3.25.88 \(\int \frac {x^2}{(x^2 (-a+x))^{2/3} (-a^2+2 a x+(-1+d) x^2)} \, dx\) [2488]

Optimal. Leaf size=205 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x-2 \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x+2 \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} x}{\sqrt [3]{-a x^2+x^3}}\right )}{a d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} x^2+\frac {\left (-a x^2+x^3\right )^{2/3}}{\sqrt [6]{d}}}{x \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}} \]

[Out]

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

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Rubi [A]
time = 0.72, antiderivative size = 397, normalized size of antiderivative = 1.94, number of steps used = 11, number of rules used = 6, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6851, 925, 129, 494, 337, 503} \begin {gather*} \frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{x-a}}}{\sqrt {3}}\right )}{2 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{x-a}}+1}{\sqrt {3}}\right )}{2 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {x^{4/3} (x-a)^{2/3} \log \left (a \left (1-\sqrt {d}\right )-(1-d) x\right )}{4 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {x^{4/3} (x-a)^{2/3} \log \left (a \left (\sqrt {d}+1\right )-(1-d) x\right )}{4 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\sqrt [6]{d} \sqrt [3]{x}-\sqrt [3]{x-a}\right )}{4 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\sqrt [3]{x-a}+\sqrt [6]{d} \sqrt [3]{x}\right )}{4 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 337

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

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 503

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

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

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Mathematica [A]
time = 0.50, size = 198, normalized size = 0.97 \begin {gather*} \frac {x^{4/3} (-a+x)^{2/3} \left (\sqrt {3} \left (\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x}-2 \sqrt [3]{-a+x}}\right )-\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x}+2 \sqrt [3]{-a+x}}\right )\right )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{-a+x}}\right )+\tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{-a+x}}+\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d} \sqrt [3]{x}}\right )\right )}{2 a d^{5/6} \left (x^2 (-a+x)\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(4/3)*(-a + x)^(2/3)*(Sqrt[3]*(ArcTan[(Sqrt[3]*d^(1/6)*x^(1/3))/(d^(1/6)*x^(1/3) - 2*(-a + x)^(1/3))] - Arc
Tan[(Sqrt[3]*d^(1/6)*x^(1/3))/(d^(1/6)*x^(1/3) + 2*(-a + x)^(1/3))]) + 2*ArcTanh[(d^(1/6)*x^(1/3))/(-a + x)^(1
/3)] + ArcTanh[(d^(1/6)*x^(1/3))/(-a + x)^(1/3) + (-a + x)^(1/3)/(d^(1/6)*x^(1/3))]))/(2*a*d^(5/6)*(x^2*(-a +
x))^(2/3))

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (x^{2} \left (-a +x \right )\right )^{\frac {2}{3}} \left (-a^{2}+2 a x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (159) = 318\).
time = 0.35, size = 521, normalized size = 2.54 \begin {gather*} -\sqrt {3} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, \sqrt {3} a^{5} d^{4} x \sqrt {\frac {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a d x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a^{5} d^{4} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{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^{5} d^{4} x \sqrt {\frac {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a d x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a^{5} d^{4} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{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 d x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{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 d x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{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 {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a d x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + {\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 {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} a d x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + {\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(x^2/(x^2*(-a+x))^(2/3)/(-a^2+2*a*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^5*d^4*x*sqrt((a^2*d^2*x^2*(1/(a^6*d^5))^(1/3) + (-a*x^2 +
x^3)^(1/3)*a*d*x*(1/(a^6*d^5))^(1/6) + (-a*x^2 + x^3)^(2/3))/x^2)*(1/(a^6*d^5))^(5/6) - 2*sqrt(3)*(-a*x^2 + x
^3)^(1/3)*a^5*d^4*(1/(a^6*d^5))^(5/6) - sqrt(3)*x)/x) - sqrt(3)*(1/(a^6*d^5))^(1/6)*arctan(1/3*(2*sqrt(3)*a^5*
d^4*x*sqrt((a^2*d^2*x^2*(1/(a^6*d^5))^(1/3) - (-a*x^2 + x^3)^(1/3)*a*d*x*(1/(a^6*d^5))^(1/6) + (-a*x^2 + x^3)^
(2/3))/x^2)*(1/(a^6*d^5))^(5/6) - 2*sqrt(3)*(-a*x^2 + x^3)^(1/3)*a^5*d^4*(1/(a^6*d^5))^(5/6) + sqrt(3)*x)/x) +
1/2*(1/(a^6*d^5))^(1/6)*log((a*d*x*(1/(a^6*d^5))^(1/6) + (-a*x^2 + x^3)^(1/3))/x) - 1/2*(1/(a^6*d^5))^(1/6)*l
og(-(a*d*x*(1/(a^6*d^5))^(1/6) - (-a*x^2 + x^3)^(1/3))/x) + 1/4*(1/(a^6*d^5))^(1/6)*log((a^2*d^2*x^2*(1/(a^6*d
^5))^(1/3) + (-a*x^2 + x^3)^(1/3)*a*d*x*(1/(a^6*d^5))^(1/6) + (-a*x^2 + x^3)^(2/3))/x^2) - 1/4*(1/(a^6*d^5))^(
1/6)*log((a^2*d^2*x^2*(1/(a^6*d^5))^(1/3) - (-a*x^2 + x^3)^(1/3)*a*d*x*(1/(a^6*d^5))^(1/6) + (-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 {x^{2}}{\left (x^{2} \left (- a + x\right )\right )^{\frac {2}{3}} \left (- a^{2} + 2 a x + d x^{2} - x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.41, size = 209, normalized size = 1.02 \begin {gather*} -\frac {\sqrt {3} \log \left (\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{4 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} \log \left (-\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{4 \, a d} - \frac {\arctan \left (\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{2 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (-\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} - 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{2 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (\frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{a \left (-d\right )^{\frac {5}{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (-a^2+2\,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(x^2/((-x^2*(a - x))^(2/3)*(2*a*x - a^2 + x^2*(d - 1))),x)

[Out]

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

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