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

Optimal. Leaf size=71 \[ -\frac{(3 x-1)^{4/3}}{x}+12 \sqrt [3]{3 x-1}+2 \log (x)-6 \log \left (\sqrt [3]{3 x-1}+1\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{3 x-1}}{\sqrt{3}}\right ) \]

[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)]

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Rubi [A]  time = 0.0275319, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {47, 50, 58, 618, 204, 31} \[ -\frac{(3 x-1)^{4/3}}{x}+12 \sqrt [3]{3 x-1}+2 \log (x)-6 \log \left (\sqrt [3]{3 x-1}+1\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{3 x-1}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[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 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

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 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, -Sim
p[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 618

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]

Rule 204

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

Rule 31

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

Rubi steps

\begin{align*} \int \frac{(-1+3 x)^{4/3}}{x^2} \, dx &=-\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 \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sqrt [3]{-1+3 x}\right )-6 \operatorname{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 \operatorname{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} \tan ^{-1}\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 [C]  time = 0.005079, size = 26, normalized size = 0.37 \[ \frac{9}{7} (3 x-1)^{7/3} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};1-3 x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 3*x)^(4/3)/x^2,x]

[Out]

(9*(-1 + 3*x)^(7/3)*Hypergeometric2F1[2, 7/3, 10/3, 1 - 3*x])/7

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Maple [A]  time = 0.013, size = 109, normalized size = 1.5 \begin{align*} 9\,\sqrt [3]{3\,x-1}- \left ( 1+\sqrt [3]{3\,x-1} \right ) ^{-1}-4\,\ln \left ( 1+\sqrt [3]{3\,x-1} \right ) +{ \left ( 1+\sqrt [3]{3\,x-1} \right ) \left ( \left ( 3\,x-1 \right ) ^{{\frac{2}{3}}}-\sqrt [3]{3\,x-1}+1 \right ) ^{-1}}+2\,\ln \left ( \left ( 3\,x-1 \right ) ^{2/3}-\sqrt [3]{3\,x-1}+1 \right ) -4\,\sqrt{3}\arctan \left ( 1/3\, \left ( 2\,\sqrt [3]{3\,x-1}-1 \right ) \sqrt{3} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x-1)^(4/3)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.43274, size = 103, normalized size = 1.45 \begin{align*} -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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.77593, size = 238, normalized size = 3.35 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [C]  time = 2.12979, size = 541, normalized size = 7.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.08422, size = 103, normalized size = 1.45 \begin{align*} -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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)