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

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

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Rubi [A]  time = 0.83, antiderivative size = 408, normalized size of antiderivative = 1.99, number of steps used = 9, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {6719, 911, 105, 59, 91} \begin {gather*} -\frac {x^{4/3} (x-a)^{2/3} \log \left (2 a \left (1-\sqrt {d}\right )-2 (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 (2 (1-d) x-2 a \left (\sqrt {d}+1\right )\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}}-\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 {\sqrt {3} x^{4/3} (x-a)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-a}}\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} \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 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/Sqrt[3] - (2*d^(1/6)*x^(1/3))/(Sqrt[3]*(-a + x)^(1/3))])/(2*a*d^(5/6)
*(-((a - x)*x^2))^(2/3)) - (Sqrt[3]*x^(4/3)*(-a + x)^(2/3)*ArcTan[1/Sqrt[3] + (2*d^(1/6)*x^(1/3))/(Sqrt[3]*(-a
+ x)^(1/3))])/(2*a*d^(5/6)*(-((a - x)*x^2))^(2/3)) - (x^(4/3)*(-a + x)^(2/3)*Log[2*a*(1 - Sqrt[d]) - 2*(1 - d
)*x])/(4*a*d^(5/6)*(-((a - x)*x^2))^(2/3)) + (x^(4/3)*(-a + x)^(2/3)*Log[-2*a*(1 + Sqrt[d]) + 2*(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 59

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

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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 911

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 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]

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 [C]  time = 0.15, size = 75, normalized size = 0.37 \begin {gather*} \frac {3 x^2 \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} x}{x-a}\right )-\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} x}{a-x}\right )\right )}{4 a \sqrt {d} \left (x^2 (x-a)\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]

(3*x^2*(-Hypergeometric2F1[2/3, 1, 5/3, (Sqrt[d]*x)/(a - x)] + Hypergeometric2F1[2/3, 1, 5/3, (Sqrt[d]*x)/(-a
+ x)]))/(4*a*Sqrt[d]*(x^2*(-a + x))^(2/3))

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IntegrateAlgebraic [A]  time = 0.66, size = 205, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\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} \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.49, 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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giac [A]  time = 0.22, 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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maple [F]  time = 0.15, 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*} \int \frac {x^{2}}{\left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}} {\left ({\left (d - 1\right )} x^{2} - a^{2} + 2 \, a x\right )}}\,{d x} \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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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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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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