3.3009 \(\int \frac{1}{a+b (c x^n)^{\frac{1}{n}}} \, dx\)

Optimal. Leaf size=30 \[ \frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.007532, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {254, 31} \[ \frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^n)^n^(-1))^(-1),x]

[Out]

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

Rule 254

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

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

Mathematica [A]  time = 0.0034746, size = 30, normalized size = 1. \[ \frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.093, size = 214, normalized size = 7.1 \begin{align*}{\frac{1}{b\sqrt [n]{c}}\ln \left ( b{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}x+a \right ){{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(b*exp(-1/2*(I*Pi*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*
csgn(I*c)+I*Pi*csgn(I*c*x^n)^3+2*n*ln(x)-2*ln(c)-2*ln(x^n))/n)*x+a)/b/(c^(1/n))*exp(1/2*(I*Pi*csgn(I*c*x^n)*cs
gn(I*c)*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*csgn(I*c*x^n)^3+2*n*l
n(x)-2*ln(x^n))/n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((c*x^n)^(1/n)*b + a), x)

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Fricas [A]  time = 1.49074, size = 46, normalized size = 1.53 \begin{align*} \frac{\log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b c^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*c^(1/n)*x + a)/(b*c^(1/n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((c*x^n)^(1/n)*b + a), x)