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

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

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {368, 36, 29, 31} \begin {gather*} \frac {\log (x)}{a}-\frac {\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a} \end {gather*}

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 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]

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 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)]

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*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.88 \begin {gather*} \frac {\log (x)-\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.16, size = 37, normalized size = 1.42 \begin {gather*} \frac {\log \left (\left (c x^n\right )^{\frac {1}{n}}\right )}{a}-\frac {\log \left (a^2+a b \left (c x^n\right )^{\frac {1}{n}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[(c*x^n)^n^(-1)]/a - Log[a^2 + a*b*(c*x^n)^n^(-1)]/a

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fricas [A]  time = 1.53, size = 21, normalized size = 0.81 \begin {gather*} -\frac {\log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) - \log \relax (x)}{a} \end {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )} x}\,{d x} \end {gather*}

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)

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maple [A]  time = 0.02, size = 35, normalized size = 1.35 \begin {gather*} \frac {\ln \left (\left (c \,x^{n}\right )^{\frac {1}{n}}\right )}{a}-\frac {\ln \left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.65, size = 40, normalized size = 1.54 \begin {gather*} \frac {\log \relax (x)}{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 {gather*}

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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mupad [B]  time = 1.22, size = 24, normalized size = 0.92 \begin {gather*} -\frac {\ln \left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )-\ln \relax (x)}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*(c*x^n)^(1/n))),x)

[Out]

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

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sympy [A]  time = 1.55, size = 56, normalized size = 2.15 \begin {gather*} \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 {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {\log {\relax (x )}}{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 {gather*}

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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