3.450 \(\int \frac{-2+3 x^6}{x (5+2 x^6)} \, dx\)

Optimal. Leaf size=19 \[ \frac{19}{60} \log \left (2 x^6+5\right )-\frac{2 \log (x)}{5} \]

[Out]

(-2*Log[x])/5 + (19*Log[5 + 2*x^6])/60

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Rubi [A]  time = 0.0167028, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 72} \[ \frac{19}{60} \log \left (2 x^6+5\right )-\frac{2 \log (x)}{5} \]

Antiderivative was successfully verified.

[In]

Int[(-2 + 3*x^6)/(x*(5 + 2*x^6)),x]

[Out]

(-2*Log[x])/5 + (19*Log[5 + 2*x^6])/60

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{-2+3 x^6}{x \left (5+2 x^6\right )} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{-2+3 x}{x (5+2 x)} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \left (-\frac{2}{5 x}+\frac{19}{5 (5+2 x)}\right ) \, dx,x,x^6\right )\\ &=-\frac{2 \log (x)}{5}+\frac{19}{60} \log \left (5+2 x^6\right )\\ \end{align*}

Mathematica [A]  time = 0.0055687, size = 19, normalized size = 1. \[ \frac{19}{60} \log \left (2 x^6+5\right )-\frac{2 \log (x)}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 3*x^6)/(x*(5 + 2*x^6)),x]

[Out]

(-2*Log[x])/5 + (19*Log[5 + 2*x^6])/60

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \begin{align*} -{\frac{2\,\ln \left ( x \right ) }{5}}+{\frac{19\,\ln \left ( 2\,{x}^{6}+5 \right ) }{60}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^6-2)/x/(2*x^6+5),x)

[Out]

-2/5*ln(x)+19/60*ln(2*x^6+5)

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Maxima [A]  time = 0.980774, size = 23, normalized size = 1.21 \begin{align*} \frac{19}{60} \, \log \left (2 \, x^{6} + 5\right ) - \frac{1}{15} \, \log \left (x^{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-2)/x/(2*x^6+5),x, algorithm="maxima")

[Out]

19/60*log(2*x^6 + 5) - 1/15*log(x^6)

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Fricas [A]  time = 0.966827, size = 47, normalized size = 2.47 \begin{align*} \frac{19}{60} \, \log \left (2 \, x^{6} + 5\right ) - \frac{2}{5} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-2)/x/(2*x^6+5),x, algorithm="fricas")

[Out]

19/60*log(2*x^6 + 5) - 2/5*log(x)

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Sympy [A]  time = 0.106576, size = 17, normalized size = 0.89 \begin{align*} - \frac{2 \log{\left (x \right )}}{5} + \frac{19 \log{\left (2 x^{6} + 5 \right )}}{60} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**6-2)/x/(2*x**6+5),x)

[Out]

-2*log(x)/5 + 19*log(2*x**6 + 5)/60

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Giac [A]  time = 1.14415, size = 23, normalized size = 1.21 \begin{align*} \frac{19}{60} \, \log \left (2 \, x^{6} + 5\right ) - \frac{1}{15} \, \log \left (x^{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-2)/x/(2*x^6+5),x, algorithm="giac")

[Out]

19/60*log(2*x^6 + 5) - 1/15*log(x^6)