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

Optimal. Leaf size=146 \[ \frac {\log \left (\sqrt [3]{d} x \sqrt [3]{x^3-a x^2}+\left (x^3-a x^2\right )^{2/3}+d^{2/3} x^2\right )}{2 a \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{x^3-a x^2}-\sqrt [3]{d} x\right )}{a \sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{2 \sqrt [3]{x^3-a x^2}+\sqrt [3]{d} x}\right )}{a \sqrt [3]{d}} \]

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Rubi [A]  time = 0.34, antiderivative size = 246, normalized size of antiderivative = 1.68, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2081, 2077, 91} \begin {gather*} \frac {\left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \log (a+(d-1) x)}{2 a^3 \sqrt [3]{d} \sqrt [3]{x^2 (x-a)}}-\frac {3 \left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \log \left (-\frac {\sqrt [3]{\frac {2}{3}} \sqrt [3]{a^2 (x-a)}}{\sqrt [3]{d}}-\sqrt [3]{\frac {2}{3}} \sqrt [3]{-a^2 x}\right )}{2 a^3 \sqrt [3]{d} \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]{a^2 (x-a)}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a^2 x}}\right )}{a^3 \sqrt [3]{d} \sqrt [3]{x^2 (x-a)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [C]  time = 0.02, size = 37, normalized size = 0.25 \begin {gather*} \frac {3 x \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {d x}{x-a}\right )}{a \sqrt [3]{x^2 (x-a)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.38, size = 146, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{-a x^2+x^3}}\right )}{a \sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{a \sqrt [3]{d}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{2 a \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.64, size = 377, normalized size = 2.58 \begin {gather*} \left [\frac {\sqrt {3} d \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \log \left (-\frac {{\left (d + 2\right )} x^{2} - 2 \, a x - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {2}{3}} x - \sqrt {3} {\left (\left (-d\right )^{\frac {1}{3}} d x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d x + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} \left (-d\right )^{\frac {2}{3}}\right )} \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{{\left (d - 1\right )} x^{2} + a x}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) + \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, a d}, -\frac {2 \, \sqrt {3} d \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \arctan \left (-\frac {\sqrt {3} {\left (\left (-d\right )^{\frac {1}{3}} x - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{3 \, x}\right ) + 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, a d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 100, normalized size = 0.68 \begin {gather*} -\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}}{3 \, d^{\frac {1}{3}}}\right )}{a d^{\frac {1}{3}}} + \frac {\log \left (d^{\frac {2}{3}} + d^{\frac {1}{3}} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}\right )}{2 \, a d^{\frac {1}{3}}} - \frac {\log \left ({\left | -d^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \right |}\right )}{a d^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

int(1/((a + 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 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+(-1+d)*x),x)

[Out]

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

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