3.3.64 \(\int \frac {1}{-x^{3/5}+x} \, dx\)

Optimal. Leaf size=14 \[ \frac {5}{2} \log \left (1-x^{2/5}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1593, 260} \begin {gather*} \frac {5}{2} \log \left (1-x^{2/5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^(3/5) + x)^(-1),x]

[Out]

(5*Log[1 - x^(2/5)])/2

Rule 260

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {1}{-x^{3/5}+x} \, dx &=\int \frac {1}{\left (-1+x^{2/5}\right ) x^{3/5}} \, dx\\ &=\frac {5}{2} \log \left (1-x^{2/5}\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {5}{2} \log \left (1-x^{2/5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^(3/5) + x)^(-1),x]

[Out]

(5*Log[1 - x^(2/5)])/2

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IntegrateAlgebraic [A]  time = 0.01, size = 25, normalized size = 1.79 \begin {gather*} \frac {5}{2} \log \left (\sqrt [5]{x}-1\right )+\frac {5}{2} \log \left (\sqrt [5]{x}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-x^(3/5) + x)^(-1),x]

[Out]

(5*Log[-1 + x^(1/5)])/2 + (5*Log[1 + x^(1/5)])/2

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fricas [A]  time = 0.40, size = 8, normalized size = 0.57 \begin {gather*} \frac {5}{2} \, \log \left (x^{\frac {2}{5}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^(3/5)+x),x, algorithm="fricas")

[Out]

5/2*log(x^(2/5) - 1)

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giac [A]  time = 0.20, size = 18, normalized size = 1.29 \begin {gather*} \frac {5}{2} \, \log \left (x^{\frac {1}{5}} + 1\right ) + \frac {5}{2} \, \log \left ({\left | x^{\frac {1}{5}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^(3/5)+x),x, algorithm="giac")

[Out]

5/2*log(x^(1/5) + 1) + 5/2*log(abs(x^(1/5) - 1))

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maple [B]  time = 0.44, size = 116, normalized size = 8.29 \begin {gather*} 2 \ln \left (x^{\frac {1}{5}}+1\right )+\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}+2 \ln \left (x^{\frac {1}{5}}-1\right )-\frac {\ln \left (2 x^{\frac {2}{5}}-\sqrt {5}\, x^{\frac {1}{5}}-x^{\frac {1}{5}}+2\right )}{2}-\frac {\ln \left (2 x^{\frac {2}{5}}-\sqrt {5}\, x^{\frac {1}{5}}+x^{\frac {1}{5}}+2\right )}{2}-\frac {\ln \left (2 x^{\frac {2}{5}}+\sqrt {5}\, x^{\frac {1}{5}}-x^{\frac {1}{5}}+2\right )}{2}-\frac {\ln \left (2 x^{\frac {2}{5}}+\sqrt {5}\, x^{\frac {1}{5}}+x^{\frac {1}{5}}+2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^(3/5)+x),x)

[Out]

1/2*ln(x+1)+1/2*ln(x-1)+2*ln(x^(1/5)-1)-1/2*ln(-5^(1/2)*x^(1/5)+2*x^(2/5)+x^(1/5)+2)-1/2*ln(5^(1/2)*x^(1/5)+2*
x^(2/5)+x^(1/5)+2)-1/2*ln(2*x^(2/5)-5^(1/2)*x^(1/5)-x^(1/5)+2)-1/2*ln(2*x^(2/5)+5^(1/2)*x^(1/5)-x^(1/5)+2)+2*l
n(x^(1/5)+1)

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maxima [A]  time = 1.30, size = 17, normalized size = 1.21 \begin {gather*} \frac {5}{2} \, \log \left (x^{\frac {1}{5}} + 1\right ) + \frac {5}{2} \, \log \left (x^{\frac {1}{5}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^(3/5)+x),x, algorithm="maxima")

[Out]

5/2*log(x^(1/5) + 1) + 5/2*log(x^(1/5) - 1)

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mupad [B]  time = 5.30, size = 8, normalized size = 0.57 \begin {gather*} \frac {5\,\ln \left (x^{2/5}-1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x - x^(3/5)),x)

[Out]

(5*log(x^(2/5) - 1))/2

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sympy [B]  time = 0.37, size = 22, normalized size = 1.57 \begin {gather*} \frac {5 \log {\left (\sqrt [5]{x} - 1 \right )}}{2} + \frac {5 \log {\left (\sqrt [5]{x} + 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**(3/5)+x),x)

[Out]

5*log(x**(1/5) - 1)/2 + 5*log(x**(1/5) + 1)/2

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