3.2777 \(\int \frac{(c x)^{-1+n}}{a+b x^n} \, dx\)

Optimal. Leaf size=28 \[ \frac{x^{-n} (c x)^n \log \left (a+b x^n\right )}{b c n} \]

[Out]

((c*x)^n*Log[a + b*x^n])/(b*c*n*x^n)

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Rubi [A]  time = 0.0109541, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {268, 260} \[ \frac{x^{-n} (c x)^n \log \left (a+b x^n\right )}{b c n} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 + n)/(a + b*x^n),x]

[Out]

((c*x)^n*Log[a + b*x^n])/(b*c*n*x^n)

Rule 268

Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracP
art[m], Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.0055718, size = 29, normalized size = 1.04 \[ \frac{x^{1-n} (c x)^{n-1} \log \left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{-1+n}}{a+b{x}^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.96295, size = 32, normalized size = 1.14 \begin{align*} \frac{c^{n - 1} \log \left (\frac{b x^{n} + a}{b}\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^(n - 1)*log((b*x^n + a)/b)/(b*n)

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Fricas [A]  time = 1.32587, size = 43, normalized size = 1.54 \begin{align*} \frac{c^{n - 1} \log \left (b x^{n} + a\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^(n - 1)*log(b*x^n + a)/(b*n)

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Sympy [A]  time = 4.32101, size = 17, normalized size = 0.61 \begin{align*} \frac{c^{n} \log{\left (1 + \frac{b x^{n}}{a} \right )}}{b c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**n*log(1 + b*x**n/a)/(b*c*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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