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

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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)]

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*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \[ \frac {x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b} \]

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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fricas [A]  time = 0.87, size = 22, normalized size = 0.73 \[ \frac {\log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{b c^{\left (\frac {1}{n}\right )}} \]

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

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)

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maple [C]  time = 0.06, size = 139, normalized size = 4.63 \[ \frac {x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}} \ln \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{a+b\,{\left (c\,x^n\right )}^{1/n}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*(c*x^n)^(1/n)),x)

[Out]

int(1/(a + b*(c*x^n)^(1/n)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a + b \left (c x^{n}\right )^{\frac {1}{n}}}\, dx \]

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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