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

Optimal. Leaf size=249 \[ \frac {\log \left (\sqrt [3]{x^3-a x^2}-\sqrt [6]{d} x\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{x^3-a x^2}+\sqrt [6]{d} x\right )}{2 a \sqrt [3]{d}}-\frac {\log \left (-\sqrt [6]{d} x \sqrt [3]{x^3-a x^2}+\left (x^3-a x^2\right )^{2/3}+\sqrt [3]{d} x^2\right )}{4 a \sqrt [3]{d}}-\frac {\log \left (\sqrt [6]{d} x \sqrt [3]{x^3-a x^2}+\left (x^3-a x^2\right )^{2/3}+\sqrt [3]{d} x^2\right )}{4 a \sqrt [3]{d}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{2 \left (x^3-a x^2\right )^{2/3}+\sqrt [3]{d} x^2}\right )}{2 a \sqrt [3]{d}} \]

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Rubi [A]  time = 0.90, antiderivative size = 408, normalized size of antiderivative = 1.64, number of steps used = 10, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {1593, 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 \sqrt [3]{d} \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 \sqrt [3]{d} \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 \sqrt [3]{d} \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 \sqrt [3]{d} \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 \sqrt [3]{d} \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 \sqrt [3]{d} \left (-\left (x^2 (a-x)\right )\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-(a*x) + 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^(1/3)
*(-((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^(1/3)*(-((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^(1/3)*(-((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^(1/3)*(-((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^(1/3)*(-((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^(1/3
)*(-((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 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

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Mathematica [C]  time = 0.17, size = 68, normalized size = 0.27 \begin {gather*} -\frac {3 x^2 \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} x}{a-x}\right )+\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} x}{x-a}\right )\right )}{4 a \left (x^2 (x-a)\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(a*x) + 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*(x^2*(-a + x))^(2/3))

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IntegrateAlgebraic [A]  time = 0.64, size = 247, normalized size = 0.99 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {x^2}{\sqrt {3}}+\frac {2 \left (-a x^2+x^3\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}}{x^2}\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (-\sqrt [6]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (\sqrt [6]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{2 a \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.67, size = 357, normalized size = 1.43 \begin {gather*} \left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {{\left (d + 2\right )} x^{2} + 2 \, a^{2} - 4 \, a x - \sqrt {3} {\left (d^{\frac {4}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )} d^{\frac {2}{3}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}}{{\left (d - 1\right )} x^{2} - a^{2} + 2 \, a x}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {1}{3}}}{x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\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}}\right )}}{3 \, d^{\frac {1}{3}} x^{2}}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {1}{3}}}{x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{4 \, a d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.25, size = 100, normalized size = 0.40 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + d^{\frac {1}{3}}\right )}}{3 \, d^{\frac {1}{3}}}\right )}{2 \, a d^{\frac {1}{3}}} - \frac {\log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + d^{\frac {1}{3}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + d^{\frac {2}{3}}\right )}{4 \, a d^{\frac {1}{3}}} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} - d^{\frac {1}{3}} \right |}\right )}{2 \, a d^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {-a x +x^{2}}{\left (x^{2} \left (-a +x \right )\right )^{\frac {2}{3}} \left (a^{2}-2 a x +\left (1-d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

Timed out

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