3.3010 \(\int \frac{1}{x (a+b (c x^n)^{\frac{1}{n}})} \, dx\)

Optimal. Leaf size=26 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a} \]

[Out]

Log[x]/a - Log[a + b*(c*x^n)^n^(-1)]/a

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Rubi [A]  time = 0.0089402, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {368, 36, 29, 31} \[ \frac{\log (x)}{a}-\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*(c*x^n)^n^(-1))),x]

[Out]

Log[x]/a - Log[a + b*(c*x^n)^n^(-1)]/a

Rule 368

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

Rule 36

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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}{x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{a}\\ &=\frac{\log (x)}{a}-\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0082195, size = 23, normalized size = 0.88 \[ \frac{\log (x)-\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*(c*x^n)^n^(-1))),x]

[Out]

(Log[x] - Log[a + b*(c*x^n)^n^(-1)])/a

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Maple [A]  time = 0.011, size = 35, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( \sqrt [n]{c{x}^{n}} \right ) }{a}}-{\frac{\ln \left ( a+b\sqrt [n]{c{x}^{n}} \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a+b*(c*x^n)^(1/n)),x)

[Out]

1/a*ln((c*x^n)^(1/n))-ln(a+b*(c*x^n)^(1/n))/a

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Maxima [A]  time = 1.02334, size = 54, normalized size = 2.08 \begin{align*} \frac{\log \left (x\right )}{a} - \frac{\log \left (\frac{b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a}{b c^{\left (\frac{1}{n}\right )}}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*(c*x^n)^(1/n)),x, algorithm="maxima")

[Out]

log(x)/a - log((b*c^(1/n)*(x^n)^(1/n) + a)/(b*c^(1/n)))/a

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Fricas [A]  time = 1.53546, size = 49, normalized size = 1.88 \begin{align*} -\frac{\log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) - \log \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*(c*x^n)^(1/n)),x, algorithm="fricas")

[Out]

-(log(b*c^(1/n)*x + a) - log(x))/a

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Sympy [A]  time = 1.53923, size = 56, normalized size = 2.15 \begin{align*} \begin{cases} \tilde{\infty } c^{- \frac{1}{n}} \left (x^{n}\right )^{- \frac{1}{n}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{c^{- \frac{1}{n}} \left (x^{n}\right )^{- \frac{1}{n}}}{b} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + c^{\frac{1}{n}} \left (x^{n}\right )^{\frac{1}{n}} \right )}}{a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*(c*x**n)**(1/n)),x)

[Out]

Piecewise((zoo*c**(-1/n)*(x**n)**(-1/n), Eq(a, 0) & Eq(b, 0)), (-c**(-1/n)*(x**n)**(-1/n)/b, Eq(a, 0)), (log(x
)/a, Eq(b, 0)), (log(x)/a - log(a/b + c**(1/n)*(x**n)**(1/n))/a, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*(c*x^n)^(1/n)),x, algorithm="giac")

[Out]

integrate(1/(((c*x^n)^(1/n)*b + a)*x), x)