3.5.24 \(\int \frac {1}{x (a^3+b^3 x)^{2/3}} \, dx\) [424]

Optimal. Leaf size=72 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {a+2 \sqrt [3]{a^3+b^3 x}}{\sqrt {3} a}\right )}{a^2}-\frac {\log (x)}{2 a^2}+\frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a^2} \]

[Out]

-1/2*ln(x)/a^2+3/2*ln(a-(b^3*x+a^3)^(1/3))/a^2-arctan(1/3*(a+2*(b^3*x+a^3)^(1/3))/a*3^(1/2))*3^(1/2)/a^2

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Rubi [A]
time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {59, 631, 210, 31} \begin {gather*} -\frac {\log (x)}{2 a^2}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt {3} a}\right )}{a^2}+\frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a^3 + b^3*x)^(2/3)),x]

[Out]

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

Rule 31

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

Rule 59

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 210

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

Rule 631

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]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 95, normalized size = 1.32 \begin {gather*} -\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {a+2 \sqrt [3]{a^3+b^3 x}}{\sqrt {3} a}\right )-2 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )+\log \left (a^2+a \sqrt [3]{a^3+b^3 x}+\left (a^3+b^3 x\right )^{2/3}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.12, size = 87, normalized size = 1.21

method	result	size
derivativedivides	\(\frac {\ln \left (a -\left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right )}{a^{2}}+\frac {-\frac {\ln \left (a^{2}+a \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}+\left (b^{3} x +a^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a^{2}}\)	\(87\)
default	\(\frac {\ln \left (a -\left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right )}{a^{2}}+\frac {-\frac {\ln \left (a^{2}+a \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}+\left (b^{3} x +a^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a^{2}}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b^3*x+a^3)^(2/3),x,method=_RETURNVERBOSE)

[Out]

ln(a-(b^3*x+a^3)^(1/3))/a^2+1/a^2*(-1/2*ln(a^2+a*(b^3*x+a^3)^(1/3)+(b^3*x+a^3)^(2/3))-3^(1/2)*arctan(1/3*(a+2*
(b^3*x+a^3)^(1/3))/a*3^(1/2)))

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Maxima [A]
time = 0.48, size = 87, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac {\log \left (a^{2} + {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}} a + {\left (b^{3} x + a^{3}\right )}^{\frac {2}{3}}\right )}{2 \, a^{2}} + \frac {\log \left (-a + {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}}\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.70, size = 86, normalized size = 1.19 \begin {gather*} -\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} a + 2 \, \sqrt {3} {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}}}{3 \, a}\right ) + \log \left (a^{2} + {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}} a + {\left (b^{3} x + a^{3}\right )}^{\frac {2}{3}}\right ) - 2 \, \log \left (-a + {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}}\right )}{2 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.89, size = 134, normalized size = 1.86 \begin {gather*} \frac {\log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} + \frac {e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x} e^{\frac {2 i \pi }{3}}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} + \frac {e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x} e^{\frac {4 i \pi }{3}}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(1 - b*(a**3/b**3 + x)**(1/3)/a)*gamma(1/3)/(3*a**2*gamma(4/3)) + exp(-2*I*pi/3)*log(1 - b*(a**3/b**3 + x)*
*(1/3)*exp_polar(2*I*pi/3)/a)*gamma(1/3)/(3*a**2*gamma(4/3)) + exp(2*I*pi/3)*log(1 - b*(a**3/b**3 + x)**(1/3)*
exp_polar(4*I*pi/3)/a)*gamma(1/3)/(3*a**2*gamma(4/3))

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Giac [A]
time = 1.12, size = 88, normalized size = 1.22 \begin {gather*} -\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac {\log \left (a^{2} + {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}} a + {\left (b^{3} x + a^{3}\right )}^{\frac {2}{3}}\right )}{2 \, a^{2}} + \frac {\log \left ({\left | -a + {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}} \right |}\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.14, size = 101, normalized size = 1.40 \begin {gather*} \frac {\ln \left (9\,a-9\,{\left (a^3+x\,b^3\right )}^{1/3}\right )}{a^2}+\frac {\ln \left (9\,{\left (a^3+x\,b^3\right )}^{1/3}-\frac {9\,a\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^2}-\frac {\ln \left (9\,{\left (a^3+x\,b^3\right )}^{1/3}+\frac {9\,a\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b^3*x + a^3)^(2/3)),x)

[Out]

log(9*a - 9*(b^3*x + a^3)^(1/3))/a^2 + (log(9*(b^3*x + a^3)^(1/3) - (9*a*(3^(1/2)*1i - 1))/2)*(3^(1/2)*1i - 1)
)/(2*a^2) - (log(9*(b^3*x + a^3)^(1/3) + (9*a*(3^(1/2)*1i + 1))/2)*(3^(1/2)*1i + 1))/(2*a^2)

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