\(\int \frac {(-1+3 x)^{4/3}}{x^2} \, dx\) [294]

Optimal result

Integrand size = 13, antiderivative size = 71 \[ \int \frac {(-1+3 x)^{4/3}}{x^2} \, dx=12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}+4 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )+2 \log (x)-6 \log \left (1+\sqrt [3]{-1+3 x}\right ) \]

[Out]

12*(-1+3*x)^(1/3)-(-1+3*x)^(4/3)/x+2*ln(x)-6*ln(1+(-1+3*x)^(1/3))+4*arctan(1/3*(1-2*(-1+3*x)^(1/3))*3^(1/2))*3
^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {43, 52, 60, 632, 210, 31} \[ \int \frac {(-1+3 x)^{4/3}}{x^2} \, dx=4 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{3 x-1}}{\sqrt {3}}\right )-\frac {(3 x-1)^{4/3}}{x}+12 \sqrt [3]{3 x-1}+2 \log (x)-6 \log \left (\sqrt [3]{3 x-1}+1\right ) \]

[In]

Int[(-1 + 3*x)^(4/3)/x^2,x]

[Out]

12*(-1 + 3*x)^(1/3) - (-1 + 3*x)^(4/3)/x + 4*Sqrt[3]*ArcTan[(1 - 2*(-1 + 3*x)^(1/3))/Sqrt[3]] + 2*Log[x] - 6*L
og[1 + (-1 + 3*x)^(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 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 60

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[-
Log[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]
&& NegQ[(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 632

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(-1+3 x)^{4/3}}{x}+4 \int \frac {\sqrt [3]{-1+3 x}}{x} \, dx \\ & = 12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}-4 \int \frac {1}{x (-1+3 x)^{2/3}} \, dx \\ & = 12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}+2 \log (x)-6 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+3 x}\right )-6 \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right ) \\ & = 12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}+2 \log (x)-6 \log \left (1+\sqrt [3]{-1+3 x}\right )+12 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+3 x}\right ) \\ & = 12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}+4 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )+2 \log (x)-6 \log \left (1+\sqrt [3]{-1+3 x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.20 \[ \int \frac {(-1+3 x)^{4/3}}{x^2} \, dx=\frac {\sqrt [3]{-1+3 x} (1+9 x)}{x}+4 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )-4 \log \left (1+\sqrt [3]{-1+3 x}\right )+2 \log \left (1-\sqrt [3]{-1+3 x}+(-1+3 x)^{2/3}\right ) \]

[In]

Integrate[(-1 + 3*x)^(4/3)/x^2,x]

[Out]

((-1 + 3*x)^(1/3)*(1 + 9*x))/x + 4*Sqrt[3]*ArcTan[(1 - 2*(-1 + 3*x)^(1/3))/Sqrt[3]] - 4*Log[1 + (-1 + 3*x)^(1/
3)] + 2*Log[1 - (-1 + 3*x)^(1/3) + (-1 + 3*x)^(2/3)]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.59 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94

