3.4.96 \(\int \frac {1}{x \sqrt [3]{a+b x}} \, dx\) [396]

Optimal. Leaf size=79 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {57, 631, 210, 31} \begin {gather*} \frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)^(1/3)),x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 95, normalized size = 1.20 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{2 \sqrt [3]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x)^(1/3)),x]

[Out]

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

________________________________________________________________________________________

Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 3.46, size = 82, normalized size = 1.04 \begin {gather*} \frac {\text {Log}\left [1-\frac {b^{\frac {1}{3}} \left (\frac {a}{b}+x\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}\right ]+\text {Log}\left [1-\frac {b^{\frac {1}{3}} \left (\frac {a}{b}+x\right )^{\frac {1}{3}} \text {exp\_polar}\left [\frac {4 I}{3} \text {Pi}\right ]}{a^{\frac {1}{3}}}\right ]-\text {Log}\left [1-\frac {b^{\frac {1}{3}} \left (\frac {a}{b}+x\right )^{\frac {1}{3}} \text {exp\_polar}\left [\frac {2 I}{3} \text {Pi}\right ]}{a^{\frac {1}{3}}}\right ]}{a^{\frac {1}{3}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 75, normalized size = 0.95

method	result	size
derivativedivides	\(\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}\)	\(75\)
default	\(\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.35, size = 76, normalized size = 0.96 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {\log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{2 \, a^{\frac {1}{3}}} + \frac {\log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.32, size = 213, normalized size = 2.70 \begin {gather*} \left [\frac {\sqrt {3} a \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x + \sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x}\right ) - a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{2 \, a}, \frac {2 \, \sqrt {3} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{2 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(3)*a*sqrt(-1/a^(2/3))*log((2*b*x + sqrt(3)*(2*(b*x + a)^(2/3)*a^(2/3) - (b*x + a)^(1/3)*a - a^(4/3)
)*sqrt(-1/a^(2/3)) - 3*(b*x + a)^(1/3)*a^(2/3) + 3*a)/x) - a^(2/3)*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/
3) + a^(2/3)) + 2*a^(2/3)*log((b*x + a)^(1/3) - a^(1/3)))/a, 1/2*(2*sqrt(3)*a^(2/3)*arctan(1/3*sqrt(3)*(2*(b*x
+ a)^(1/3) + a^(1/3))/a^(1/3)) - a^(2/3)*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) + 2*a^(2/3)
*log((b*x + a)^(1/3) - a^(1/3)))/a]

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.94, size = 155, normalized size = 1.96 \begin {gather*} \frac {2 \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac {5}{3}\right )} + \frac {2 e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac {5}{3}\right )} + \frac {2 e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac {5}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(2/3)/(3*a**(1/3)*gamma(5/3)) + 2*exp(2*I*pi/3)*log(1 - b**
(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(2/3)/(3*a**(1/3)*gamma(5/3)) + 2*exp(-2*I*pi/3)*log
(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(2/3)/(3*a**(1/3)*gamma(5/3))

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 137, normalized size = 1.73 \begin {gather*} 3 \left (-\frac {\left (a^{\frac {1}{3}}\right )^{2} \ln \left (\left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2}+a^{\frac {1}{3}} \left (a+b x\right )^{\frac {1}{3}}+a^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a}+\frac {\left (a^{\frac {1}{3}}\right )^{2} \arctan \left (\frac {2 \left (\left (a+b x\right )^{\frac {1}{3}}+\frac {a^{\frac {1}{3}}}{2}\right )}{\sqrt {3} a^{\frac {1}{3}}}\right )}{\sqrt {3} a}+\frac {a^{\frac {1}{3}} a^{\frac {1}{3}} \ln \left |\left (a+b x\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right |}{3 a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.09, size = 99, normalized size = 1.25 \begin {gather*} \frac {\ln \left (9\,{\left (a+b\,x\right )}^{1/3}-9\,a^{1/3}\right )}{a^{1/3}}+\frac {\ln \left (9\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{1/3}}-\frac {\ln \left (9\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x)^(1/3)),x)

[Out]

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

________________________________________________________________________________________