3.5.20 \(\int \frac {1}{x \sqrt [3]{a^3+b^3 x}} \, dx\) [420]

Optimal. Leaf size=71 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {a+2 \sqrt [3]{a^3+b^3 x}}{\sqrt {3} a}\right )}{a}-\frac {\log (x)}{2 a}+\frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 71, 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 = {57, 631, 210, 31} \begin {gather*} \frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt {3} a}\right )}{a}-\frac {\log (x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a^3 + b^3*x)^(1/3)),x]

[Out]

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

Rule 31

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

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), 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 \sqrt [3]{a^3+b^3 x}} \, dx &=-\frac {\log (x)}{2 a}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{a^2+a x+x^2} \, dx,x,\sqrt [3]{a^3+b^3 x}\right )-\frac {3 \text {Subst}\left (\int \frac {1}{a-x} \, dx,x,\sqrt [3]{a^3+b^3 x}\right )}{2 a}\\ &=-\frac {\log (x)}{2 a}+\frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a}-\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}\\ &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a^3+b^3 x}}{a}}{\sqrt {3}}\right )}{a}-\frac {\log (x)}{2 a}+\frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 97, normalized size = 1.37 \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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)])/(2*a)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 3.70, size = 95, normalized size = 1.34 \begin {gather*} \frac {\text {Gamma}\left [-\frac {1}{3}\right ] \left (-\text {Log}\left [1-\frac {a \text {exp\_polar}\left [\frac {2 I}{3} \text {Pi}\right ]}{b {\left (\frac {a^3}{b^3}+x\right )}^{\frac {1}{3}}}\right ]-\text {Log}\left [1-\frac {a \text {exp\_polar}\left [2 I \text {Pi}\right ]}{b {\left (\frac {a^3}{b^3}+x\right )}^{\frac {1}{3}}}\right ]\right )}{3 a \text {Gamma}\left [\frac {2}{3}\right ]} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Gamma[-1 / 3] (-1 ^ (2 / 3) Log[1 - a exp_polar[4 I / 3 Pi] / (b (a ^ 3 / b ^ 3 + x) ^ (1 / 3))] - Log[1 - a e
xp_polar[2 I Pi] / (b (a ^ 3 / b ^ 3 + x) ^ (1 / 3))] + -1 ^ (1 / 3) Log[1 - a exp_polar[2 I / 3 Pi] / (b (a ^
3 / b ^ 3 + x) ^ (1 / 3))]) / (3 a Gamma[2 / 3])

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Maple [A]
time = 0.12, size = 86, normalized size = 1.21

method	result	size
derivativedivides	\(\frac {\ln \left (a -\left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right )}{a}+\frac {-\frac {\ln \left (a^{2}+a \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}+\left (b^{3} x +a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a}\)	\(86\)
default	\(\frac {\ln \left (a -\left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right )}{a}+\frac {-\frac {\ln \left (a^{2}+a \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}+\left (b^{3} x +a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(a-(b^3*x+a^3)^(1/3))/a+1/a*(-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 = 0.34, size = 86, 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} - \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} + \frac {\log \left (-a + {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.32, size = 88, normalized size = 1.24 \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [C] Result contains complex when optimal does not.
time = 1.05, size = 138, normalized size = 1.94 \begin {gather*} \frac {e^{\frac {i \pi }{3}} \log {\left (- \frac {a e^{\frac {2 i \pi }{3}}}{b \sqrt [3]{\frac {a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a \Gamma \left (\frac {2}{3}\right )} + \frac {e^{- \frac {i \pi }{3}} \log {\left (- \frac {a e^{\frac {4 i \pi }{3}}}{b \sqrt [3]{\frac {a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a \Gamma \left (\frac {2}{3}\right )} - \frac {\log {\left (- \frac {a e^{2 i \pi }}{b \sqrt [3]{\frac {a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a \Gamma \left (\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 103, normalized size = 1.45 \begin {gather*} 3 \left (\frac {\ln \left |\left (a^{3}+b^{3} x\right )^{\frac {1}{3}}-a\right |}{3 a}-\frac {\ln \left (\left (\left (a^{3}+b^{3} x\right )^{\frac {1}{3}}\right )^{2}+\left (a^{3}+b^{3} x\right )^{\frac {1}{3}} a+a^{2}\right )}{6 a}+\frac {\arctan \left (\frac {a+2 \left (a^{3}+b^{3} x\right )^{\frac {1}{3}}}{a \sqrt {3}}\right )}{\sqrt {3} a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.10, size = 105, normalized size = 1.48 \begin {gather*} \frac {\ln \left (9\,{\left (a^3+x\,b^3\right )}^{1/3}-9\,a\right )}{a}+\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}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a}-\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}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b^3*x + a^3)^(1/3)),x)

[Out]

log(9*(b^3*x + a^3)^(1/3) - 9*a)/a + (log(9*(b^3*x + a^3)^(1/3) - (9*a*(3^(1/2)*1i - 1)^2)/4)*(3^(1/2)*1i - 1)
)/(2*a) - (log(9*(b^3*x + a^3)^(1/3) - (9*a*(3^(1/2)*1i + 1)^2)/4)*(3^(1/2)*1i + 1))/(2*a)

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