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

Optimal. Leaf size=438 \[ \frac {(x-a)^{2/3} (x-b) \left (-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}+\sqrt [3]{d} (b-x)^{2/3}\right ) \left (\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}+\sqrt [3]{d} (b-x)^{2/3}\right ) \left ((x-a)^{2/3} \sqrt [3]{b-x}+b \left (-\sqrt [3]{d}\right )+\sqrt [3]{d} x\right ) \left (\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 d^{5/6} (a-b)}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}+2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 d^{5/6} (a-b)}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{x-a} (b-x)^{2/3}}{\sqrt [6]{d} (x-b)}\right )}{d^{5/6} (a-b)}-\frac {\tanh ^{-1}\left (\frac {\frac {(x-a)^{2/3}}{\sqrt [6]{d}}+\sqrt [6]{d} (b-x)^{2/3}}{\sqrt [3]{x-a} \sqrt [3]{b-x}}\right )}{2 d^{5/6} (a-b)}\right )}{\left ((x-a) (b-x)^2\right )^{2/3} \left (-a^2+2 a x+b^2 d-2 b d x+(d-1) x^2\right )} \]

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Rubi [A]  time = 1.32, antiderivative size = 513, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 5, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6719, 911, 105, 59, 91} \begin {gather*} -\frac {(x-a)^{2/3} (x-b)^{4/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {(x-a)^{2/3} (x-b)^{4/3} \log \left (2 (1-d) x-2 \left (\sqrt {d}+1\right ) \left (a-b \sqrt {d}\right )\right )}{4 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 (x-a)^{2/3} (x-b)^{4/3} \log \left (-\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{x-b}\right )}{4 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {3 (x-a)^{2/3} (x-b)^{4/3} \log \left (\sqrt [6]{d} \sqrt [3]{x-b}-\sqrt [3]{x-a}\right )}{4 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {\sqrt {3} (x-a)^{2/3} (x-b)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}\right )}{2 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {\sqrt {3} (x-a)^{2/3} (x-b)^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [C]  time = 0.28, size = 95, normalized size = 0.22 \begin {gather*} -\frac {3 (b-x)^2 \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} (b-x)}{x-a}\right )-\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} (x-b)}{x-a}\right )\right )}{4 \sqrt {d} (a-b) \left ((x-a) (b-x)^2\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 14.42, size = 438, normalized size = 1.00 \begin {gather*} \frac {(-a+x)^{2/3} (-b+x) \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right ) \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right ) \left (-b \sqrt [3]{d}+\sqrt [3]{d} x+\sqrt [3]{b-x} (-a+x)^{2/3}\right ) \left (\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {(b-x)^{2/3} \sqrt [3]{-a+x}}{\sqrt [6]{d} (-b+x)}\right )}{(a-b) d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} (b-x)^{2/3}+\frac {(-a+x)^{2/3}}{\sqrt [6]{d}}}{\sqrt [3]{b-x} \sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6}}\right )}{\left ((b-x)^2 (-a+x)\right )^{2/3} \left (-a^2+b^2 d+2 a x-2 b d x+(-1+d) x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.68, size = 2217, normalized size = 5.06

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6)*arctan(-1/3*(2*
sqrt(3)*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2
)*x)^(1/3)*d^4*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(5/6) - 2*sqrt
(3)*((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d^4*x - (a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^
2*b^4 + 5*a*b^5 - b^6)*d^4)*sqrt(((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*((a - b)*d*x - (a*b -
b^2)*d)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6) + ((a^2 - 2*a*
b + b^2)*d^2*x^2 - 2*(a^2*b - 2*a*b^2 + b^3)*d^2*x + (a^2*b^2 - 2*a*b^3 + b^4)*d^2)*(1/((a^6 - 6*a^5*b + 15*a^
4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/3) + (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x
)^(2/3))/(b^2 - 2*b*x + x^2))*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))
^(5/6) - sqrt(3)*(b - x))/(b - x)) + sqrt(3)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b
^5 + b^6)*d^5))^(1/6)*arctan(-1/3*(2*sqrt(3)*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*(-a*b^2
- (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*d^4*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4
- 6*a*b^5 + b^6)*d^5))^(5/6) - 2*sqrt(3)*((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d^4*x - (a
^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*d^4)*sqrt(-((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b
+ b^2)*x)^(1/3)*((a - b)*d*x - (a*b - b^2)*d)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a
*b^5 + b^6)*d^5))^(1/6) - ((a^2 - 2*a*b + b^2)*d^2*x^2 - 2*(a^2*b - 2*a*b^2 + b^3)*d^2*x + (a^2*b^2 - 2*a*b^3
+ b^4)*d^2)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/3) - (-a*b^2 -
(a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3))/(b^2 - 2*b*x + x^2))*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b
^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(5/6) + sqrt(3)*(b - x))/(b - x)) + 1/4*(1/((a^6 - 6*a^5*b + 15*a^4*b^2
- 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6)*log(((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^
(1/3)*((a - b)*d*x - (a*b - b^2)*d)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)
*d^5))^(1/6) + ((a^2 - 2*a*b + b^2)*d^2*x^2 - 2*(a^2*b - 2*a*b^2 + b^3)*d^2*x + (a^2*b^2 - 2*a*b^3 + b^4)*d^2)
*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/3) + (-a*b^2 - (a + 2*b)*
x^2 + x^3 + (2*a*b + b^2)*x)^(2/3))/(b^2 - 2*b*x + x^2)) - 1/4*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 +
15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6)*log(-((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*((a - b)*
d*x - (a*b - b^2)*d)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6) -
((a^2 - 2*a*b + b^2)*d^2*x^2 - 2*(a^2*b - 2*a*b^2 + b^3)*d^2*x + (a^2*b^2 - 2*a*b^3 + b^4)*d^2)*(1/((a^6 - 6*a
^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/3) - (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*
a*b + b^2)*x)^(2/3))/(b^2 - 2*b*x + x^2)) + 1/2*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*
a*b^5 + b^6)*d^5))^(1/6)*log(-(((a - b)*d*x - (a*b - b^2)*d)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15
*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6) + (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3))/(b - x)) - 1/2
*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6)*log((((a - b)*d*x - (a
*b - b^2)*d)*(1/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^5))^(1/6) - (-a*b^2
- (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3))/(b - x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b - x\right )}^{2}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\left (-b +x \right )^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (-a^{2}+b^{2} d +2 \left (-b d +a \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((-b+x)^2/((-a+x)*(-b+x)^2)^(2/3)/(-a^2+b^2*d+2*(-b*d+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 {{\left (b - x\right )}^{2}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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