method	result	size
meijerg	\(-\frac {4 \operatorname {signum}\left (x -\frac {1}{3}\right )^{\frac {4}{3}} \left (\frac {3 \Gamma \left (\frac {2}{3}\right )}{4 x}+3 \left (2+\frac {\pi \sqrt {3}}{6}-\frac {\ln \left (3\right )}{2}+\ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )-\frac {3 \Gamma \left (\frac {2}{3}\right ) x {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (\frac {2}{3},1,1;2,3;3 x \right )}{2}\right )}{3 \Gamma \left (\frac {2}{3}\right ) \left (-\operatorname {signum}\left (x -\frac {1}{3}\right )\right )^{\frac {4}{3}}}\)	\(67\)
pseudoelliptic	\(\frac {\left (27 x +3\right ) \left (-1+3 x \right )^{\frac {1}{3}}-6 x \left (2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+3 x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )-\ln \left (\left (-1+3 x \right )^{\frac {2}{3}}-\left (-1+3 x \right )^{\frac {1}{3}}+1\right )+2 \ln \left (1+\left (-1+3 x \right )^{\frac {1}{3}}\right )\right )}{\left (\left (-1+3 x \right )^{\frac {2}{3}}-\left (-1+3 x \right )^{\frac {1}{3}}+1\right ) \left (1+\left (-1+3 x \right )^{\frac {1}{3}}\right )}\)	\(106\)
derivativedivides	\(9 \left (-1+3 x \right )^{\frac {1}{3}}+\frac {1+\left (-1+3 x \right )^{\frac {1}{3}}}{\left (-1+3 x \right )^{\frac {2}{3}}-\left (-1+3 x \right )^{\frac {1}{3}}+1}+2 \ln \left (\left (-1+3 x \right )^{\frac {2}{3}}-\left (-1+3 x \right )^{\frac {1}{3}}+1\right )-4 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+3 x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )-\frac {1}{1+\left (-1+3 x \right )^{\frac {1}{3}}}-4 \ln \left (1+\left (-1+3 x \right )^{\frac {1}{3}}\right )\)	\(109\)
default	\(9 \left (-1+3 x \right )^{\frac {1}{3}}+\frac {1+\left (-1+3 x \right )^{\frac {1}{3}}}{\left (-1+3 x \right )^{\frac {2}{3}}-\left (-1+3 x \right )^{\frac {1}{3}}+1}+2 \ln \left (\left (-1+3 x \right )^{\frac {2}{3}}-\left (-1+3 x \right )^{\frac {1}{3}}+1\right )-4 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+3 x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )-\frac {1}{1+\left (-1+3 x \right )^{\frac {1}{3}}}-4 \ln \left (1+\left (-1+3 x \right )^{\frac {1}{3}}\right )\)	\(109\)
risch	\(\frac {\left (-1+3 x \right )^{\frac {1}{3}}}{x}+\frac {\left (-\frac {4 \left (-1+3 x \right )^{\frac {2}{3}} \left (-\operatorname {signum}\left (x -\frac {1}{3}\right )\right )^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {\ln \left (3\right )}{2}+\ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+2 \Gamma \left (\frac {2}{3}\right ) x {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (1,1,\frac {5}{3};2,2;3 x \right )\right )}{\left (\left (-1+3 x \right )^{2}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x -\frac {1}{3}\right )^{\frac {2}{3}}}+\frac {9 \left (-1+3 x \right )^{\frac {2}{3}} \left (-\operatorname {signum}\left (x -\frac {1}{3}\right )\right )^{\frac {2}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {2}{3},1;2;3 x \right )}{\left (\left (-1+3 x \right )^{2}\right )^{\frac {1}{3}} \operatorname {signum}\left (x -\frac {1}{3}\right )^{\frac {2}{3}}}\right ) \left (\left (-1+3 x \right )^{2}\right )^{\frac {1}{3}}}{\left (-1+3 x \right )^{\frac {2}{3}}}\)	\(146\)
trager	\(\frac {\left (1+9 x \right ) \left (-1+3 x \right )^{\frac {1}{3}}}{x}-4 \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-1+3 x \right )^{\frac {2}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +\left (-1+3 x \right )^{\frac {1}{3}}}{x}\right )+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-1+3 x \right )^{\frac {2}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-1+3 x \right )^{\frac {1}{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -\left (-1+3 x \right )^{\frac {2}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{x}\right )\)	\(195\)

[In]

int((-1+3*x)^(4/3)/x^2,x,method=_RETURNVERBOSE)

[Out]

-4/3/GAMMA(2/3)*signum(x-1/3)^(4/3)/(-signum(x-1/3))^(4/3)*(3/4*GAMMA(2/3)/x+3*(2+1/6*Pi*3^(1/2)-1/2*ln(3)+ln(
x)+I*Pi)*GAMMA(2/3)-3/2*GAMMA(2/3)*x*hypergeom([2/3,1,1],[2,3],3*x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.13 \[ \int \frac {(-1+3 x)^{4/3}}{x^2} \, dx=-\frac {4 \, \sqrt {3} x \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (3 \, x - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 2 \, x \log \left ({\left (3 \, x - 1\right )}^{\frac {2}{3}} - {\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) + 4 \, x \log \left ({\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) - {\left (9 \, x + 1\right )} {\left (3 \, x - 1\right )}^{\frac {1}{3}}}{x} \]

[In]

integrate((-1+3*x)^(4/3)/x^2,x, algorithm="fricas")

[Out]

-(4*sqrt(3)*x*arctan(2/3*sqrt(3)*(3*x - 1)^(1/3) - 1/3*sqrt(3)) - 2*x*log((3*x - 1)^(2/3) - (3*x - 1)^(1/3) +
1) + 4*x*log((3*x - 1)^(1/3) + 1) - (9*x + 1)*(3*x - 1)^(1/3))/x

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.29 (sec) , antiderivative size = 541, normalized size of antiderivative = 7.62 \[ \int \frac {(-1+3 x)^{4/3}}{x^2} \, dx=\frac {189 \cdot \sqrt [3]{3} \left (x - \frac {1}{3}\right )^{\frac {4}{3}} e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {84 \cdot \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {84 \left (x - \frac {1}{3}\right ) \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} - \frac {84 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{i \pi } + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {84 \left (x - \frac {1}{3}\right ) e^{\frac {2 i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {28 \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} - \frac {28 e^{\frac {i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{i \pi } + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {28 e^{\frac {2 i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} \]

[In]

integrate((-1+3*x)**(4/3)/x**2,x)

[Out]

