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

Optimal. Leaf size=139 \[ -\frac {\log \left (d^{2/3} \left (x^3-a x^2\right )^{2/3}+\sqrt [3]{d} x \sqrt [3]{x^3-a x^2}+x^2\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{x^3-a x^2}\right )}{a \sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{d} \sqrt [3]{x^3-a x^2}+x}\right )}{a \sqrt [3]{d}} \]

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Rubi [A]  time = 0.36, antiderivative size = 194, normalized size of antiderivative = 1.40, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {6719, 91} \begin {gather*} -\frac {x^{4/3} (x-a)^{2/3} \log (-a d-(1-d) x)}{2 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 (\frac {\sqrt [3]{x}}{\sqrt [3]{d}}-\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 [3]{x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{d} \left (-\left (x^2 (a-x)\right )\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [C]  time = 0.03, size = 46, normalized size = 0.33 \begin {gather*} -\frac {3 x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x}{d (x-a)}\right )}{2 a d \left (x^2 (x-a)\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.40, size = 139, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}}\right )}{a \sqrt [3]{d}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{-a x^2+x^3}\right )}{a \sqrt [3]{d}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{-a x^2+x^3}+d^{2/3} \left (-a x^2+x^3\right )^{2/3}\right )}{2 a \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 338, normalized size = 2.43 \begin {gather*} \left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (\frac {2 \, a d x - {\left (2 \, d + 1\right )} x^{2} - \sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x - 2 \, {\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 {1}{3}} d^{\frac {1}{3}} x}{a d x - {\left (d - 1\right )} x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {2}{3}} x - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right )}{2 \, a d}, -\frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} x + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}}\right )}}{3 \, d^{\frac {1}{3}} x}\right ) - 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {2}{3}} x - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x}\right ) + d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right )}{2 \, a d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 107, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {3} {\left | d \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} d^{\frac {1}{3}} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + \frac {1}{d^{\frac {1}{3}}}\right )}\right )}{a d} - \frac {{\left | d \right |}^{\frac {2}{3}} \log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{d^{\frac {1}{3}}} + \frac {1}{d^{\frac {2}{3}}}\right )}{2 \, a d} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} - \frac {1}{d^{\frac {1}{3}}} \right |}\right )}{a d^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {x}{\left (a\,d-x\,\left (d-1\right )\right )\,{\left (-x^2\,\left (a-x\right )\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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