3.2765 \(\int \frac{(c x)^{-1-n}}{a+b x^n} \, dx\)

Optimal. Leaf size=69 \[ -\frac{b x^n \log (x) (c x)^{-n}}{a^2 c}+\frac{b x^n (c x)^{-n} \log \left (a+b x^n\right )}{a^2 c n}-\frac{(c x)^{-n}}{a c n} \]

[Out]

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

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Rubi [A]  time = 0.037959, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {268, 266, 44} \[ -\frac{b x^n \log (x) (c x)^{-n}}{a^2 c}+\frac{b x^n (c x)^{-n} \log \left (a+b x^n\right )}{a^2 c n}-\frac{(c x)^{-n}}{a c n} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 - n)/(a + b*x^n),x]

[Out]

-(1/(a*c*n*(c*x)^n)) - (b*x^n*Log[x])/(a^2*c*(c*x)^n) + (b*x^n*Log[a + b*x^n])/(a^2*c*n*(c*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 266

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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{\left (x^n (c x)^{-n}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,x^n\right )}{c n}\\ &=\frac{\left (x^n (c x)^{-n}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx,x,x^n\right )}{c n}\\ &=-\frac{(c x)^{-n}}{a c n}-\frac{b x^n (c x)^{-n} \log (x)}{a^2 c}+\frac{b x^n (c x)^{-n} \log \left (a+b x^n\right )}{a^2 c n}\\ \end{align*}

Mathematica [A]  time = 0.0266034, size = 42, normalized size = 0.61 \[ -\frac{(c x)^{-n} \left (-b x^n \log \left (a+b x^n\right )+a+b n x^n \log (x)\right )}{a^2 c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.062, 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 = 0.96795, size = 85, normalized size = 1.23 \begin{align*} -\frac{b c^{-n - 1} \log \left (x\right )}{a^{2}} + \frac{b c^{-n - 1} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{2} n} - \frac{c^{-n - 1}}{a n x^{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]

-b*c^(-n - 1)*log(x)/a^2 + b*c^(-n - 1)*log((b*x^n + a)/b)/(a^2*n) - c^(-n - 1)/(a*n*x^n)

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Fricas [A]  time = 1.4321, size = 123, normalized size = 1.78 \begin{align*} -\frac{b c^{-n - 1} n x^{n} \log \left (x\right ) - b c^{-n - 1} x^{n} \log \left (b x^{n} + a\right ) + a c^{-n - 1}}{a^{2} n x^{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]

-(b*c^(-n - 1)*n*x^n*log(x) - b*c^(-n - 1)*x^n*log(b*x^n + a) + a*c^(-n - 1))/(a^2*n*x^n)

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Sympy [A]  time = 5.06794, size = 34, normalized size = 0.49 \begin{align*} - \frac{c^{- n} x^{- n}}{a c n} + \frac{b c^{- n} \log{\left (\frac{a x^{- n}}{b} + 1 \right )}}{a^{2} 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)*x**(-n)/(a*c*n) + b*c**(-n)*log(a*x**(-n)/b + 1)/(a**2*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)