3.25.14 \(\int \frac {-a+x}{\sqrt [3]{x^2 (-a+x)} (a^2 d-2 a d x+(-1+d) x^2)} \, dx\) [2414]

Optimal. Leaf size=194 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x-2 \sqrt [6]{d} \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [6]{d} \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}}\right )}{a d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [6]{d}}+\sqrt [6]{d} \left (-a x^2+x^3\right )^{2/3}}{x \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}} \]

[Out]

1/2*3^(1/2)*arctan(3^(1/2)*x/(x-2*d^(1/6)*(-a*x^2+x^3)^(1/3)))/a/d^(5/6)-1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2*d^(
1/6)*(-a*x^2+x^3)^(1/3)))/a/d^(5/6)-arctanh(x/d^(1/6)/(-a*x^2+x^3)^(1/3))/a/d^(5/6)-1/2*arctanh((x^2/d^(1/6)+d
^(1/6)*(-a*x^2+x^3)^(2/3))/x/(-a*x^2+x^3)^(1/3))/a/d^(5/6)

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(405\) vs. \(2(194)=388\).
time = 0.59, antiderivative size = 405, normalized size of antiderivative = 2.09, number of steps used = 11, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6851, 925, 129, 399, 245, 384} \begin {gather*} \frac {\sqrt {3} x^{2/3} \sqrt [3]{x-a} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x-a}}}{\sqrt {3}}\right )}{2 a d^{5/6} \sqrt [3]{-\left (x^2 (a-x)\right )}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x-a} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x-a}}+1}{\sqrt {3}}\right )}{2 a d^{5/6} \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 d^{5/6} \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 d^{5/6} \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 d^{5/6} \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 d^{5/6} \sqrt [3]{-\left (x^2 (a-x)\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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^(5/6)*(-((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^(5/6)*(-((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^(5/6)*(-((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^
(5/6)*(-((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^(5/6)*
(-((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^(5/6)*(-((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 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

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

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Mathematica [A]
time = 0.42, size = 188, normalized size = 0.97 \begin {gather*} \frac {x^{2/3} \sqrt [3]{-a+x} \left (\sqrt {3} \left (\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [6]{d} \sqrt [3]{-a+x}}\right )-\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [6]{d} \sqrt [3]{-a+x}}\right )\right )-2 \tanh ^{-1}\left (\frac {\sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}\right )-\tanh ^{-1}\left (\frac {\sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{x}}\right )\right )}{2 a d^{5/6} \sqrt [3]{x^2 (-a+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-a + 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)*(Sqrt[3]*(ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) - 2*d^(1/6)*(-a + x)^(1/3))] - ArcTan[(Sqr
t[3]*x^(1/3))/(x^(1/3) + 2*d^(1/6)*(-a + x)^(1/3))]) - 2*ArcTanh[x^(1/3)/(d^(1/6)*(-a + x)^(1/3))] - ArcTanh[x
^(1/3)/(d^(1/6)*(-a + x)^(1/3)) + (d^(1/6)*(-a + x)^(1/3))/x^(1/3)]))/(2*a*d^(5/6)*(x^2*(-a + x))^(1/3))

