3.20.70 \(\int \frac {1}{\sqrt [3]{x^2 (-a+x)} (-a d+(-1+d) x)} \, dx\)

Optimal. Leaf size=140 \[ \frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{x^3-a x^2}\right )}{a d^{2/3}}-\frac {\log \left (d^{2/3} \left (x^3-a x^2\right )^{2/3}+\sqrt [3]{d} x \sqrt [3]{x^3-a x^2}+x^2\right )}{2 a d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{d} \sqrt [3]{x^3-a x^2}+x}\right )}{a d^{2/3}} \]

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Rubi [A]  time = 0.38, antiderivative size = 248, normalized size of antiderivative = 1.77, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2081, 2077, 91} \begin {gather*} -\frac {\left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \log ((d-1) x-a d)}{2 a^3 d^{2/3} \sqrt [3]{x^2 (x-a)}}+\frac {3 \left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \log \left (-\sqrt [3]{\frac {2}{3}} \sqrt [3]{d} \sqrt [3]{a^2 (x-a)}-\sqrt [3]{\frac {2}{3}} \sqrt [3]{-a^2 x}\right )}{2 a^3 d^{2/3} \sqrt [3]{x^2 (x-a)}}+\frac {\sqrt {3} \left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{d} \sqrt [3]{a^2 (x-a)}}{\sqrt {3} \sqrt [3]{-a^2 x}}\right )}{a^3 d^{2/3} \sqrt [3]{x^2 (x-a)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2077

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> Dist[(a + b*x + d*x^3)^p/
((3*a - b*x)^p*(3*a + 2*b*x)^(2*p)), Int[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b,
d, e, f, m, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]

Rule 2081

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (-a d+(-1+d) x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{3} (a (-1+d)-3 a d)+(-1+d) x\right ) \sqrt [3]{-\frac {2 a^3}{27}-\frac {a^2 x}{3}+x^3}} \, dx,x,-\frac {a}{3}+x\right )\\ &=\frac {\left (2^{2/3} \left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (-a+x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {2 a^3}{9}-\frac {2 a^2 x}{3}\right )^{2/3} \sqrt [3]{-\frac {2 a^3}{9}+\frac {a^2 x}{3}} \left (\frac {1}{3} (a (-1+d)-3 a d)+(-1+d) x\right )} \, dx,x,-\frac {a}{3}+x\right )}{3 \sqrt [3]{-a x^2+x^3}}\\ &=\frac {\sqrt {3} \sqrt [3]{-a^2 (a-x)} \left (-a^2 x\right )^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{d} \sqrt [3]{-a^2 (a-x)}}{\sqrt {3} \sqrt [3]{-a^2 x}}\right )}{a^3 d^{2/3} \sqrt [3]{-a x^2+x^3}}-\frac {\sqrt [3]{-a^2 (a-x)} \left (-a^2 x\right )^{2/3} \log (a d+(1-d) x)}{2 a^3 d^{2/3} \sqrt [3]{-a x^2+x^3}}+\frac {3 \sqrt [3]{-a^2 (a-x)} \left (-a^2 x\right )^{2/3} \log \left (\sqrt [3]{d} \sqrt [3]{-a^2 (a-x)}+\sqrt [3]{-a^2 x}\right )}{2 a^3 d^{2/3} \sqrt [3]{-a x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 42, normalized size = 0.30 \begin {gather*} -\frac {3 x \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x}{d (x-a)}\right )}{a d \sqrt [3]{x^2 (x-a)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.35, size = 140, 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 d^{2/3}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{-a x^2+x^3}\right )}{a d^{2/3}}-\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 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [A]  time = 0.67, size = 153, normalized size = 1.09 \begin {gather*} \frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} {\left (2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d + {\left (d^{2}\right )}^{\frac {1}{3}} x\right )}}{3 \, d x}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d - {\left (d^{2}\right )}^{\frac {1}{3}} x}{x}\right ) - {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}} d x + {\left (d^{2}\right )}^{\frac {2}{3}} x^{2}}{x^{2}}\right )}{2 \, a d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*sqrt(3)*(d^2)^(1/6)*d*arctan(1/3*sqrt(3)*(d^2)^(1/6)*(2*(-a*x^2 + x^3)^(1/3)*d + (d^2)^(1/3)*x)/(d*x))
+ 2*(d^2)^(2/3)*log(((-a*x^2 + x^3)^(1/3)*d - (d^2)^(1/3)*x)/x) - (d^2)^(2/3)*log(((-a*x^2 + x^3)^(2/3)*d^2 +
(-a*x^2 + x^3)^(1/3)*(d^2)^(1/3)*d*x + (d^2)^(2/3)*x^2)/x^2))/(a*d^2)

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giac [A]  time = 0.44, size = 103, normalized size = 0.74 \begin {gather*} \frac {\sqrt {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 {\left | d \right |}^{\frac {2}{3}}} - \frac {{\left | d \right |}^{\frac {4}{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^{2}} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} - \frac {1}{d^{\frac {1}{3}}} \right |}\right )}{a d^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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