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

Optimal. Leaf size=267 \[ -\frac {\log \left (d^{2/3} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{2/3}+\sqrt [3]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3} \left (\sqrt [3]{d} x-b \sqrt [3]{d}\right )+b^2-2 b x+x^2\right )}{2 \sqrt [3]{d} (a-b)}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}+b-x\right )}{\sqrt [3]{d} (a-b)}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} b-\sqrt {3} x}{-2 \sqrt [3]{d} \sqrt [3]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}+b-x}\right )}{\sqrt [3]{d} (a-b)} \]

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Rubi [A]  time = 0.63, antiderivative size = 239, normalized size of antiderivative = 0.90, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6719, 91} \begin {gather*} -\frac {(x-a)^{2/3} (x-b)^{4/3} \log (-a d+b-(1-d) x)}{2 \sqrt [3]{d} (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 (\frac {\sqrt [3]{x-b}}{\sqrt [3]{d}}-\sqrt [3]{x-a}\right )}{2 \sqrt [3]{d} (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 [3]{x-b}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{d} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [C]  time = 0.06, size = 62, normalized size = 0.23 \begin {gather*} -\frac {3 (b-x)^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {b-x}{a d-d x}\right )}{2 d (a-b) \left ((x-a) (b-x)^2\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 5.81, size = 267, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} b-\sqrt {3} x}{b-x-2 \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{(a-b) \sqrt [3]{d}}+\frac {\log \left (b-x+\sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{(a-b) \sqrt [3]{d}}-\frac {\log \left (b^2-2 b x+x^2+\left (-b \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}+d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{2 (a-b) \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 715, normalized size = 2.68 \begin {gather*} \left [-\frac {\sqrt {3} d \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \log \left (\frac {2 \, a b d + {\left (2 \, d + 1\right )} x^{2} + b^{2} - 2 \, {\left ({\left (a + b\right )} d + b\right )} x + \sqrt {3} {\left ({\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} \left (-d\right )^{\frac {2}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d + {\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}} - 3 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} \left (-d\right )^{\frac {1}{3}}}{a b d + {\left (d - 1\right )} x^{2} - b^{2} - {\left ({\left (a + b\right )} d - 2 \, b\right )} x}\right ) + \left (-d\right )^{\frac {2}{3}} \log \left (-\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} \left (-d\right )^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d + {\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (-\frac {{\left (b - x\right )} \left (-d\right )^{\frac {2}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b - x}\right )}{2 \, {\left (a - b\right )} d}, -\frac {2 \, \sqrt {3} d \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \arctan \left (-\frac {\sqrt {3} {\left ({\left (b - x\right )} \left (-d\right )^{\frac {1}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {2}{3}}\right )} \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{3 \, {\left (b - x\right )}}\right ) + \left (-d\right )^{\frac {2}{3}} \log \left (-\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} \left (-d\right )^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d + {\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (-\frac {{\left (b - x\right )} \left (-d\right )^{\frac {2}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b - x}\right )}{2 \, {\left (a - b\right )} d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {-b +x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (b -a d +\left (-1+d \right ) x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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