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

Optimal. Leaf size=136 \[ -\frac {\log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{3 a^{2/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{2 \sqrt [3]{a x^3-b}+\sqrt [3]{a} x}\right )}{\sqrt {3} a^{2/3}}+\frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{a x^3-b}+\left (a x^3-b\right )^{2/3}\right )}{6 a^{2/3}} \]

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Rubi [A]  time = 0.07, antiderivative size = 130, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {331, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {\log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}\right )}{3 a^{2/3}}-\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3}}+\frac {\log \left (\frac {a^{2/3} x^2}{\left (a x^3-b\right )^{2/3}}+\frac {\sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1\right )}{6 a^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(-b + a*x^3)^(2/3),x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {x}{\left (-b+a x^3\right )^{2/3}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{1-a x^3} \, dx,x,\frac {x}{\sqrt [3]{-b+a x^3}}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{a} x} \, dx,x,\frac {x}{\sqrt [3]{-b+a x^3}}\right )}{3 \sqrt [3]{a}}-\frac {\operatorname {Subst}\left (\int \frac {1-\sqrt [3]{a} x}{1+\sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-b+a x^3}}\right )}{3 \sqrt [3]{a}}\\ &=-\frac {\log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{3 a^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{a}+2 a^{2/3} x}{1+\sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-b+a x^3}}\right )}{6 a^{2/3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-b+a x^3}}\right )}{2 \sqrt [3]{a}}\\ &=-\frac {\log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{3 a^{2/3}}+\frac {\log \left (1+\frac {a^{2/3} x^2}{\left (-b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{6 a^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{a^{2/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3}}-\frac {\log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{3 a^{2/3}}+\frac {\log \left (1+\frac {a^{2/3} x^2}{\left (-b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{6 a^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 44, normalized size = 0.32 \begin {gather*} \frac {x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {a x^3}{a x^3-b}\right )}{2 \left (a x^3-b\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(-b + a*x^3)^(2/3),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.23, size = 136, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{\sqrt [3]{a} x+2 \sqrt [3]{-b+a x^3}}\right )}{\sqrt {3} a^{2/3}}-\frac {\log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{3 a^{2/3}}+\frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 a^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.56, size = 181, normalized size = 1.33 \begin {gather*} \frac {2 \, \sqrt {3} a \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} a x - 2 \, \sqrt {3} {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}}}{3 \, a^{2} x}\right ) - 2 \, \left (-a^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-a^{2}\right )^{\frac {2}{3}} x - {\left (a x^{3} - b\right )}^{\frac {1}{3}} a}{x}\right ) + \left (-a^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-a^{2}\right )^{\frac {1}{3}} a x^{2} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}} x - {\left (a x^{3} - b\right )}^{\frac {2}{3}} a}{x^{2}}\right )}{6 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(a*x^3 - b)^(2/3), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (a \,x^{3}-b \right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.43, size = 108, normalized size = 0.79 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a^{\frac {1}{3}} + \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {2}{3}}} + \frac {\log \left (a^{\frac {2}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} a^{\frac {1}{3}}}{x} + \frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a^{\frac {2}{3}}} - \frac {\log \left (-a^{\frac {1}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}{3 \, a^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a*x^3 - b)^(2/3),x)

[Out]

int(x/(a*x^3 - b)^(2/3), x)

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sympy [C]  time = 0.94, size = 41, normalized size = 0.30 \begin {gather*} \frac {x^{2} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {a x^{3}}{b}} \right )}}{3 b^{\frac {2}{3}} \Gamma \left (\frac {5}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*exp(-2*I*pi/3)*gamma(2/3)*hyper((2/3, 2/3), (5/3,), a*x**3/b)/(3*b**(2/3)*gamma(5/3))

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