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

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

[Out]

(5*Log[1 - x^(2/5)])/2

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Rubi [A]  time = 0.0046696, antiderivative size = 14, normalized size of antiderivative = 1., 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} \[ \frac{5}{2} \log \left (1-x^{2/5}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-x^(3/5) + x)^(-1),x]

[Out]

(5*Log[1 - x^(2/5)])/2

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]

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]

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*}

Mathematica [A]  time = 0.0024372, size = 14, normalized size = 1. \[ \frac{5}{2} \log \left (1-x^{2/5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-x^(3/5) + x)^(-1),x]

[Out]

(5*Log[1 - x^(2/5)])/2

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Maple [B]  time = 0.096, size = 116, normalized size = 8.3 \begin{align*}{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{1}{2}\ln \left ( -\sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}+\sqrt [5]{x}+2 \right ) }-{\frac{1}{2}\ln \left ( \sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}+\sqrt [5]{x}+2 \right ) }+2\,\ln \left ( -1+\sqrt [5]{x} \right ) +2\,\ln \left ( 1+\sqrt [5]{x} \right ) -{\frac{1}{2}\ln \left ( -\sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}-\sqrt [5]{x}+2 \right ) }-{\frac{1}{2}\ln \left ( \sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}-\sqrt [5]{x}+2 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0348, size = 23, normalized size = 1.64 \begin{align*} \frac{5}{2} \, \log \left (x^{\frac{1}{5}} + 1\right ) + \frac{5}{2} \, \log \left (x^{\frac{1}{5}} - 1\right ) \end{align*}

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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Fricas [A]  time = 0.787572, size = 30, normalized size = 2.14 \begin{align*} \frac{5}{2} \, \log \left (x^{\frac{2}{5}} - 1\right ) \end{align*}

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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Sympy [B]  time = 0.385201, size = 22, normalized size = 1.57 \begin{align*} \frac{5 \log{\left (\sqrt [5]{x} - 1 \right )}}{2} + \frac{5 \log{\left (\sqrt [5]{x} + 1 \right )}}{2} \end{align*}

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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Giac [A]  time = 1.17703, size = 24, normalized size = 1.71 \begin{align*} \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{align*}

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))