*x^2+2*(-x^2*(a-x))^(2/3))/(1/d)^(1/3)/x^2)-ln(((1/d)^(1/3)*(-x
^2*(a-x))^(2/3)+(-a+x)*(-x^2*(a-x))^(1/3)+(1/d)^(2/3)*x^2)/x^2)+2*ln((-(1/d)^(1/3)*x^2+(-x^2*(a-x))^(2/3))/x^2
))/(1/d)^(2/3)/a/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.53 \[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (\frac {2 \, a^{2} d - 4 \, a d x + {\left (2 \, d + 1\right )} x^{2} + \sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a d - d x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {1}{3}}}{a^{2} d - 2 \, a d x + {\left (d - 1\right )} x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a d - d x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}}{x^{2}}\right )}{4 \, a d}, -\frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}\right )}}{3 \, d^{\frac {1}{3}} x^{2}}\right ) - 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right ) + d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a d - d x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}}{x^{2}}\right )}{4 \, a d}\right ] \]

[In]

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

[Out]

[1/4*(sqrt(3)*d*sqrt(-1/d^(2/3))*log((2*a^2*d - 4*a*d*x + (2*d + 1)*x^2 + sqrt(3)*(d^(1/3)*x^2 + 2*(-a*x^2 + x
^3)^(1/3)*(a*d - d*x) + (-a*x^2 + x^3)^(2/3)*d^(2/3))*sqrt(-1/d^(2/3)) - 3*(-a*x^2 + x^3)^(2/3)*d^(1/3))/(a^2*
d - 2*a*d*x + (d - 1)*x^2)) + 2*d^(2/3)*log(-(d^(2/3)*x^2 - (-a*x^2 + x^3)^(2/3)*d)/x^2) - d^(2/3)*log((d^(1/3
)*x^2 - (-a*x^2 + x^3)^(1/3)*(a*d - d*x) + (-a*x^2 + x^3)^(2/3)*d^(2/3))/x^2))/(a*d), -1/4*(2*sqrt(3)*d^(2/3)*
arctan(1/3*sqrt(3)*(d^(1/3)*x^2 + 2*(-a*x^2 + x^3)^(2/3)*d^(2/3))/(d^(1/3)*x^2)) - 2*d^(2/3)*log(-(d^(2/3)*x^2
- (-a*x^2 + x^3)^(2/3)*d)/x^2) + d^(2/3)*log((d^(1/3)*x^2 - (-a*x^2 + x^3)^(1/3)*(a*d - d*x) + (-a*x^2 + x^3)
^(2/3)*d^(2/3))/x^2))/(a*d)]

Sympy [F]

\[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int \frac {x}{\sqrt [3]{x^{2} \left (- a + x\right )} \left (a^{2} d - 2 a d x + d x^{2} - x^{2}\right )}\, dx \]

[In]

integrate(x/(x**2*(-a+x))**(1/3)/(a**2*d-2*a*d*x+(-1+d)*x**2),x)

[Out]

Integral(x/((x**2*(-a + x))**(1/3)*(a**2*d - 2*a*d*x + d*x**2 - x**2)), x)

Maxima [F]

\[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int { \frac {x}{{\left (a^{2} d - 2 \, a d x + {\left (d - 1\right )} x^{2}\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integrate(x/((a^2*d - 2*a*d*x + (d - 1)*x^2)*(-(a - x)*x^2)^(1/3)), x)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.44 \[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} {\left | d \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} d^{\frac {1}{3}} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \frac {1}{d^{\frac {1}{3}}}\right )}\right )}{2 \, a d} - \frac {{\left | d \right |}^{\frac {2}{3}} \log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{d^{\frac {1}{3}}} + \frac {1}{d^{\frac {2}{3}}}\right )}{4 \, a d} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} - \frac {1}{d^{\frac {1}{3}}} \right |}\right )}{2 \, a d^{\frac {1}{3}}} \]

[In]

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

[Out]

-1/2*sqrt(3)*abs(d)^(2/3)*arctan(1/3*sqrt(3)*d^(1/3)*(2*(-a/x + 1)^(2/3) + 1/d^(1/3)))/(a*d) - 1/4*abs(d)^(2/3
)*log((-a/x + 1)^(4/3) + (-a/x + 1)^(2/3)/d^(1/3) + 1/d^(2/3))/(a*d) + 1/2*log(abs((-a/x + 1)^(2/3) - 1/d^(1/3
)))/(a*d^(1/3))

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int \frac {x}{{\left (-x^2\,\left (a-x\right )\right )}^{1/3}\,\left (d\,a^2-2\,d\,a\,x+\left (d-1\right )\,x^2\right )} \,d x \]

[In]

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

[Out]

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