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {-a +x}{\left (x^{2} \left (-a +x \right )\right )^{\frac {1}{3}} \left (a^{2} d -2 a d x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (152) = 304\).
time = 0.54, size = 535, normalized size = 2.76 \begin {gather*} -\sqrt {3} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, \sqrt {3} a d x \sqrt {\frac {{\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a^{5} d^{4} x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} + a^{4} d^{3} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {2}{3}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - 2 \, \sqrt {3} {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a d \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - \sqrt {3} x}{3 \, x}\right ) - \sqrt {3} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, \sqrt {3} a d x \sqrt {-\frac {{\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a^{5} d^{4} x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - a^{4} d^{3} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {2}{3}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - 2 \, \sqrt {3} {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a d \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + \sqrt {3} x}{3 \, x}\right ) - \frac {1}{2} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a^{5} d^{4} x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {a^{5} d^{4} x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{4} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a^{5} d^{4} x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} + a^{4} d^{3} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {2}{3}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {1}{4} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a^{5} d^{4} x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - a^{4} d^{3} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {2}{3}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*(1/(a^6*d^5))^(1/6)*arctan(1/3*(2*sqrt(3)*a*d*x*sqrt(((-a*x^2 + x^3)^(1/3)*a^5*d^4*x*(1/(a^6*d^5))^(5
/6) + a^4*d^3*x^2*(1/(a^6*d^5))^(2/3) + (-a*x^2 + x^3)^(2/3))/x^2)*(1/(a^6*d^5))^(1/6) - 2*sqrt(3)*(-a*x^2 + x
^3)^(1/3)*a*d*(1/(a^6*d^5))^(1/6) - sqrt(3)*x)/x) - sqrt(3)*(1/(a^6*d^5))^(1/6)*arctan(1/3*(2*sqrt(3)*a*d*x*sq
rt(-((-a*x^2 + x^3)^(1/3)*a^5*d^4*x*(1/(a^6*d^5))^(5/6) - a^4*d^3*x^2*(1/(a^6*d^5))^(2/3) - (-a*x^2 + x^3)^(2/
3))/x^2)*(1/(a^6*d^5))^(1/6) - 2*sqrt(3)*(-a*x^2 + x^3)^(1/3)*a*d*(1/(a^6*d^5))^(1/6) + sqrt(3)*x)/x) - 1/2*(1
/(a^6*d^5))^(1/6)*log((a^5*d^4*x*(1/(a^6*d^5))^(5/6) + (-a*x^2 + x^3)^(1/3))/x) + 1/2*(1/(a^6*d^5))^(1/6)*log(
-(a^5*d^4*x*(1/(a^6*d^5))^(5/6) - (-a*x^2 + x^3)^(1/3))/x) - 1/4*(1/(a^6*d^5))^(1/6)*log(((-a*x^2 + x^3)^(1/3)
*a^5*d^4*x*(1/(a^6*d^5))^(5/6) + a^4*d^3*x^2*(1/(a^6*d^5))^(2/3) + (-a*x^2 + x^3)^(2/3))/x^2) + 1/4*(1/(a^6*d^
5))^(1/6)*log(-((-a*x^2 + x^3)^(1/3)*a^5*d^4*x*(1/(a^6*d^5))^(5/6) - a^4*d^3*x^2*(1/(a^6*d^5))^(2/3) - (-a*x^2
+ x^3)^(2/3))/x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {- a + x}{\sqrt [3]{x^{2} \left (- a + x\right )} \left (a^{2} d - 2 a d x + d x^{2} - x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.48, size = 233, normalized size = 1.20 \begin {gather*} -\frac {\sqrt {3} \log \left (\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{4 \, \left (-d^{5}\right )^{\frac {1}{6}} a} + \frac {\sqrt {3} \log \left (-\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{4 \, \left (-d^{5}\right )^{\frac {1}{6}} a} + \frac {\arctan \left (\frac {\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{2 \, \left (-d^{5}\right )^{\frac {1}{6}} a} + \frac {\arctan \left (-\frac {\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{6}} - 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{2 \, \left (-d^{5}\right )^{\frac {1}{6}} a} + \frac {\arctan \left (\frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{\left (-d^{5}\right )^{\frac {1}{6}} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(3)*log(sqrt(3)*(-a/x + 1)^(1/3)*(-1/d)^(1/6) + (-a/x + 1)^(2/3) + (-1/d)^(1/3))/((-d^5)^(1/6)*a) + 1
/4*sqrt(3)*log(-sqrt(3)*(-a/x + 1)^(1/3)*(-1/d)^(1/6) + (-a/x + 1)^(2/3) + (-1/d)^(1/3))/((-d^5)^(1/6)*a) + 1/
2*arctan((sqrt(3)*(-1/d)^(1/6) + 2*(-a/x + 1)^(1/3))/(-1/d)^(1/6))/((-d^5)^(1/6)*a) + 1/2*arctan(-(sqrt(3)*(-1
/d)^(1/6) - 2*(-a/x + 1)^(1/3))/(-1/d)^(1/6))/((-d^5)^(1/6)*a) + arctan((-a/x + 1)^(1/3)/(-1/d)^(1/6))/((-d^5)
^(1/6)*a)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {a-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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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