3.20.35 \(\int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx\)

Optimal. Leaf size=135 \[ \frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{a x^3-b}+\left (a x^3-b\right )^{2/3}\right )}{6 \sqrt [3]{a}}-\frac {\log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{2 \sqrt [3]{a x^3-b}+\sqrt [3]{a} x}\right )}{\sqrt {3} \sqrt [3]{a}} \]

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Rubi [A]  time = 0.01, antiderivative size = 74, normalized size of antiderivative = 0.55, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {239} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {\log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{2 \sqrt [3]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-b + a*x^3)^(-1/3),x]

[Out]

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

Rule 239

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

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 118, normalized size = 0.87 \begin {gather*} \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 )-2 \log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{6 \sqrt [3]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-b + a*x^3)^(-1/3),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.23, size = 135, 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} \sqrt [3]{a}}-\frac {\log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{3 \sqrt [3]{a}}+\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 \sqrt [3]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 350, normalized size = 2.59 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} a \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (-3 \, a x^{3} + 3 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} x^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (\left (-a\right )^{\frac {1}{3}} a x^{3} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} a x^{2} + 2 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} + 2 \, b\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} \log \left (\frac {\left (-a\right )^{\frac {1}{3}} x + {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right ) + \left (-a\right )^{\frac {2}{3}} \log \left (\frac {\left (-a\right )^{\frac {2}{3}} x^{2} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} x + {\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a}, -\frac {6 \, \sqrt {\frac {1}{3}} a \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-a\right )^{\frac {1}{3}} x - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}}{x}\right ) + 2 \, \left (-a\right )^{\frac {2}{3}} \log \left (\frac {\left (-a\right )^{\frac {1}{3}} x + {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right ) - \left (-a\right )^{\frac {2}{3}} \log \left (\frac {\left (-a\right )^{\frac {2}{3}} x^{2} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} x + {\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^3 - b)^(-1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.43, size = 108, normalized size = 0.80 \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 {1}{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 {1}{3}}} - \frac {\log \left (-a^{\frac {1}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}{3 \, a^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x^3-b)^(1/3),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^(1/3) + 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^(1/3) - 1/3*log(-a^(1/3) + (a*x^3 - b)^(1/3)/x)/a^(1/3)

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mupad [B]  time = 0.97, size = 39, normalized size = 0.29 \begin {gather*} \frac {x\,{\left (1-\frac {a\,x^3}{b}\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ \frac {a\,x^3}{b}\right )}{{\left (a\,x^3-b\right )}^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^3 - b)^(1/3),x)

[Out]

(x*(1 - (a*x^3)/b)^(1/3)*hypergeom([1/3, 1/3], 4/3, (a*x^3)/b))/(a*x^3 - b)^(1/3)

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sympy [C]  time = 0.86, size = 37, normalized size = 0.27 \begin {gather*} \frac {x e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {a x^{3}}{b}} \right )}}{3 \sqrt [3]{b} \Gamma \left (\frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(-I*pi/3)*gamma(1/3)*hyper((1/3, 1/3), (4/3,), a*x**3/b)/(3*b**(1/3)*gamma(4/3))

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