189*3**(1/3)*(x - 1/3)**(4/3)*exp(I*pi/3)*gamma(7/3)/(9*(x - 1/3)*exp(I*pi/3)*gamma(10/3) + 3*exp(I*pi/3)*gamm
a(10/3)) + 84*3**(1/3)*(x - 1/3)**(1/3)*exp(I*pi/3)*gamma(7/3)/(9*(x - 1/3)*exp(I*pi/3)*gamma(10/3) + 3*exp(I*
pi/3)*gamma(10/3)) + 84*(x - 1/3)*log(-3**(1/3)*(x - 1/3)**(1/3)*exp_polar(I*pi/3) + 1)*gamma(7/3)/(9*(x - 1/3
)*exp(I*pi/3)*gamma(10/3) + 3*exp(I*pi/3)*gamma(10/3)) - 84*(x - 1/3)*exp(I*pi/3)*log(-3**(1/3)*(x - 1/3)**(1/
3)*exp_polar(I*pi) + 1)*gamma(7/3)/(9*(x - 1/3)*exp(I*pi/3)*gamma(10/3) + 3*exp(I*pi/3)*gamma(10/3)) + 84*(x -
1/3)*exp(2*I*pi/3)*log(-3**(1/3)*(x - 1/3)**(1/3)*exp_polar(5*I*pi/3) + 1)*gamma(7/3)/(9*(x - 1/3)*exp(I*pi/3
)*gamma(10/3) + 3*exp(I*pi/3)*gamma(10/3)) + 28*log(-3**(1/3)*(x - 1/3)**(1/3)*exp_polar(I*pi/3) + 1)*gamma(7/
3)/(9*(x - 1/3)*exp(I*pi/3)*gamma(10/3) + 3*exp(I*pi/3)*gamma(10/3)) - 28*exp(I*pi/3)*log(-3**(1/3)*(x - 1/3)*
*(1/3)*exp_polar(I*pi) + 1)*gamma(7/3)/(9*(x - 1/3)*exp(I*pi/3)*gamma(10/3) + 3*exp(I*pi/3)*gamma(10/3)) + 28*
exp(2*I*pi/3)*log(-3**(1/3)*(x - 1/3)**(1/3)*exp_polar(5*I*pi/3) + 1)*gamma(7/3)/(9*(x - 1/3)*exp(I*pi/3)*gamm
a(10/3) + 3*exp(I*pi/3)*gamma(10/3))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07 \[ \int \frac {(-1+3 x)^{4/3}}{x^2} \, dx=-4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + 9 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} + \frac {{\left (3 \, x - 1\right )}^{\frac {1}{3}}}{x} + 2 \, \log \left ({\left (3 \, x - 1\right )}^{\frac {2}{3}} - {\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) - 4 \, \log \left ({\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) \]

[In]

integrate((-1+3*x)^(4/3)/x^2,x, algorithm="maxima")

[Out]

-4*sqrt(3)*arctan(1/3*sqrt(3)*(2*(3*x - 1)^(1/3) - 1)) + 9*(3*x - 1)^(1/3) + (3*x - 1)^(1/3)/x + 2*log((3*x -
1)^(2/3) - (3*x - 1)^(1/3) + 1) - 4*log((3*x - 1)^(1/3) + 1)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07 \[ \int \frac {(-1+3 x)^{4/3}}{x^2} \, dx=-4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + 9 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} + \frac {{\left (3 \, x - 1\right )}^{\frac {1}{3}}}{x} + 2 \, \log \left ({\left (3 \, x - 1\right )}^{\frac {2}{3}} - {\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) - 4 \, \log \left ({\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) \]

[In]

integrate((-1+3*x)^(4/3)/x^2,x, algorithm="giac")

[Out]

-4*sqrt(3)*arctan(1/3*sqrt(3)*(2*(3*x - 1)^(1/3) - 1)) + 9*(3*x - 1)^(1/3) + (3*x - 1)^(1/3)/x + 2*log((3*x -
1)^(2/3) - (3*x - 1)^(1/3) + 1) - 4*log((3*x - 1)^(1/3) + 1)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.27 \[ \int \frac {(-1+3 x)^{4/3}}{x^2} \, dx=9\,{\left (3\,x-1\right )}^{1/3}-4\,\ln \left (144\,{\left (3\,x-1\right )}^{1/3}+144\right )+\frac {{\left (3\,x-1\right )}^{1/3}}{x}+\ln \left (18-36\,{\left (3\,x-1\right )}^{1/3}+\sqrt {3}\,18{}\mathrm {i}\right )\,\left (2+\sqrt {3}\,2{}\mathrm {i}\right )-\ln \left (36\,{\left (3\,x-1\right )}^{1/3}-18+\sqrt {3}\,18{}\mathrm {i}\right )\,\left (-2+\sqrt {3}\,2{}\mathrm {i}\right ) \]

[In]

int((3*x - 1)^(4/3)/x^2,x)

[Out]

9*(3*x - 1)^(1/3) - 4*log(144*(3*x - 1)^(1/3) + 144) + (3*x - 1)^(1/3)/x + log(3^(1/2)*18i - 36*(3*x - 1)^(1/3
) + 18)*(3^(1/2)*2i + 2) - log(3^(1/2)*18i + 36*(3*x - 1)^(1/3) - 18)*(3^(1/2)*2i - 2)